We prove that there are only $O(H^{3+\epsilon})$ quartic integer polynomials with height at most $H$ and a Galois group which is a proper subgroup of $S_4$. This improves in the special case of degree four a bound by Gallagher that yielded $O(H^{7/2} \log H)$.

For a closed topological $n$-manifold $X$, the surgery exact sequence contains the set of manifold structures and the set of tangential structures of $X$. In the case of a compact topological $n$-manifold with boundary $(X$, $\partial X)$, the classical surgery theory usually considers two different types of structures. The first one concerns structures whose restrictions are fixed on the boundary. The second one uses two similar structures on the manifold pair. In his classical book, Wall mentioned the possibility of introducing a mixed type of structure on a manifold with boundary. Following this suggestion, we introduce mixed structures on a topological manifold with boundary, and describe their properties. Then we obtain connections between these structures and the classical ones, and prove that they fit in some surgery exact sequences. The relationships can be described by using certain braids of exact sequences. Finally, we discuss explicitly several geometric examples.

Let $\chi(n)$ be a quadratic character modulo a prime $p$. For a fixed integer $s\ne 0$, we estimate certain exponential sums with truncated $L$-functions \[L_{s,p}(n) = \sum_{j=1}^n \frac{\chi(\,j)}{j^s}\qquad (n =1, 2, \ldots)\]. Our estimate implies certain uniformly of distribution properties of reductions of $L_{s,p}(n)$ in the residue classes modulo $p$.

A ring $R$ with identity is called strongly clean if every element of $R$ is the sum of an idempotent and a unit that commute. For a commutative local ring $R$, $n=3,4$, and $m, k, s \in {\mathbb N}$ it is proved that ${\mathbb M}_n(R)$ is strongly clean if and only if ${\mathbb M}_n(R[[x]])$ is strongly clean if and only if ${\mathbb M}_n(R[[x_1, x_2, \ldots, x_m]])$ is strongly clean if and only if $ {\mathbb M}_n(\frac{R[x]}{(x^{k})})$ is strongly clean if and only if $ {\mathbb M}_n(\dfrac{R[x_{1}, x_{2}, \ldots , x_{s}]}{(x^{n_1}_{1}, x^{n_{2}}_{2}, \ldots , x^{n_{s}}_{s})}) $ is strongly clean if and only if ${\mathbb M}_n(R \propto R)$ is strongly clean where $ R\propto R=\{\scriptsize(\begin{array}{@{}c@{\quad}c@{}} a& b \\ 0& a \end{array}): a, b \in R \}$ is the trivial extension of $R$. This extends a result of J. Chen, X. Yang and Y. Zhou [$\mathbf{5}$] from $n=2$ to 3 and 4.

We show that the spherical subalgebra $U_{k,c}$ of the rational Cherednik algebra associated to $S_n C_{\ell}$, the wreath product of the symmetric group and the cyclic group of order $\ell$, is isomorphic to a quotient of the ring of invariant differential operators on a space of representations of the cyclic quiver of size $\ell$. This confirms a version of [5Conjecture 11.22] in the case of cyclic groups. The proof is a straightforward application of work of Oblomkov [12] on the deformed Harish–Chandra homomorphism, and of Crawley–Boevey, [3] and [4], and Gan and Ginzburg [7] on preprojective algebras.

We give a necessary and sufficient condition for $k$-step nilmanifolds associated with graphs $(k \geq 3)$ to admit Anosov automorphisms. We also prove the nonexistence of Anosov automorphisms on certain classes of 2-step and 3-step nilmanifolds.

If bounded linear operators $A$ and $B$ are each reguloid, and have the single valued extension property, then Weyl's theorem holds for all holomorphic functions of all operator matrices $M_{C}=\scriptsize\scriptsize(\begin{array}{@{}cc@{}}A&C\\0&B\end{array})$.

A number of integral Hopf algebras have been studied that have, as their underlying modules, the free ${\bf Z}$-module generated by finite words in a certain alphabet. For example, the tensor algebra, the rings of quasisymmetric functions and of noncommutative symmetric functions, the Solomon descent algebra, the Malvenuto-Reutenauer algebra and the homology and cohomology of $\Omega\Sigma{\bf C} P^{\infty}$ are all of this type. Some of these are known to be isomorphic or dual to each other, some are known only to be rationally isomorphic, some have been stated in the literature to be isomorphic when they are only rationally isomorphic.

This paper is, in part, an attempt to find order in this chaos of word Hopf algebras. We consider three multiplications on such modules, and their dual comultiplications, and clarify which of these operations can be combined to obtain Hopf structures. We discuss when the results are isomorphic, integrally or rationally, and study the resulting structures. We are not attempting a classification of Hopf algebras of words, merely an organization of some of the Hopf algebras of this type that have been studied in the literature.

We provide a complete local classification of pseudo-parallel sub-manifolds with flat normal bundle of space forms, extending the classification by Dillen-Nölker for the semi-parallel case.

In this note we answer two question posed by Berkani and Koliha [Acta Sci. Math.69 (2003), 359–376]. We show that generalized Browder's (resp. generalized $a$-Browder's) theorem holds for a Banach space operator if and only if Browder's (resp. $a$-Browder's) theorem does. We also give condition under which generalized Weyl's (resp. generalized $a$-Weyl's) theorem is equivalent to Weyl's (resp. $a$-Weyl's) theorem.

We define “star reducible” Coxeter groups to be those Coxeter groups for which every fully commutative element (in the sense of Stembridge) is equivalent to a product of commuting generators by a sequence of length-decreasing star operations (in the sense of Lusztig). We show that the Kazhdan–Lusztig bases of these groups have a nice projection property to the Temperley–Lieb type quotient, and furthermore that the images of the basis elements $C'_w$ (for fully commutative $w$) in the quotient have structure constants in ${\mathbb Z}^{\geq 0}[v, v^{-1}]$. We also classify the star reducible Coxeter groups and show that they form nine infinite families with two exceptional cases.