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.

We prove the existence of a power series having radius of convergence 0, whose partial sums have universal approximation properties on any compact set with connected complement that is contained in a finite union of circles centred at 0 and having rational radii, but do not have such properties on any compact set with nonempty interior. This relates to a theorem of A. I. Seleznev.

In this paper, we give an elementary proof of a curious identity of elliptic functions. It is very similar to a beautiful proof given by Coates of a different identity. The result was strongly motivated by Wildeshaus' generalisation of Zagier's conjecture.

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.)

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.

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 ].

It is proved that a ring (≠0) is projective-free precisely when it has invariant basis number and every idempotent matrix not equal to I is non-full.

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.

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.

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.

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

A classical theorem of Valiron states that a function which is holomorphic in the unit disk, unbounded, and bounded on a spiral that accumulates at all points of the unit circle, has asymptotic value $\infty$. This property, and various other properties of such functions, are shown to hold for more general classes of functions which are bounded on a subset of the disk that has a suitably large set of nontangential limit points on the unit circle.

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 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).

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.

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.

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 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.

Let $C$ be a germ at $O \in {\mathbb R}^2$ of a real analytic plane curve, and $C^{\mathbb C}$ its complexification; let $C_t \subset B_{\varepsilon}$ be a fiber of a real smooth deformation of $C$ in the ball $B_{\varepsilon} = B(O, \varepsilon)$. The following inequality is proved between the integrals of real curvature $k$ of $C_t$ and those of Gaussian curvature $K$ of $C_{t}^{\mathbb C}$: $$ 2 \lim_{\varepsilon, t \rightarrow 0} \int_{C_t^\varepsilon} \vert k \vert \leq \lim_{\varepsilon, t \rightarrow 0} \int_{C_t^{\varepsilon {\mathbb C}}} \vert K \vert.$$ The sharpness of this inequality is proved in the case where $C$ is a real irreducible germ. Similar results are proved for an affine algebraic curve $C \subset {\mathbb R}^2$ of degree $d$.