We rectify two omissions in the list of generators and include a brief discussion of the localization theorem of Kosniowski and Stong.

We define an equivalence relation between bimodules over maximal abelian self-adjoint algebras (MASA bimodules), which we call spatial Morita equivalence. We prove that two reflexive MASA bimodules are spatially Morita equivalent if and only if their (essential) bilattices are isomorphic. We also prove that if are bilattices that correspond to reflexive MASA bimodules , and is an onto bilattice homomorphism, then

(i) if is synthetic, then is synthetic;

(ii) if contains a non-zero compact (or a finite or a rank 1) operator, then also contains a non-zero compact (or a finite or a rank 1) operator.

In this paper we study the multiplicity of non-trivial solutions to a class of nonlinear boundary-value problems of Kirchhoff type. We prove existence results when the problem has nonlinearities with subcritical and with critical Caffarelli–Kohn–Nirenberg exponent.

In this paper we consider the problem of explicitly finding canonical ideals of one-dimensional Cohen–Macaulay local rings. We show that Gorenstein ideals contained in a high power of the maximal ideal are canonical ideals. In the codimension 2 case, from a Hilbert–Burch resolution, we show how to construct canonical ideals of curve singularities. Finally, we translate the problem of the analytic classification of curve singularities to the classification of local Artin Gorenstein rings with suitable length.

We answer a question of Masser by showing that for the Weierstrass zeta function ζ corresponding to a given lattice Λ, the density of algebraic points of absolute multiplicative height bounded by T and degree bounded by k lying on the graph of ζ, restricted to an appropriate domain, does not exceed c(log T)15 for an effective constant c > 0 depending on k and on Λ. Using different methods, we also give two bounds of the same form for the density of algebraic points of bounded height in a fixed number field lying on the graph of ζ restricted to an appropriate subset of (0, 1). In one case the constant c can be shown not to depend on the choice of lattice; in the other, the exponent can be improved to 12.

Let A and B be operators acting on infinite-dimensional complex Banach spaces. We say that the Weyl spectral identity holds for the tensor product A⊗B if σw(A⊗B) = σw(A)·σ(B)∪σ(A)·σw(B), where σ(·) and σw(·) stand for the spectrum and the Weyl spectrum, respectively. Conditions on A and B for which the Weyl spectral identity holds are investigated. Especially, it is shown that if A and B are biquasitriangular (in particular, if the spectra of A and B have empty interior), then the Weyl spectral identity holds. It is also proved that if A and B are biquasitriangular, then the tensor product A ⊗ B is biquasitriangular.

We show that a weakly holomorphic modular function can be written as a sum of modular units of higher level. Furthermore, we find a necessary and sufficient condition for a meromorphic Siegel modular function of degree g to have neither a zero nor a pole on a certain subset of the Siegel upper half-space .

We study two notions of purity in categories of sheaves: the categorical and the geometric. It is shown that pure injective envelopes exist in both cases under very general assumptions on the scheme. Finally, we introduce the class of locally absolutely pure (quasi-coherent) sheaves with respect to the geometrical purity, and characterize locally Noetherian closed subschemes of a projective scheme in terms of the new class.

For two modules M and N, iM (N) stands for the largest submodule of N relative to which M is injective. For any module M, iM : Mod-R → Mod-R thus defines a left exact preradical, and iM (M) is quasi-injective. Classes of ring including strongly prime, semi-Artinian rings and those with no middle class are characterized using this functor: a ring R is semi-simple or right strongly prime if and only if for any right R-module M, iM (R) = R or 0, extending a result of Rubin; R is a right QI-ring if and only if R has the ascending chain condition (a.c.c.) on essential right ideals and iM is a radical for each M ∈ Mod-R (the a.c.c. is not redundant), extending a partial answer of Dauns and Zhou to a long-standing open problem. Also discussed are rings close to those with no middle class.

Let be a field and let Q be a minimal Hopf quiver, i.e. a cyclic quiver or the infinite linear quiver, and let repln(Q) denote the category of locally nilpotent finite-dimensional -representations of Q. The category repln(Q) has natural tensor structures induced from graded Hopf structures on the path coalgebra . Tensor categories of the form repln(Q) are an interesting class of tame hereditary pointed tensor categories that are not finite. The aim of this paper is to compute the Clebsch–Gordan formulae and Green rings of such tensor categories.

Let τ be the involution changing the sign of two coordinates in ℙ4. We prove that τ induces the identity action on the second Chow group of the intersection of a τ-invariant cubic with a τ-invariant quadric hypersurface in ℙ4. Let lτ and Πτ be the one- and two-dimensional components of the fixed locus of the involution τ. We describe the generalized Prymian associated with the projection of a τ-invariant cubic 𝓵 ⊂ P4 from lτ onto Πτ in terms of the Prymians 𝓅2 and 𝓅3 associated with the double covers of two irreducible components, of degree 2 and 3, respectively, of the reducible discriminant curve. This gives a precise description of the induced action of the involution τ on the continuous part of the Chow group CH2 (𝓵). The action on the subgroup corresponding to 𝓅3 is the identity, and the action on the subgroup corresponding to 𝓅2 is the multiplication by —1.

