We prove that the group of automorphisms of the Lie algebra DerK(Qn) of derivations of the field of rational functions Qn = K(x1, . . ., xn) over a field of characteristic zero is canonically isomorphic to the group of automorphisms of the K-algebra Qn.

In this note, we have obtained a Whitehead-like tower of a module by considering a suitable set of morphisms and shown that the different stages of the tower are the Adams cocompletions of the module with respect to the suitably chosen set of morphisms.

Let $\mathcal{M}$ and G denote, respectively, the maximal operator and the geometric maximal operator associated with the dyadic lattice on $\mathbb{R}^d$.

1. (i) We prove that for any 0 < p < ∞, any weight w on $\mathbb{R}^d$ and any measurable f on $\mathbb{R}^d$, we have Fefferman–Stein-type estimate

   $$\begin{equation*} ||G(f)||_{L^p(w)}\leq e^{1/p}||f||_{L^p(\mathcal{M}w)}. \end{equation*} $$
   For each p, the constant e1/p is the best possible.

2. (ii) We show that for any weight w on $\mathbb{R}^d$ and any measurable f on $\mathbb{R}^d$,

   $$\begin{equation*} \int_{\mathbb{R}^d} G(f)^{1/\mathcal{M}w}w\mbox{d}x\leq e\int_{\mathbb{R}^d} |f|^{1/w}w\mbox{d}x \end{equation*} $$
   and prove that the constant e is optimal.

Over a Cohen–Macaulay (CM) local ring, we characterize those modules that can be obtained as a direct limit of finitely generated maximal CM modules. We point out two consequences of this characterization: (1) Every balanced big CM module, in the sense of Hochster, can be written as a direct limit of small CM modules. In analogy with Govorov and Lazard's characterization of flat modules as direct limits of finitely generated free modules, one can view this as a “structure theorem” for balanced big CM modules. (2) Every finitely generated module has a pre-envelope with respect to the class of finitely generated maximal CM modules. This result is, in some sense, dual to the existence of maximal CM approximations, which has been proved by Auslander and Buchweitz.

We classify all non-affine Hopf algebras H over an algebraically closed field k of characteristic zero that are integral domains of Gelfand–Kirillov dimension two and satisfy the condition Ext1H(k, k) ≠ 0. The affine ones were classified by the authors in 2010 (Goodearl and Zhang, J. Algebra324 (2010), 3131–3168).

We provide ℕ-filtrations on the negative part Uq($\mathfrak{n}$−) of the quantum group associated to a finite-dimensional simple Lie algebra $\mathfrak{g}$, such that the associated graded algebra is a skew-polynomial algebra on $\mathfrak{n}$−. The filtration is obtained by assigning degrees to Lusztig's quantum PBW root vectors. The possible degrees can be described as lattice points in certain polyhedral cones. In the classical limit, such a degree induces an ℕ-filtration on any finite-dimensional simple $\mathfrak{g}$-module. We prove for type An, Cn, B3, D4 and G2 that a degree can be chosen such that the associated graded modules are defined by monomial ideals, and conjecture that this is true for any $\mathfrak{g}$.

Voevodsky has conjectured that numerical equivalence and smash-equivalence coincide for algebraic cycles on any smooth projective variety. Building on work of Vial and Kahn–Sebastian, we give some new examples of varieties where Voevodsky's conjecture is verified.

In this paper, we study the multiplicity of solutions for the following problem:

$$\begin{equation*} \begin{cases} -\Delta u-\Delta(|u|^{\alpha})|u|^{\alpha-2}u=g(x,u)+\theta h(x,u), \ \ x\in \Omega,\\ u=0, \ \ x\in \partial\Omega, \end{cases} \end{equation*}$$

${\mathbb{R}}$

$\bar{\Omega}$

${\mathbb{R}}$

$\mathbb{N}$

We show that every fibrewise map from a Serre microfibration to a Serre fibration is n-connected if it is fibrewise n-connected. This generalises a result of M. Weiss and related results by Bökstedt–Madsen and Galatius–Randal–Williams. We also discuss an application to configuration spaces.

