Let $W$ be a finite-dimensional ${\bb Z}/p$ -module over a field, ${\bf k}$ , of characteristic $p$ . The maximum degree of an indecomposable element of the algebra of invariants, ${\bf k}[W]^{{\bb Z}/p}$ , is called the Noether number of the representation, and is denoted by $\beta(W)$ . A lower bound for $\beta(W)$ is derived, and it is shown that if $U$ is a ${\bb Z}/p$ submodule of $W$ , then $\beta(U)\le \beta(W)$ . A set of generators, in fact a SAGBI basis, is constructed for ${\bf k}[V_2\oplus V_3]^{{\bb Z}/p}$ , where $V_n$ is the indecomposable ${\bb Z}/p$ -module of dimension $n$ .

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.

Minimal complex surfaces of general type with $p_g = 0$ and $K^2 = 7$ or $8$ whose bicanonical map is not birational are studied. It is shown that if $S$ is such a surface, then the bicanonical map has degree 2 (see Bulletin of the London Mathematical Society 33 (2001) 1–10) and there is a fibration $f : S \rightarrow {\bb P}^1$ such that (i) the general fibre $F$ of $f$ is a genus 3 hyperelliptic curve; (ii) the involution induced by the bicanonical map of $S$ restricts to the hyperelliptic involution of $F$.

Furthermore, if $K^2_S = 8$, then $f$ is an isotrivial fibration with six double fibres, and if $K^2_S = 7$, then $f$ has five double fibres and it has precisely one fibre with reducible support, consisting of two components.

The paper gives lists of all the primitive permutation groups of squarefree degree. All such groups are either solvable and act on a prime number of points, or are almost simple. Among the almost simple examples, the groups of Lie type have rank at most $2$, or the point stabilizer is a parabolic subgroup.

In 1956, A. Selberg introduced weakly symmetric spaces in the framework of his development of the trace formula, and proved that in a weakly symmetric space, the algebra of all invariant (with respect to the full isometry group) differential operators is commutative [22]. In this paper, Selberg asked whether the converse holds. In the present work, we shall answer this question by presenting examples of commutative spaces which are not weakly symmetric. These examples arise in the quaternionic analogues to the Heisenberg group, endowed with certain special metrics (see Theorem 5 and the explicit realization after it).

We prove that a proper holomorphic map on the unit disk in the complex plane is uniquely determined up to post-composition with a Möbius transformation by its critical points.

The main purpose of this paper is to use the properties of Gaussian sums and the analytic method to study the problem of calculating one kind of character sums modulo $q$ , and to give an interesting identity.

An explicit, asymptotically good, tower of curves over the field with eight elements is constructed. The genus and the number of rational points are calculated explicitly.

For $f$ meromorphic in the complex plane and $\varphi$ meromorphic in the unit disc, sharp upper bounds are obtained for $$m\left(r,\frac{f^{(k)}}{f^{(j)}}\right)=\frac{1}{2\pi}\int_0^{2\pi} \log^{+}\left|\frac{f^{(k)}(re^{i\theta})}{f^{(j)}(re^{i\theta})}\right |\, d\theta,\qquad r<\infty,$$ and $$m\left(r,\frac{\vp^{(k)}}{\vp^{(j)}}\right)=\frac{1}{2\pi}\int_0^{2\pi} \log^{+}\left|\frac{\vp^{(k)}(re^{i\theta})}{\vp^{(j)}(re^{i\theta})}\right |\, d\theta,\qquad r<1,$$ where $k$ and $j$ are integers satisfying $k>j\geq 0$. The results generalize the logarithmic derivative estimate due to Gol'dberg and Grinshtein to derivatives of higher order.

Hyperbolic invariant sets Λ of C1+γ diffeomorphisms where either the stable or unstable leaves are 1-dimensional are considered in this paper. Under the assumption that the Λ has local product structure, the authors prove that the holonomies between the 1-dimensional leaves are C1+α for some 0 < α < 1.

For a transcendental meromorphic function f, various properties of the set

formula here

were obtained in [8] and [9]. Here we establish analogous properties for the smaller sets

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introduced in [5], and

formula here

We deduce a symmetry result for Julia sets J(f), and also indicate some techniques for showing that certain invariant curves lie in I′(f), Z(f) and J(f).

We prove that a finite solvable group whose order is divisible by a prime p has at least 2√p−1 conjugacy classes.

Nearly four hundred years ago, the cubic close-packing of equal spheres in R3 was discovered by Kepler [21], in which each sphere touches 12 others. In 1694, Gregory and Newton discussed the following thirteen spheres problem. Can a rigid material sphere be brought into contact with 13 other such spheres of the same size? Gregory believed ‘yes’, while Newton thought ‘no’.

In this paper, a proof is given that there exists a universal Taylor series – in the sense of Nestoridis – in the complement of a disk with respect to every center. This answers a question asked by G. Costakis.

The metric entropy of absolute convex hulls of sets in Hilbert spaces is studied for the general case when the metric entropy of the sets is arbitrary. Under some regularity assumptions, the results are sharp.

Co-Hopfian finitely generated torsion-free nilpotent groups are characterized in this paper in terms of their Lie algebra automorphisms. Many examples of such groups are also constructed.Partially supported by NSF Grant # DMS-0203979.

We study zeros of elliptic integrals I(h)=∫∫H[les ]hR(x, y)dx dy, where H(x, y) is a real cubic polynomial with a symmetry of order three, and R(x, y) is a real polynomial of degree at most n. It turns out that the vector space [Ascr ]n formed by such integrals is a Chebishev system: the number of zeros of each elliptic integral I(h)∈[Ascr ]n is less than the dimension of the vector space [Ascr ]n.

If there are infinitely many p-Mersenne prime numbers, there is no bad field of positive characteristic p.

We give here a detailed description of the Brill–Noether loci of stable vector bundles of slope 2 on a smooth projective curve, which is assumed to be nonhyperelliptic.

We present a simple proof of the fact that every countable group ${\Gamma}$ is weak Rohlin, that is, there is in the Polish space $mathbb{A}_{\Gamma}$ of measure preserving ${\Gamma}$-actions an action $\mathbf{T}$ whose orbit in $\mathbb{A}_{\Gamma}$ under conjugations is dense. In conjunction with earlier results this in turn yields a new characterization of non-Kazhdan groups as those groups which admit such an action $\mathbf{T}$ which is also ergodic.