For a non-negative and non-trivial real-valued continuous function hΩ × [0, ∞) such that h(x, 0) = 0 for all x ∈ Ω, we study the boundary-value problem

where Ω ⊆ ℝN, N ⩾ 2, is a bounded smooth domain and Δp:= div(|Du|p–2DDu) is the p-Laplacian. This work investigates growth conditions on h(x, t) that would lead to the existence or non-existence of distributional solutions to (BVP). In a major departure from past works on similar problems, in this paper we do not impose any special structure on the inhomogeneous term h(x, t), nor do we require any monotonicity condition on h in the second variable. Furthermore, h(x, t) is allowed to vanish in either of the variables.

We investigate topological realizations of higher-rank graphs. We show that the fundamental group of a higher-rank graph coincides with the fundamental group of its topological realization. We also show that topological realization of higher-rank graphs is a functor and that for each higher-rank graph Λ, this functor determines a category equivalence between the category of coverings of Λ and the category of coverings of its topological realization. We discuss how topological realization relates to two standard constructions for k-graphs: projective limits and crossed products by finitely generated free abelian groups.

Let Φ be a reduced irreducible root system of rank greater than or equal to 2, let R be a commutative ring and let I, J be two ideals of R. In the present paper we describe generators of the commutator groups of relative elementary subgroups [E(Φ,R,I),E(Φ,R,J)] both as normal subgroups of the elementary Chevalley group E(Φ,R), and as groups. Namely, let xα(ξ), α ∈ Φ ξ ∈ R, be an elementary generator of E(Φ,R). As a normal subgroup of the absolute elementary group E(Φ,R), the relative elementary subgroup is generated by xα(ξ), α ∈ Φ, ξ ∈ I. Classical results due to Stein, Tits and Vaserstein assert that as a group E(Φ,R,I) is generated by zα(ξ,η), where α ∈ Φ, ξ ∈ I, η ∈ R. In the present paper, we prove the following birelative analogues of these results. As a normal subgroup of E(Φ,R) the relative commutator subgroup [E(Φ,R,I),E(Φ,R,J)] is generated by the following three types of generators: (i) [xα(ξ),zα(ζ,η)], (ii) [xα(ξ),x_α(ζ)] and (iii) xα(ξζ), where α ∈ Φ, ξ ∈ I, ζ ∈ J, η ∈ R. As a group, the generators are essentially the same, only that type (iii) should be enlarged to (iv) zα(ξζ,η). For classical groups, these results, with many more computational proofs, were established in previous papers by the authors. There is already an amazing application of these results in the recent work of Stepanov on relative commutator width.

We classify all (saturated) fusion systems on bicyclic 2-groups. Here, a bicyclic group is a product of two cyclic subgroups. This extends previous work on fusion systems on metacyclic 2-groups. As an application we prove Olsson's conjecture for all blocks with bicyclic defect groups.

In this paper we study a quasi-linear elliptic problem coupled with Dirichlet boundary conditions. We propose a new set of assumptions ensuring the existence of infinitely many solutions.

If a locally compact group G acts on a C*-algebra B, we have both full and reduced crossed products and each has a coaction of G. We investigate ‘exotic’ coactions in between the two, which are determined by certain ideals E of the Fourier–Stieltjes algebra B(G); an approach that is inspired by recent work of Brown and Guentner on new C*-group algebra completions. We actually carry out the bulk of our investigation in the general context of coactions on a C*-algebra A. Buss and Echterhoff have shown that not every coaction comes from one of these ideals, but nevertheless the ideals do generate a wide array of exotic coactions. Coactions determined by these ideals E satisfy a certain ‘E-crossed product duality’, intermediate between full and reduced duality. We give partial results concerning exotic coactions with the ultimate goal being a classification of which coactions are determined by ideals of B(G).

Let be a non-constant elliptic function. We prove that the Hausdorff dimension of the escaping set of f equals 2q/(q+1), where q is the maximal multiplicity of poles of f. We also consider the escaping parameters in the family fβ = βf, i.e. the parameters β for which the orbit of one critical value of fβ escapes to infinity. Under additional assumptions on f we prove that the Hausdorff dimension of the set of escaping parameters ε in the family fβ is greater than or equal to the Hausdorff dimension of the escaping set in the dynamical space. This demonstrates an analogy between the dynamical plane and the parameter plane in the class of transcendental meromorphic functions.

We present a new approach to Davie’s theorem on the uniqueness of solutions to the equation

for almost all Brownian paths. A generalization of this result and a discussion of some closely related problems are given.

Symmetry groups of Lie algebras and superalgebras constructed from (∈, δ)-Freudenthal-Kantor triple systems have been studied. In particular, for a special (ε, ε)-Freudenthal–Kantor triple, it is the SL(2) group. Also, the relationship between two constructions of Lie algebras from structurable algebras has been investigated.