it is shown that, for all large $x$, there are more than $x^{0.33}$ carmichael numbers up to $x$, improving on the ground-breaking work of alford, granville and pomerance, who were the first to demonstrate that there are infinitely many such numbers. the same basic construction as that employed by these authors is used, but a slight modification enables a stronger result on primes in arithmetic progressions based on a sieve method to be employed.

a proof is given to show that for an inner form of $\gl_n$ over a global field of zero characteristic, there exist only a finite number of automorphic representations with fixed local factor (up to equivalence) at almost every place. what is new in comparison to earlier work (see a. i. badulescu and p. broussous, ‘un théorème de finitude’, compositio math. 132 (2002) 177–190) is the case when the local factors are not fixed at the infinite places, as well as the statement of the result for the automorphic spectrum, rather than the cuspidal one.

let $e/\mathbb{q}$ be an elliptic curve. for a prime $p$ of good reduction, let $e(\mathbb{f}_p)$ be the set of rational points defined over the finite field $\mathbb{f}_p$. denote by $\omega(\#e(\mathbb{f}_p))$ the number of distinct prime divisors of $\#e(\mathbb{f}_p)$. for an elliptic curve with complex multiplication, the normal order of $\omega(\#e(\mathbb{f}_p))$ is shown to be $\log \log p$. the normal order of the number of distinct prime factors of the exponent of $e(\mathbb{f}_p)$ is also studied.

for a commutative ring $r$, let $\nil(r)$ be the set of all nilpotent elements of $r$, $z(r)$ the set of all zero divisors of $r$, and $t(r)$ the total quotient ring of $r$. set $\mathcal{h} = \{r \mid r$ is a commutative ring and $\nil (r)$ is a divided prime ideal of $ r \}$. for a ring $r \in \mathcal{h}$, let $\phi : t(r) \longrightarrow r_{\nil (r)}$ be such that $\phi(a/b) = a/b$ for every $a \in r$ and $b \in r\backslash z(r)$. a ring $r$ is called a zpui ring if every proper ideal of $r$ can be written as a finite product of invertible and prime ideals of $r$. this paper gives a generalization of the concept of zpui domains (which was extensively studied by olberding) to the context of rings that are in the class $\mathcal{h}$. let $r \in \mathcal{h}$. if every nonnil ideal of $r$ can be written as a finite product of invertible and prime ideals of $r$, then $r$ is called a nonnil zpui ring; also, if every nonnil ideal of $\phi(r)$ can be written as a finite product of invertible and prime ideals of $\phi(r)$, then $r$ is called a nonnil$\phi$-zpui ring. the theory of $\phi$-zpui rings is shown here to resemble that of zpui domains.

a structure theorem is given for $n$-dimensional smooth subvarieties of the grassmannian $g(1,n)$, with $n\geq n+3$, that can be isomorphically projected to $g(1,n+1)$. a complete classification in the cases $n=2n+1$ and $n=2n$ follows, as a corollary.

let $r$ be a ring and $t$ a 1-tilting right $r$-module. then $t$ is of countable type. moreover, $t$ is of finite type in the case where $r$ is a prüfer domain.

a conjecture is proposed, bounding the number of cycles with label $w^n$ in a labeled directed graph. some partial results towards this conjecture are established. as a consequence, it is proved that $\langle a_1, a_2, \ldots\,{\mid}\,w^n\rangle$ is coherent for $n\,{\geq}\,4$. furthermore, it is coherent for $n\,{\geq}\,2$, provided that the strengthened hanna neumann conjecture holds.

a tits alternative theorem is proved in this paper for groups acting on cat(0) cubical complexes. that is, a proof is given to show that if $g$ is assumed to be a group for which there is a bound on the orders of its finite subgroups, and if $g$ acts properly on a finite-dimensional cat(0) cubical complex, then either $g$ contains a free subgroup of rank 2, or $g$ is finitely generated and virtually abelian. in particular, the above conclusion holds for any group $g$ with a free action on a finite-dimensional cat(0) cubical complex.

the cohomology of a right-angled artin group with group ring coefficients is explicitly presented in terms of the cohomology of its defining flag complex.

in two papers, littlewood studied seemingly unrelated constants: (i) the best $\alpha$ such that for any polynomial $f$, of degree $n$, the areal integral of its spherical derivative is at most $\const\cdot n^\alpha$, and (ii) the extremal growth rate $\beta$ of the length of green's equipotentials for simply connected domains. these two constants are shown to coincide, thus greatly improving known estimates on $\alpha$.

it is proved in this paper that super-exponentially decaying, possibly non-selfadjoint perturbations of the free jacobi operator are uniquely determined by the location of all their eigenvalues and resonances.

in this paper, the space $\mathcal{a}_\psi(\mathbb{d})$ is considered, consisting of those holomorphic functions $f$ on the unit disk $\mathbb{d}$ such that $\|f\|_\psi=\sup_{z\in\mathbb{d}}|f(z)|\psi(|z|)<+\infty$, with $\psi(1)=0$. the sampling problem is studied for weights satisfying $\ln\psi(r)/\ln(1-r)\to0$. möbius stability of sampling is shown to fail in this space.

an index of a collection of 1-forms on a complex isolated complete intersection singularity corresponding to a chern number is defined and – in the case when the 1-forms are complex analytic – expressed as the dimension of a certain algebra.

functional identities for the classical rogers dilogarithm associated to simply laced dynkin diagrams are proved uniformly in the setting of $y$-systems related to clusters.

let $e$ be a subset of the boundary of the open unit disc $d$, and let $a$ be the algebra of bounded holomorphic functions on $d$ that extend continuously to $d \cup e$. it is shown that if $f$ is a bounded harmonic function on $d$ that extends continuously to $d \cup e$ and is not holomorphic, then the uniformly closed algebra $a[f]$ generated by $a$ and $f$ contains $c({\overline{d}})$. this result contains as special cases a result on the disc algebra due to čirka and a result on $h^{\infty}(d)$ due to axler and shields. a stronger form of the result, in which $f$ is allowed to have discontinuities on a small subset of $e$, is also established.

a general, albeit simple, approach to obtaining rearrangement inequalities of maximal operators is given. being applied to specific operators it leads not only to new proofs of well-known relations in a simpler and shorter way, but also to new, profitable results.

the degree defined by p. benevieri and m. furi (see ‘a simple notion of orientability for fredholm maps of index zero between banach manifolds and degree theory’, ann. sci. math. québec 22 (1998) 131–148) is used here to obtain some sharp lower bounds for the number of solutions of the $\lambda$-sections of the compact components of the set of non-trivial solutions of $\mf{f}(\lambda,x)=0$, where $\mf{f}$ is a $\cal{c}^1$ fredholm map of index 1 such that $\mf{f}(\lambda,0)=0$ for all $\lambda\in\mathbb{r}$. these bounds are given in terms of the parity of the linearized fredholm family $d_2\mf{f}(\cdot,0)$. the parity is a local invariant measuring the change of the orientation of $d_2\mf{f}(\lambda,0)$ as $\lambda$ crosses an interval. the set of eigenvalues of $d_2\mf{f}(\cdot,0)$ is not assumed to be discrete. therefore, even in the classical situation when $\mf{f}$ is a compact perturbation of the identity, the results presented here are completely new.

v. paulsen and d. singh gave a characterization of certain hilbert spaces contractively contained in $l^p$, $p \geq 2$. here, the analog of their theorem is proved for $1 \leq p < 2$.