Let ${\mathcal C}[{\mathcal X}]$ be any class of operators on a Banach space ${\mathcal X}$, and let ${Holo}^{-1}({\mathcal C})$ denote the class of operators A for which there exists a holomorphic function f on a neighbourhood ${\mathcal N}$ of the spectrum σ(A) of A such that f is non-constant on connected components of ${\mathcal N}$ and f(A) lies in ${\mathcal C}$. Let ${{\mathcal R}[{\mathcal X}]}$ denote the class of Riesz operators in ${{\mathcal B}[{\mathcal X}]}$. This paper considers perturbation of operators $A\in\Phi_{+}({\mathcal X})\Cup\Phi_{-}({\mathcal X})$ (the class of all upper or lower [semi] Fredholm operators) by commuting operators in $B\in{Holo}^{-1}({\mathcal R}[{\mathcal X}])$. We prove (amongst other results) that if πB(B) = ∏mi = 1(B − μi) is Riesz, then there exist decompositions ${\mathcal X}=\oplus_{i=1}^m{{\mathcal X}_i}$ and $B=\oplus_{i=1}^m{B|_{{\mathcal X}_i}}=\oplus_{i=1}^m{B_i}$ such that: (i) If λ ≠ 0, then $\pi_B(A,\lambda)=\prod_{i=1}^m{(A-\lambda\mu_i)^{\alpha_i}} \in\Phi_{+}({\mathcal X})$ (resp., $\in\Phi_{-}({\mathcal X})$) if and only if $A-\lambda B_0-\lambda\mu_i\in\Phi_{+}({\mathcal X})$ (resp., $\in\Phi_{-}({\mathcal X})$), and (ii) (case λ = 0) $A\in\Phi_{+}({\mathcal X})$ (resp., $\in\Phi_{-}({\mathcal X})$) if and only if $A-B_0\in\Phi_{+}({\mathcal X})$ (resp., $\in\Phi_{-}({\mathcal X})$), where B0 = ⊕mi = 1(Bi − μi); (iii) if $\pi_B(A,\lambda)\in\Phi_{+}({\mathcal X})$ (resp., $\in\Phi_{-}({\mathcal X})$) for some λ ≠ 0, then $A-\lambda B\in\Phi_{+}({\mathcal X})$ (resp., $\in\Phi_{-}({\mathcal X})$).

We characterise the elements of the (maximum) idempotent-generated subsemigroup of the Kauffman monoid in terms of combinatorial data associated with certain normal forms. We also calculate the smallest size of a generating set and idempotent generating set.

Given a complete hereditary cotorsion pair $(\mathcal{X}, \mathcal{Y})$, we introduce the concept of $(\mathcal{X}, \mathcal{X} \cap \mathcal{Y})$-Gorenstein projective modules and study its stability properties. As applications, we first get two model structures related to Gorenstein flat modules over a right coherent ring. Secondly, for any non-negative integer n, we construct a cofibrantly generated model structure on Mod(R) in which the class of fibrant objects are the modules of Gorenstein injective dimension ≤ n over a left Noetherian ring R. Similarly, if R is a left coherent ring in which all flat left R-modules have finite projective dimension, then there is a cofibrantly generated model structure on Mod(R) such that the cofibrant objects are the modules of Gorenstein projective dimension ≤ n. These structures have their analogous in the category of chain complexes.

Let $\mathbb{K}$ be a field and S = ${\mathbb{K}}$[x1, . . ., xn] be the polynomial ring in n variables over the field $\mathbb{K}$. For every monomial ideal I ⊂ S, we provide a recursive formula to determine a lower bound for the Stanley depth of S/I. We use this formula to prove the inequality sdepth(S/I) ≥ size(I) for a particular class of monomial ideals.

The purpose of this paper is to investigate sphere theorems for submanifolds with positive biorthogonal (sectional) curvature. We provide some upper bounds for the full norm of the second fundamental form under which a compact submanifold must be diffeomorphic to a sphere.

Let (A, ${\mathfrak{m}$) be a Cohen–Macaulay local ring of dimension d and let I ⊆ J be two ${\mathfrak{m}$-primary ideals with I a reduction of J. For i = 0,. . .,d, let eiJ(A) (eiI(A)) be the ith Hilbert coefficient of J (I), respectively. We call the number ci(I, J) = eiJ(A) − eiI(A) the ith relative Hilbert coefficient of J with respect to I. If GI(A) is Cohen–Macaulay, then ci(I, J) satisfy various constraints. We also show that vanishing of some ci(I, J) has strong implications on depth GJn(A) for n ≫ 0.

Let (M, g(t)) be a compact Riemannian manifold and the metric g(t) evolve by the Ricci flow. In the paper, we prove that the eigenvalues of geometric operator −Δφ + $\frac{R}{2}$ are non-decreasing under the Ricci flow for manifold M with some curvature conditions, where Δφ is the Witten Laplacian operator, φ ∈ C2(M), and R is the scalar curvature with respect to the metric g(t). We also derive the evolution of eigenvalues under the normalized Ricci flow. As a consequence, we show that compact steady Ricci breather with these curvature conditions must be trivial.