In this paper, we apply Moriwaki's arithmetic height functions to obtain an analogue of Silverman's specialization theorem for families of Abelian varieties over $K$, where $K$ is any field finitely generated over $\mathbb{Q}$.

In answer to a question of Macpherson and Neumann related to the classification of maximal subgroups of infinite symmetric groups, we characterize set ideals on an infinite set X whose stabilizers in the symmetric group Sym(X) are maximal subgroups. Specifically, for an ideal I containing all subsets of cardinality <[mid ]X[mid ], the stabilizer S{I} is maximal if and only if the corresponding Boolean dynamical system (X, I, S{I}) is minimal, that in turn can be expressed as an intrinsic structural property of the Boolean space (X, I). The general case is reduced to a minimal dynamical system on a subset. This characterization clarifies many earlier results, and reveals an interesting link between the theory of infinite permutation groups, topological dynamics and ergodic theory.

The paper introduces a new grading on the preprojective algebra of an arbitrary locally finite quiver. Viewing the algebra as a left module over the path algebra, the author uses the grading to give an explicit geometric construction of a canonical collection of exact sequences of its submodules. If a vertex of the quiver is a source, the above submodules behave nicely with respect to the corresponding reflection functor. It follows that when the quiver is finite and without oriented cycles, the canonical exact sequences are the almost split sequences with preprojective terms, and the indecomposable direct summands of the submodules are the non-isomorphic indecomposable preprojective modules. The proof extends that given by Gelfand and Ponomarev in the case when the finite quiver is a tree.

Several results are proved, related to an old problem posed by G. R. MacLane, namely whether functions in the class [Ascr ] that are locally univalent can have arc tracts. In particular, a proof is given of an assertion of MacLane that if f ∈ [Ascr ] is locally univalent and has no arc tracts, then f′ ∈ [Ascr ].

Regenerative phenomena were introduced some forty years ago to address problems in the theory of continuous-time Markov processes. The early work in the theory left a number of difficult unsolved problems, in the classification of $p$-functions, oscillation and inequalities, the multiplicative theory, and the theory of unbounded semi-$p$-functions. Recent years have shown progress on all of these fronts, and this paper surveys these results, while drawing attention to significant problems that remain open.

Let Γ be a group having finite virtual cohomological dimension. Let BΓ be its classifying space, and let C(Γ) be its Quillen category, whose objects are elementary abelian 2-subgroups V of Γ and whose morphisms are restrictions of conjugacy of Γ. The group Γ is called a Quillen group if the natural map

formula here

is an isomorphism. The symmetric groups (see [8]) and the Coxeter groups of finite order (see [7]) are examples of Quillen groups.

J. H. Gunawardena, J. Lannes and S. Zarati obtained the characterization of Quillen groups in terms of unstable modules over the mod 2 Steenrod algebra [8] (see Proposition 2 below). By making use of this result, M. Errokh and F. Grazzini obtained the classification of Coxeter groups on three generators which are Quillen groups [8]. The purpose of the present paper is to obtain a necessary and sufficient condition for aspherical Coxeter groups to be Quillen groups (Theorem 1). Our result is very simple, and the previously mentioned result by Errokh and Grazzini follows easily from ours (see Example 1).

The paper is organized as follows. In Section 2, after giving relevant definitions concerning Coxeter groups, we shall state our main result. The next two sections are devoted to preliminaries. In Section 3, we shall review necessary definitions and facts concerning unstable modules over the mod 2 Steenrod algebra. In Section 4, we shall discuss the Leray spectral sequences converging to the cohomology of Coxeter groups. The proof of Theorem 1 will be given in Section 5. In Section 6, we shall give examples illustrating Theorem 1.

Notation. The cardinality of a finite set X is denoted by [mid ]X[mid ].

A compact subset of the unit circle is an interpolation set for conformal mappings if and only if it has logarithmic capacity zero.

The local Caratheodory conjecture on the index of an isolated singularity of the principal foliations in surface theory is equivalent to a conjecture of Loewner on the index of the isolated singularities of the Hessian operator of a smooth function in the plane. Here we prove the latter conjecture for a special class of functions.

