We give a complete computation of the Bieri–Neumann–Strebel–Renz invariants Σm(Hn) of the Houghton groups Hn. Partial results were previously obtained by the author, with a conjecture about the full picture, which we now confirm. The proof involves covering relevant subcomplexes of an associated CAT (0) cube complex by their intersections with certain locally convex subcomplexes, and then applying a strong form of the Nerve Lemma. A consequence of the full computation is that for each 1 ≤ m ≤ n − 1, Hn admits a map onto ℤ whose kernel is of type Fm−1 but not Fm; moreover, no such kernel is ever of type Fn−1.

Let φ : ℝn × [0, ∞) → [0, ∞) satisfy that φ(x, · ), for any given x ∈ ℝn, is an Orlicz function and φ( · , t) is a Muckenhoupt A∞ weight uniformly in t ∈ (0, ∞). The (weak) Musielak–Orlicz Hardy space Hφ(ℝn) (WHφ(ℝn)) generalizes both the weighted (weak) Hardy space and the (weak) Orlicz Hardy space and hence has wide generality. In this paper, two boundedness criteria for both linear operators and positive sublinear operators from Hφ(ℝn) to Hφ(ℝn) or from Hφ(ℝn) to WHφ(ℝn) are obtained. As applications, we establish the boundedness of Bochner–Riesz means from Hφ(ℝn) to Hφ(ℝn), or from Hφ(ℝn) to WHφ(ℝn) in the critical case. These results are new even when φ(x, t): = Φ(t) for all (x, t) ∈ ℝn × [0, ∞), where Φ is an Orlicz function.

We introduce a notion of Koszul A∞-algebra that generalizes Priddy's notion of a Koszul algebra and we use it to construct small A∞-algebra models for Hochschild cochains. As an application, this yields new techniques for computing free loop space homology algebras of manifolds that are either formal or coformal (over a field or over the integers). We illustrate these techniques in two examples.

Balanced pairs appear naturally in the realm of relative homological algebra associated with the balance of right-derived functors of the Hom functor. Cotorsion triplets are a natural source of such pairs. In this paper, we study the connection between balanced pairs and cotorsion triplets by using recent quiver representation techniques. In doing so, we find a new characterization of abelian categories that have enough projectives and injectives in terms of the existence of complete hereditary cotorsion triplets. We also provide a short proof of the lack of balance for derived functors of Hom computed using flat resolutions, which extends the one given by Enochs in the commutative case.

We give an equality condition for a symmetrization inequality for condensers proved by F.W. Gehring regarding elliptic areas. We then use this to obtain a monotonicity result involving the elliptic area of the image of a holomorphic function f.

We construct prime amphicheiral knots that have free period 2. This settles an open question raised by the second-named author, who proved that amphicheiral hyperbolic knots cannot admit free periods and that prime amphicheiral knots cannot admit free periods of order > 2.

Let $\mathcal {P}(\mathbf{N})$ be the power set of N. We say that a function $\mu ^\ast : \mathcal {P}(\mathbf{N}) \to \mathbf{R}$ is an upper density if, for all X, Y ⊆ N and h, k ∈ N+, the following hold: (f1) $\mu ^\ast (\mathbf{N}) = 1$; (f2) $\mu ^\ast (X) \le \mu ^\ast (Y)$ if X ⊆ Y; (f3) $\mu ^\ast (X \cup Y) \le \mu ^\ast (X) + \mu ^\ast (Y)$; (f4) $\mu ^\ast (k\cdot X) = ({1}/{k}) \mu ^\ast (X)$, where k · X : = {kx: x ∈ X}; and (f5) $\mu ^\ast (X + h) = \mu ^\ast (X)$. We show that the upper asymptotic, upper logarithmic, upper Banach, upper Buck, upper Pólya and upper analytic densities, together with all upper α-densities (with α a real parameter ≥ −1), are upper densities in the sense of our definition. Moreover, we establish the mutual independence of axioms (f1)–(f5), and we investigate various properties of upper densities (and related functions) under the assumption that (f2) is replaced by the weaker condition that $\mu ^\ast (X)\le 1$ for every X ⊆ N. Overall, this allows us to extend and generalize results so far independently derived for some of the classical upper densities mentioned above, thus introducing a certain amount of unification into the theory.

