Let $(W,S)$ be a finite Coxeter system, and let $J\,{\subseteq}\, S$. Any $w\,{\in}\,W$ has a unique factorization $w\,{=}\,w^Jw_J$, where $w_J$ belongs to the parabolic subgroup $W_J$ generated by $J$, and $w^J$ is of minimal length in the coset $wW_J$. It is shown here that $w_I\,{=}\,w^J$ if and only if $w^I\,{=}\,w_J$, for all $I,J\,{\subseteq}\,S$. Furthermore, a similar symmetry property in arbitrary $(W_I,W_J)$-double cosets is conjectured, which links this result to the Solomon descent algebra of $W$.

For an irreducible, crystallographic root system $\Phi$ in a Euclidean space $V$ and a positive integer $m$, the arrangement of hyperplanes in $V$ given by the affine equations $(\alpha, x)\,{=}\,k$, for $\alpha\,{\in}\,\Phi$ and $k\,{=}\,0, 1,\dots,m$, is denoted here by ${\mathcal A}_{\Phi}^m$. The characteristic polynomial of ${\mathcal A}_{\Phi}^m$ is related in the paper to that of the Coxeter arrangement ${\mathcal A}_{\Phi}$ (corresponding to $m\,{=}\,0$), and the number of regions into which the fundamental chamber of ${\mathcal A}_{\Phi}$ is dissected by the hyperplanes of ${\mathcal A}_{\Phi}^m$ is deduced to be equal to the product $\prod_{i=1}^{\ell} ({e_i\,{+}\,m h\,{+}\,1})/({e_i\,{+}\,1})$, where $e_1, e_2,\dots,e_\ell$ are the exponents of $\Phi$ and $h$ is the Coxeter number. A similar formula for the number of bounded regions follows. Applications to the enumeration of antichains in the root poset of $\Phi$ are included.

It is proved that if $R$ is an associative ring that is cotorsion as a left module over itself, and $J$ is the Jacobson radical of $R$, then the quotient ring $R/J$ is a left self-injective von Neumann regular ring and idempotents lift modulo $J$. In particular, if $R$ is indecomposable, then it is a local ring.

Let $K$ be a $p$-adic field, let $Z_{\Phi }(s,f)$, $s\,{\in}\,\mathbb{C}$, with Re$(s)\,{>}\,0$, be the Igusa local zeta function associated to $f(x)\,{=}\,(f_{1}(x),\ldots,f_{l}(x))\,{\in}\,[ K( x_{1},\ldots,x_{n})]^{l}$, and let $\Phi $ be a Schwartz–Bruhat function. The aim of this paper is to describe explicitly the poles of the meromorphic continuation of $Z_{\Phi }(s,f)$. Using resolution of singularities it is possible to express $Z_{\Phi }(s,f)$ as a finite sum of $p$-adic monomial integrals. These monomial integrals are computed explicitly by using techniques of toroidal geometry. In this way, an explicit list of the candidates for poles of $Z_{\Phi }(s,f)$ is obtained.

Algebraic higher-rank actions on connected groups are often remarkably rigid in their topological and measurable structure. In contrast to this, the author of this paper constructs uncountably many closed invariant sets and uncountably many invariant measures with positive entropy for irreducible algebraic ${\mathbb{Z}^d}$-actions on zero-dimensional groups.

The minimum value of the Mahler measure of a nonreciprocal polynomial whose coefficients are all odd integers is proved here to be the golden ratio. The smallest measures of reciprocal polynomials with $\pm1$ coefficients and degree at most 72 are also determined.

A real-valued function $f$ defined on an open, convex set $D$ of a real normed space is called $(\varepsilon,\delta)$-midconvex if it satisfies $$f\left(\frac{x+y}{2}\right) \le \frac{f(x)+f(y)}{2} + \varepsilon|x-y| + \delta, \quad\hbox{for } x,y\in D.$$ The main result of the paper states that if $f$ is locally bounded from above at a point of $D$ and is $(\varepsilon,\delta)$-midconvex, then it satisfies the convexity-type inequality $$f(\lambda x+(1-\lambda) y) \leq \lambda f(x)+(1-\lambda)f(y)+2\delta +2\varepsilon \varphi(\lambda)|x-y| \quad\hbox{for } x,y\in D, \, \lambda\in[0,1],$$ where $\varphi:[0,1]\to{\mathbb R}$ is a continuous function satisfying $$\max(-\lambda\log_2\lambda,\,-(1-\lambda)\log_2(1-\lambda)) \le\varphi(\lambda)\le 1.4\max(-\lambda\log_2\lambda,\,-(1-\lambda)\log_2(1-\lambda))$$. The particular case $\varepsilon=0$ of this result is due to Ng and Nikodem (Proc. Amer. Math. Soc. 118 (1993) 103–108), while the specialization $\varepsilon=\delta=0$ yields the theorem of Bernstein and Doetsch (Math. Ann. 76 (1915) 514–526).

An inequality is established involving colengths of the tight closure of ideals of systems of parameters in local rings with some mild conditions. As an application, a proof is given of a result due to Goto and Nakamura (first conjectured by Watanabe and Yoshida), which states that the Hilbert–Samuel multiplicity of a parameter ideal is greater than or equal to the colength of the tight closure of the ideal. The result is also further refined.

Let ${\cal A}$ be a unital maximal full algebra of operator fields with base space $[0, 1]^k$ and fibre algebras $\{{\cal A}_t\}_{t\in[0, 1]}^{k}$. It is shown in this paper that the stable rank of ${\cal A}$ is bounded above by the quantity sup$_{t\in[0, 1]^k}\,{\rm sr}(C([0, 1]^k)\,{\otimes}\,{\cal A}_t)$, where ‘sr’ means stable rank. Using the above estimate, the stable ranks of the C$^*$-algebras of the (possibly higher rank) discrete Heisenberg groups are computed.

Let $S$ be a compact Riemann surface of genus ${\rm g}\,{>}\,1$. By using Teichmüller theory, a new approach is obtained to the existence of a Jenkins–Strebel differential with prescribed type and heights on $S$. Moreover, the geometric structures of horizontal trajectories of certain classes of quadratic differentials are discussed.

Let $\idemp$ be the set of idempotents in a finite-dimensional real algebra $A$. Let $p$ and $q$ be idempotents that lie in the same component of $\idemp$. Then, among the continuous paths connecting $p$ and $q$ in $\idemp$, there exist a polynomial path of degree at most $3$ and a polygonal path consisting of at most three segments.

The integration-by-parts methods introduced in this paper improve upon the $L^p$ estimates on transport densities given in the recent paper by L. De Pascale and A. Pratelli (Calc. Var. Partial Differential Equations 14 (2002) 249–274).

The following property of a normalized basis in a Banach space is considered: any normalized block sequence of the basis has a subsequence equivalent to the basis. Under uniformity or other natural assumptions, a basis with this property is equivalent to the unit vector basis of $c_0$ or $\ell_p$. An analogous problem concerning spreading models is also addressed.

Sharp decay estimates are provided in this paper for spherical averages of a certain multilinear extension operator on $L^{2}(\mathbb{S}^{n-1})\times \ldots\times L^{2}(\mathbb{S}^{n-1})$.