A long-standing problem is the characterization of subsets of the range of a vector measure. It is known that the range of a countably additive vector measure is relatively weakly compact and, in addition, possesses several interesting properties (see [2]). In [6] it is proved that if m: Σ → Χ is a countably additive vector measure, then the range of m has not only the Banach–Saks property, but even the alternate Banach-Saks property. A tantalizing conjecture, which we shall disprove in this article, is that the range of m has to have, for some p > 1, the p-Banach–Saks property. Another conjecture, which has been around for some time (see [2]) and is also disproved in this paper, is that weakly null sequences in the range of a vector measure admit weakly-2-summable sub-sequences. In fact, we shall show a weakly null sequence in the range of a countably additive vector measure having, for every p < ∞, no weakly-p-summable sub-sequences.

If A(z) belongs to the Bergman space , then the differential equation f″+A(z)f=0 is Blaschke-oscillatory, meaning that the zero sequence of every nontrivial solution satisfies the Blaschke condition. Conversely, if A(z) is analytic in the unit disc such that the differential equation is Blaschke-oscillatory, then A(z) almost belongs to . It is demonstrated that certain “nice” Blaschke sequences can be zero sequences of solutions in both cases when A ∈ or A ∉ . In addition, no condition regarding only the number of zeros of solutions is sufficient to guarantee that A ∈ .

In the theory of self-adjoint operators in Hilbert space and of formally self-adjoint linear differential equations there are many situations involving analytic functions on the complex plane whose singularities are confined to the real axis and where the growth of the function at such singular points is strictly limited.

In this paper we obtain the general solution of the equation

whose kernel is

where 0<α, β<1, μ, η, σ > 0, δ>0, are real parameters, z = max(x, y), φ(t) and g(y) are prescribed functions and ƒ(x) is to be determined.

The cone length Cl(f) of a map f: X → Y is defined to be the least number of attaching maps possible in a conic (or iterated mapping cone) structure for f. Cone length is a homotopy invariant in the sense that if φ: X → X and ρ: Y → Y are homotopy equivalences then Cl (ρ°f°φ) = Cl(f). Furthermore Cl(f) depends only on the homotopy class of f. It was shown by Ganea [8] that the cone length of the map * → X coincides with the strong Lusternik-Schnirelmann category of X as a space (see Proposition 1.6 below). Recent work of Cornea ([3]–[6]) is much concerned with cone length and its role in critical point theory. For example, let f be a smooth real valued function on a manifold triad (M; V0, V1) with V0 ≠ θ. Under certain conditions, if f has only “reasonable” critical points then it must have at least Cl(V0↪M) of them (see [6]).

The aim of this note is to present a new, elementary proof of a result of Baas and Madsen on the mod p cohomology of certain quotients of the spectrum BP.

An element a in a ring R is left annihilator-stable (or left AS) if, whenever $Ra+{\rm l}(b)=R$ with $b\in R$ , $a-u\in {\rm l}(b)$ for a unit u in R, and the ring R is a left AS ring if each of its elements is left AS. In this paper, we show that the left AS elements in a ring form a multiplicatively closed set, giving an affirmative answer to a question of Nicholson [J. Pure Appl. Alg.221 (2017), 2557–2572.]. This result is used to obtain a necessary and sufficient condition for a formal triangular matrix ring to be left AS. As an application, we provide examples of left AS rings R over which the triangular matrix rings ${\mathbb T}_n(R)$ are not left AS for all $n\ge 2$ . These examples give a negative answer to another question of Nicholson [J. Pure Appl. Alg.221 (2017), 2557–2572.] whether R/J(R) being left AS implies that R is left AS.

For a locally compact Hausdorff space X and a C*-algebra A with only finitely many closed ideals, we discuss a characterization of closed ideals of C0(X,A) in terms of closed ideals of A and a class of closed subspaces of X. We further use this result to prove that a closed ideal of C0(X)⊗minA is a finite sum of product ideals. We also establish that for a unital C*-algebra A, C0(X,A) has the centre-quotient property if and only if A has the centre-quotient property. As an application, we characterize the closed Lie ideals of C0(X,A) and identify all the closed Lie ideals of HC0(X)⊗minB(H), H being a separable Hilbert space.

In this paper, we prove a theorem concerning asymptotic estimates of the eigenvalues of certain positive integral operators with Laplace transform type kernels.

A theorem of Brady and Meier states that a right-angled Artin group is a duality group if and only if the flag complex of the defining graph is Cohen–Macaulay. We use this to give an example of a RAAG with the property that its outer automorphism group is not a virtual duality group. This gives a partial answer to a question of Vogtmann. In an appendix, Brück describes how he used a computer-assisted search to find further examples.