In this paper the authors consider the weighted estimates for the Calderón commutator defined by

\mathcal{C}_{m+1, A}(a_1,\ldots,a_{m};f)(x)={\rm p. v.} \displaystyle\int_{\mathbb{R}}\displaystyle\frac{P_2(A; x, y)\prod\nolimits_{j=1}^m(A_j(x)-A_j(y))}{(x-y)^{m+2}}f(y){\rm d}y,

$L^{p_1}(\mathbb {R},w_1) \times \cdots \times L^{p_{m+1}}(\mathbb {R},w_{m+1})$

$L^{p}(\mathbb {R},\nu _{\vec {\kern 1pt w}})$

$\vec {\kern 1pt w}=(w_1, \ldots , w_{m+1})\in A_{\vec {P}}(\mathbb {R}^{m+1})$

$\mathcal {C}_{m+1, A}$

Let R be a semiprime ring with the extended centroid C and Q the maximal right ring of quotients of R. Set [y, x]1 = [y, x] = yx − xy for x, y ∈ Q and inductively [y, x]k = [[y, x]k−1, x] for k > 1. Suppose that f : R → Q is an additive map satisfying [f(x), x]n = 0 for all x ∈ R, where n is a fixed positive integer. Then it can be shown that there exist λ ∈ C and an additive map μ : R → C such that f(x) = λx + μ(x) for all x ∈ R. This gives the affirmative answer to the unsolved problem of such functional identities initiated by Brešar in 1996.

We begin by defining a homoclinic class for homeomorphisms. Then we prove that if a topological homoclinic class Λ associated with an area-preserving homeomorphism f on a surface M is topologically hyperbolic (i.e. has the shadowing and expansiveness properties), then Λ = M and f is an Anosov homeomorphism.

In this note we give simple proofs of several results involving maximal truncated Calderón–Zygmund operators in the general setting of rearrangement-invariant quasi-Banach function spaces by sparse domination. Our techniques allow us to track the dependence of the constants in weighted norm inequalities; additionally, our results hold in ℝn as well as in many spaces of homogeneous type.

We show quantifier elimination theorems for real closed valued fields with separated analytic structure and overconvergent analytic structure in their natural one-sorted languages and deduce that such structures are weakly o-minimal. We also provide a short proof that algebraically closed valued fields with separated analytic structure (in any rank) are C-minimal.

We characterize the topological spaces of minimum cardinality which are weakly contractible but not contractible. This is equivalent to finding the non-dismantlable posets of minimum cardinality such that the geometric realization of their order complexes are contractible. Specifically, we prove that all weakly contractible topological spaces with fewer than nine points are contractible. We also prove that there exist (up to homeomorphism) exactly two topological spaces of nine points which are weakly contractible but not contractible.

We show how to find higher generating families of subgroups, in the sense of Abels and Holz, for groups acting on Cohen–Macaulay complexes. We apply this to groups with a BN-pair to prove higher generation by parabolic and Levi subgroups and describe higher generating families of parabolic subgroups in Aut(Fn).

In this paper we study a general eigenvalue problem for the so called (p, 2)-Laplace operator on a smooth bounded domain Ω ⊂ ℝN under a nonlinear Steklov type boundary condition, namely

\[\left\{ \begin{aligned} -\Delta_pu-\Delta u & =\lambda a(x)u \quad {\rm in}\ \Omega,\\ (|\nabla u|^{p-2}+1)\dfrac{\partial u}{\partial\nu} & =\lambda b(x)u \quad {\rm on}\ \partial\Omega . \end{aligned} \right.\]