Let G be a finite p-solvable group. We describe the structure of the p-complements of G when the set of p-regular conjugacy classes has exactly three class sizes. For instance, when the set of p-regular class sizes of G is {1, pa, pam} or {1, m, pam} with (m, p) = 1, then we show that m = qb for some prime q and the structure of the p-complements of G is determined.

C*-algebras form a 2-category with *-homomorphisms or correspondences as morphisms and unitary intertwiners as 2-morphisms. We use this structure to define weak actions of 2-categories, weakly equivariant maps between weak actions and modifications between weakly equivariant maps. In the group case, we identify the resulting notions with known ones, including Busby–Smith twisted actions and the equivalence of such actions, covariant representations and saturated Fell bundles. For 2-groups, weak actions combine twists in the sense of Green, and Busby and Smith.

The Packer–Raeburn Stabilization Trick implies that all Busby–Smith twisted group actions of locally compact groups are Morita equivalent to classical group actions. We generalize this to actions of strict 2-groupoids.

We prove the compactness of critical Sobolev embeddings with applications to nonlinear singular Schrödinger equations and provide a unified treatment in dimensions N > 2 and N = 2, based on a somewhat unexpectedly broad array of parallel properties between spaces and H10 of the unit disc. These properties include Leray inequality for N = 2 as a counterpart of Hardy inequality for N > 2, pointwise estimates by ground states r(2−N)/2 and of the respective Hardy-type inequalities, as well as compactness of the limiting Sobolev embeddings once the Sobolev norm is appended by a potential term whose growth at singularities exceeds that of the corresponding Hardy-type potential.

We give two generalizations of Kac's Theorem on representations of quivers. One is to representations of equipped graphs by relations, in the sense of Gelfand and Ponomarev. The other is to representations of quivers in which certain of the linear maps are required to have maximal rank.

We consider classifying spaces of a family of p-groups and prove that mod p cohomology enriched with Bockstein spectral sequences determines their homotopy type among p-completed CW-complexes.

We introduce different notions of invertibility for generalized functions in the sense of Colombeau. Several necessary conditions for (left, right) invertibility are derived, giving rise to the concepts of compactly asymptotic injectivity and surjectivity. We analyse the extent to which these properties are also sufficient to guarantee the existence of a (left, right) inverse of a generalized function. Finally, we establish several Inverse Function Theorems in this setting and study the relation to their classical counterparts.

We prove a vanishing theorem for the twisted de Rham cohomology of a compact manifold.

We obtain a sharp bound for p-elementary subgroups in the Cremona group Cr2(k) over an arbitrary perfect field k.

Let A be a unital C*-algebra with the canonical (H) C*-bundle over the maximal ideal space of the centre of A, and let E(A) be the set of all elementary operators on A. We consider derivations on A which lie in the completely bounded norm closure of E(A), and show that such derivations are necessarily inner in the case when each fibre of is a prime C*-algebra. We also consider separable C*-algebras A for which is an (F) bundle. For these C*-algebras we show that the following conditions are equivalent: E(A) is closed in the operator norm; A as a Banach module over its centre is topologically finitely generated; fibres of have uniformly finite dimensions, and each restriction bundle of over a set where its fibres are of constant dimension is of finite type as a vector bundle.

We present, given an odd integer d, a decomposition of the multiset of bar lengths of a bar partition λ as the union of two multisets, one consisting of the bar lengths in its d-core partition cd(λ) and the other consisting of modified bar lengths in its d-quotient partition. In particular, we obtain that the multiset of bar lengths in cd(λ) is a sub-multiset of the multiset of bar lengths in λ. Also, we obtain a relative bar formula for the degrees of spin characters of the Schur extensions of . The proof involves a recent similar result for partitions, proved by Bessenrodt and the authors.

We establish a defect relation for algebraically non-degenerate meromorphic maps over generalized p-parabolic manifolds that intersect hypersurfaces in smooth projective algebraic varieties, extending certain results of H. Cartan, L. Ahlfors, W. Stoll, M. Ru, P. M. Wong and Philip P. W. Wong and others.

Results of Fowler and Sims show that every k-graph is completely determined by its k-coloured skeleton and collection of commuting squares. Here we give an explicit description of the k-graph associated with a given skeleton and collection of squares and show that two k-graphs are isomorphic if and only if there is an isomorphism of their skeletons which preserves commuting squares. We use this to prove directly that each k-graph Λ is isomorphic to the quotient of the path category of its skeleton by the equivalence relation determined by the commuting squares, and show that this extends to a homeomorphism of infinite-path spaces when the k-graph is row finite with no sources. We conclude with a short direct proof of the characterization, originally due to Robertson and Sims, of simplicity of the C*-algebra of a row-finite k-graph with no sources.

In this paper we study Lp−Lr estimates of both the extension operator and the averaging operator associated with the algebraic variety S = {x ∈ : Q(x) = 0}, where Q(x) is a non-degenerate quadratic form over the finite field with q elements. We show that the Fourier decay estimate on S is good enough to establish the sharp averaging estimates in odd dimensions. In addition, the Fourier decay estimate enables us to simply extend the sharp L2−L4 conical extension result in , due to Mockenhaupt and Tao, to the L2−L2(d+1)/(d−1) estimate in all odd dimensions d ≥ 3. We also establish a sharp estimate of the mapping properties of the average operators in the case when the variety S in even dimensions d ≥ 4 contains a d/2-dimensional subspace.

We prove the classification of the real vector subspaces of a quaternionic vector space by using a covariant functor which associates, to any pair formed of a quaternionic vector space and a real subspace, a coherent sheaf over the sphere.

The following rather surprising result is noted.

(1) A function f(z) = ∑anzn such that an ↓ 0 (n → ∞) belongs to H1 if and only if ∑(an/(n + 1)) < ∞.

A more subtle analysis is needed to prove that assertion (2) remains true if H1 is replaced by the predual, 1(⊂ H1), of the Bloch space. Assertion (1) extends the Hardy–Littlewood theorem, which says the following.

(2) f belongs to Hp (1 < p < ∞) if and only if ∑(n + 1)p−2anp < ∞.

A new proof of (2) is given and applications of (1) and (2) to the Libera transform of functions with positive coefficients are presented. The fact that the Libera operator does not map H1 to H1 is improved by proving that it does not map 1 into H1.

We give two sufficient criteria for schlichtness of envelopes of holomorphy in terms of topology. These are weakened converses of results of Kerner and Royden. Our first criterion generalizes a result of Hammond in dimension 2. Along the way, we also prove a generalization of Royden's theorem.