An element k of a unital Banach algebra A is said to be Hermitian if its numerical range

is contained in ℝ; equivalently, ∥eitk∥ = 1(t ∈ ℝ)—see Bonsall and Duncan [3] and [4]. Here we find the largest possible extent of V(kn), n ∈ ℕ, given V(k) ⊆ [−1, 1], and so ∥k∥ ≤ 1: previous knowledge is in Bollobás [2] and Crabb, Duncan and McGregor [7]. The largest possible sets all occur in a single example. Surprisingly, they all have straight line segments in their boundaries. The example is in [2] and [7], but here we give A. Browder's construction from [5], partly published in [6]. We are grateful to him for a copy of [5], and for discussions which led to the present work. We are also grateful to J. Duncan for useful discussions.

For the Kronecker algebra, Zwara found in [14] an example of a module whose orbit closure is neither unibranch nor Cohen-Macaulay. In this paper, we explain how to extend this example to all representation-infinite algebras with a preprojective component.

We prove some inequalities for the spectral radius of positive operators on Banach function spaces. In particular, we prove the following extension of Levinger's theorem. Let $K$ be a positive compact kernel operator on $L^2(X, \mu)$ with the spectral radius $r(K)$. Then the function $\phi$ defined by $\phi(t) = r(t K + (1-t) K^*)$ is non-decreasing on $[0, \frac{1}{2}]$. We also prove that $\| A + B^* \| \ge 2 \cdot \sqrt{r(A B)}$ for any positive operators $A$ and $B$ on $L^2(X, \mu)$.

Strata in manifold stratified spaces are shown to have neighborhoods that are teardrops of manifold stratified approximate fibrations (under dimension and compactness assumptions). This is the best possible version of the tubular neighborhood theorem for strata in the topological setting. Applications are given to replacement of singularities, to the structure of neighborhoods of points in manifold stratified spaces, and to spaces of manifold stratified approximate fibrations.

We calculate the moments of the characteristic polynomials of $N\times N$ matrices drawn from the Hermitian ensembles of Random Matrix Theory, at a position t in the bulk of the spectrum, as a series expansion in powers of t. We focus in particular on the Gaussian Unitary Ensemble. We employ a novel approach to calculate the coefficients in this series expansion of the moments, appropriately scaled. These coefficients are polynomials in N. They therefore grow as $N\to\infty$ , meaning that in this limit the radius of convergence of the series expansion tends to zero. This is related to oscillations as t varies that are increasingly rapid as N grows. We show that the $N\to\infty$ asymptotics of the moments can be derived from this expansion when $t=0$ . When $t\ne 0$ we observe a surprising cancellation when the expansion coefficients for N and $N+1$ are formally averaged: this procedure removes all of the N-dependent terms leading to values that coincide with those expected on the basis of previously established asymptotic formulae for the moments. We obtain as well formulae for the expectation values of products of the secular coefficients.

We give an algebraic proof of a class number formula for dihedral extensions of number fields of degree 2q, where q is any odd integer. Our formula expresses the ratio of class numbers as a ratio of orders of cohomology groups of units and allows one to recover similar formulas which have appeared in the literature. As a corollary of our main result, we obtain explicit bounds on the (finitely many) possible values which can occur as ratio of class numbers in dihedral extensions. Such bounds are obtained by arithmetic means, without resorting to deep integral representation theory.

We characterize the pairs of operator spaces that occur as pairs of Morita equivalence bimodules between non-selfadjoint operator algebras in terms of the mutual relation between the spaces. We obtain a characterization of the operator spaces which are completely isometrically isomorphic to imprimitivity bimodules between some strongly Morita equivalent (in the sense of Rieffel) C*-algebras. As corollaries, we give representation results for such operator spaces.

Let C be the Cayley algebra denned over the real field. If, for given elements α, β, and γ of a quaternion subalgebra of C, α = βγ, it follows, by associativity, that for any nonzero element δ of the same quaternion subalgebra, α = (βδ)(δ-1γ). For Cayley numbers ζ ξ, and η with ζ = ξη, the relation ζ = (ξδ)(δ-1η) in general only holds when δ is a nonzero real number. Because of the existence of factorization results [1, 2] in the orders of C, the question naturally arises of whether it is possible to choose one-to-one mappings, θ and φ, of C onto itself such that ζ = θξ. φη whenever ζ = ξη. To discuss this question, we make the following definition.

Let $p$ be a prime number, $\Q_p$ the field of $p$-adic numbers, $K$ a finite field extension of $\Q_p$, $\skew4\bar K$ a fixed algebraic closure of $K$, and $\C_p$ the completion of $\skew4\bar K$ with respect to the $p$-adic valuation. We discuss some properties of Lipschitzian elements, which are elements $T$ of $\C_p$ defined by a certain metric condition that allows one to integrate Lipschitzian functions along the Galois orbit of $T$ over $K$ with respect to the Haar distribution.