In this paper, the relation between algebraic shifting and join which was conjectured by Eran Nevo will be proved. Let σ and τ be simplicial complexes and σ*τ be their join. Let J_σ be the exterior face ideal of σ and Δ(σ) the exterior algebraic shifted complex of σ. Assume that σ*τ is a simplicial complex on [n]={1,2,..., n}. For any d-subset S⊂[n], let $m_{\preceq_{\textrm{rev}}S}(\sigma)$ denote the number of d-subsets R∈σ which are equal to or smaller than S with respect to the reverse lexicographic order. We will prove that $m_{\preceq_{\textrm{rev}}S}(\Delta(\sigma*\tau))\geq m_{\preceq_{\textup{rev}}S}(\Delta(\Delta(\sigma) *\Delta(\tau)))$ for all S⊂[n]. To prove this fact, we also prove that $m_{\preceq_{\textrm{rev}}S}(\Delta(\sigma))\geq m_{\preceq_{\textup{rev}}S}(\Delta(\Delta_{\varphi}(\sigma)))$ for all S⊂[n] and for all nonsingular matrices ϕ, where Δ_ϕ(σ) is the simplicial complex defined by $J_{\Delta_{\varphi}(\sigma)}=\textup{in}(\varphi(J_{\sigma}))$.

After recalling briefly the main properties of the amalgamated duplication of a ring R along an ideal I, denoted by $R\bowtie I$, see M. D’Anna and M. Fontana, to appear in J. Algebra Appl., we restrict our attention to the study of the properties of $R\bowtie I$, when I is a multiplicative canonical ideal of R, see W. J. Heinzer, J. A. Huckaba and I. J. Papick, Comm. Algebra. In particular, we study when every regular fractional ideal of $R\bowtie I$ is divisorial.

In this paper, we construct some unirational Calabi–Yau threefolds in characteristic 3. We adopt the method by Schoen, but we use quasi-elliptic surfaces instead of elliptic surfaces. We find new examples which do not admit a lifting to characteristic zero.

The purpose of this erratum is to point out a mistake in a previous article by the authors and to explain how this affects the results of that article.

We prove the vanishing of the Koszul homology group H_μ(Kos(M)_μ), where μ is the minimal number of generators of M. We give a counterexample that the Koszul complex of a module is not always acyclic and show its relationship with the homology of commutative rings.

If n is a non-negative integer, then denote by ∂^-nH^∞ the space of all complex-valued functions f defined on $\mathbb{D}$ such that f, f⁽¹⁾, f⁽²⁾,..., f⁽ⁿ⁾ belong to H^∞, with the norm$\|f\|=\sum_{j=0}^{n}\frac{1}{j!}\|f^{(j)}\|_{\infty}.$We prove bounds on the solution in the corona problem for ∂^-nH^∞. As corollaries, we obtain estimates in the corona theorem also for some other subalgebras of the Hardy space H^∞.

In this paper we consider the computational complexity of uniformizing a domain with a given computable boundary. We give nontrivial upper and lower bounds in two settings: when the approximation of the boundary is given either as a list of pixels, or by a Turing machine.

The topological group $\mathcal{D}^k(\mathbb{S})$ of diffeomorphisms of the unit circle $\mathbb{S}$ of Sobolev class H^k, for k large enough, is a Banach manifold modeled on the Hilbert space $H^k(\mathbb{S})$. In this paper we show that the H¹ right-invariant metric obtained by right-translation of the H¹ inner product on $T_{\rm id}\mathcal{D}^k(\mathbb{S})\simeq H^k(\mathbb{S})$ defines a smooth Riemannian metric on $\mathcal{D}^k(\mathbb{S})$, and we explicitly construct a compatible smooth affine connection. Once this framework has been established results from the general theory of affine connections on Banach manifolds can be applied to study the exponential map, geodesic flow, parallel translation, curvature etc. The diffeomorphism group of the circle provides the natural geometric setting for the Camassa–Holm equation – a nonlinear wave equation that has attracted much attention in recent years – and in this context it has been remarked in various papers how to construct a smooth Riemannian structure compatible with the H¹ right-invariant metric. We give a self-contained presentation that can serve as a detailed mathematical foundation for the future study of geometric aspects of the Camassa–Holm equation.

We obtain integral boundary decay estimates for solutions of fourth-order elliptic equations on a bounded domain with regular boundary. We apply these estimates to obtain stability bounds for the corresponding eigenvalues under small perturbations of the boundary.

We classify the ordinary differential equations that correspond to elliptic CR-manifolds with maximal isotropy. It follows that the dimension of the isotropy group of an elliptic CR-manifold can only be 10 (for the quadric), 4 (for the listed examples) or less. This is in contrast with the situation of hyperbolic CR-manifolds, where the dimension can be 10 (for the quadric), 6 or 5 (for semi-quadrics) or less than 4. We also prove that, for all elliptic CR-manifolds with non-linearizable isotropy group, except for two special manifolds, the points with non-linearizable isotropy form exactly some complex curve on the manifold.

We show that the Luzin area integral or the square function on the unit ball of ℂⁿ, regarded as an operator in the weighted space L²(w) has a linear bound in terms of the invariant A₂ characteristic of the weight. We show a dimension-free estimate for the “area-integral” associated with the weighted L²(w) norm of the square function. We prove the equivalence of the classical and the invariant A₂ classes.

Let Ω⊂ℝⁿ be an arbitrary open set. We characterize the space W^1,1_loc(Ω) using variants of the classical area and coarea formulas. We use these characterizations to obtain a norm approximation and trace theorems for functions in the space W^1,1(ℝⁿ).