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 $1 \leq p < 2$.

The methods of $p$-adic analysis are used to prove a congruence for $(pn)!/(p^{n}n!)$ modulo a power of a prime $p$.

The notion of an asymptotic cycle is extended to flows on non-compact spaces, and it is proved that for a suitable class of flows on manifolds it is true that the set of points with asymptotic cycles defined has measure one with respect with every invariant probability measure.

An upper bound on operator norms of compound matrices is presented, and special cases that involve the l1, l2 and l∞ norms are investigated. The results are then used to obtain bounds on products of the largest or smallest eigenvalues of a matrix.

Let $X$ be a non-singular real algebraic curve in ${\rm C\!\!\!I}P^2$ of even degree. In this paper a refinement is proved of a theorem of Kharlamov about ($M - 2$)-curves that are invariants under the projective involution. In particular, if the ($M - 2$)-symmetric curve $X$ satisfies the Arnold congruence, then either $X$ or its twin is a separating curve.

A boundary point of a domain $D$ in $\Bbb{R}^n$ is said to be broadly accessible if it ‘almost lies’ on the boundary of a round ball contained in $D$. If $f$ is a quasiconformal mapping of the unit ball $B^n$ onto $D$, then it is shown that broadly accessible boundary points on $\partial D$ correspond under $f$ to a set of full measure on $\partial B^n$.This research was carried out while R. Näkki was visiting The University of Texas at Austin in 1985–86.

The norm of a group G is the subgroup of elements of G which normalise every subgroup of G. We shall denote it κ(G). An ascending series of subgroups κi(G) in G may be defined recursively by: κ0(G) = 1 and, for i[ges ]0, κi+1(G)/κi(G) = κ(G/κi(G)). For each i, the section κi+1(G)/κi(G) clearly contains the centre of the group G/κi(G). A result of Schenkman [8] gives a very close connection between this norm series and the upper central series: ζi(G)⊆κi(G) ⊆ζ2i(G).

In this paper, the authors study intersections of a special class of curves with algebraic subgroups of the multiplicative group of complex dimension at least 2. They show how results of Khovanskii on fewnomials can be used to derive finiteness results and bounds for the degrees of algebraic points for such intersections from more general results on intersections of curves with non-algebraic subgroups. They thereby generalise their earlier results, and recover in some cases, using different methods, more uniform bounds than those given in related work of Bombieri, Masser and Zannier.

It is shown that, for λ > ω, λ-orthogonality classes in locally λ-presentable categories coincide with full subcategories that are closed under limits and under λ-directed colimits. This comes as a surprise, since the statement is known to be false for λ = ω.

One important invariant of a closed Riemannian 3-manifold is the Chern–Simons invariant [1]. The concept was generalized to hyperbolic 3-manifolds with cusps in [11], and to geometric (spherical, euclidean or hyperbolic) 3-orbifolds, as particular cases of geometric cone-manifolds, in [7]. In this paper, we study the behaviour of this generalized invariant under change of orientation, and we give a method to compute it for hyperbolic 3-manifolds using virtually regular coverings [10]. We confine ourselves to virtually regular coverings because a covering of a geometric orbifold is a geometric manifold if and only if the covering is a virtually regular covering of the underlying space of the orbifold, branched over the singular locus. Therefore our work is the most general for the applications in mind; namely, computing volumes and Chern–Simons invariants of hyperbolic manifolds, using the computations for cone-manifolds for which a convenient Schläfli formula holds (see [7]). Among other results, we prove that every hyperbolic manifold obtained as a virtually regular covering of a figure-eight knot hyperbolic orbifold has rational Chern–Simons invariant. We give explicit examples with computations of volumes and Chern–Simons invariants for some hyperbolic 3-manifolds, to show the efficiency of our method. We also give examples of different hyperbolic manifolds with the same volume, whose Chern–Simons invariants (mod ½) differ by a rational number, as well as pairs of different hyperbolic manifolds with the same volume and the same Chern–Simons invariant (mod ½). (Examples of this type were also obtained in [12] and [9], but using mutation and surgery techniques, respectively, instead of coverings as we do here.)