Let Γ be an undirected graph, V(Γ) the vertex set of Γ and G a subgroup of aut(Γ). For each vertex x ↦ V(Γ), let Γx denote the set of vertices adjacent to x in Γ and the permutation group induced on Γx. by the stabilizer Gx. For each i ≥ 1, will denote the pointwise stabilizer in Gx of the set of vertices at distance at most i from x in Γ. Let

for each i ≥ 1 and any set of vertices x, y, …, z of Γ. An s-path (or s-arc) is an (s + 1)-tuple (x0, x1, … xs) of vertices such that xi ↦ Γxi–1 for 1 ≤ i ≤ s and xi ╪ xi–2 for 2 ≤ i ≤ s.

Let G be a finite group of automorphisms of an associative ring R. Then the inner automorphisms (x↦ u−1xu = xu, for some unit u of R) contained in G form a normal subgroup G0 of G. In general, the Galois theory associated with the outer automorphism group G/G0 is quit well behaved (e.g. [7], 2·3–2·7, 2·10), while little group-theoretic restriction on the structure of G/G0 may be expected (even when R is a commutative field). The structure of the inner automorphism groups G0 does not seem to have received much attention so far. Here we classify the finite groups of inner automorphisms of division rings, i.e. the finite subgroups of PGL (1, D), where D is a division ring. Such groups also arise in the study of finite collineation groups of projective spaces (via the fundamental theorem of projective geometry, cf. [1], 2·26), and provide examples of finite groups having faithful irreducible projective representations over fields.

Let K be an arbitrary field, G a polycyclic-by-finite group and A a prime ideal of the group ring KG. It is well known that the quotient ring (KG)/A is a Goldie ring; we denote by R its ring of fractions. Let U be a subgroup of units of the matrix ring Rn×n let K[U] be the linear envelope of U and let rad (K[U]) be the nilpotent radical of K [U].

In the past fifteen years the theory of complexity and varieties of modules has become a standard tool in the modular representation theory of finite groups. Moreover the techniques have been used in the study of integral representations [8] and have been extended to the representation theories of objects such as groups of finite virtual cohomological dimension [1], infinitesimal subgroups of algebraic groups and restricted Lie algebras [14, 16]. In all cases some sort of finiteness condition on the module category has been required to make the theory work. Usually this comes in the form of stipulating that all modules under consideration be finitely generated. While the restrictions have been efficient for most applications to date, there are very good reasons for wanting to develop a theory that will accommodate infinitely generated modules. One reason might be the possibility of extending the techniques of representations to other classes of infinite groups. Another reason is that some recent work has revealed a few of the defects of the finiteness requirement. One such problem can be summarized as follows.

This paper was initiated by a question of Eisenbud who asked whether the entries of the matrices in a minimal free resolution of a monomial ideal (which, after a suitable choice of bases, are monomials) divide the least common multiple of the generators of the ideal. We will see that this is indeed the case, and prove it by lifting the multigraded resolution of an ideal, or more generally of a multigraded module, keeping track of how the shifts ‘deform’' in such a lifting; see Theorem 2·1 and Corollary 2·2.

We study here the connections between the well known Puppe-Verdier notion of triangulated category and an abstract setting for homotopical algebra, based on homotopy kernels and cokernels, which was expounded by the author in [11, 13[.

We show that a right-homotopical category A (having well-behaved homotopy cokernels, i.e. mapping cones) has a sort of weak triangulated structure with regard to the suspension endofunctor σ, called σ-homotopical category. If A is homotopical and h-stable (in a sense related to the suspension-loop adjunction), this structure is also h-stable, i.e. satisfies ‘up to homotopy’ the axioms of Verdier[29[ for a triangulated category, excepting the octahedral one which depends on some further elementary conditions on the cone endofunctor of A. Every σ-homotopical category can be stabilized, by two universal procedures, respectively initial and terminal.

A Banach space is c0-saturated if all of its closed infinite-dimensional subspaces contain an isomorph of c0. In this article, we study the stability of this property under the formation of direct sums and tensor products. Some of the results are: (1) a slightly more general version of the fact that c0-sums of c0-saturated spaces are c0-saturated; (2) C(K, E) is c0-saturated if both C(K) and E are; (3) the tensor product is c0-saturated, where JH is the James Hagler space

An open subgroup U of a topological group G is always closed, since U is the complement of the open set . An arbitrary closed subgroup C of G is almost never open, unless G belongs to a small family of exceptional groups. In fact, if G is a locally compact abelian group in which every non-trivial subgroup is open, then G is the additive group δp of p-adic integers or the additive group Ωp of p-adic rationale (cf. Robertson and Schreiber[5[, proposition 7). The fact that δp has interesting properties as a topological group has many roots. One is that its character group is the Prüfer group ℤp∞, which makes it unique inside the category of compact abelian groups. But even within the bigger class of not necessarily abelian compact groups the p-adic group δp is distinguished: it is the only one all of whose non-trivial subgroups are isomorphic (cf. Morris and Oates-Williams[2[), and it is also the only one all of whose non-trivial closed subgroups have finite index (cf. Morris, Oates-Williams and Thompson [3[).

An analytic operator-valued function A is an analytic map A: D → L(E, E), where D = D(A) is an open subset of the complex plane C and E = E(A) is a complex Banach space. For such a function A the singular set σ(A) of A is defined as the set of points z ∈ D such that A(z) is not invertible. It is a relatively closed subset of D.

Let be an open Riemann surface with finite genus and finite number of boundary components, and let be a closed Riemann surface. An open continuous function from to is termed a (p, q)-map, 0 < q < p, if it has a finite number of branch points and assumes every point in either p or q times, counting multiplicity, with possibly a finite number of exceptions. These comprise the most general class of all non-trivial functions having two valences between and .

The object of this paper is to study the geometry of (p, q)-maps and establish a generalized embedding theorem which asserts that the image surfaces of (p, q)-maps embed in p-fold closed coverings possibly having branch points off the image surfaces.

We prove that each countable differential field of characteristic 0 can be embedded into the differential ring of infinitely differentiable functions from the algebraic numbers to the complex numbers. This statement remains valid when we replace the set of algebraic numbers by an arbitrary non-empty countable set of complex numbers without any isolated points.

The main result of this paper is an oscillation theorem on linear self-adjoint differential systems and a corresponding eigenvalue problem. It establishes a formula between the number of focal points of a so-called conjoined basis of the differential system on a given compact interval and the number of eigenvalues which are less than the given eigenvalue parameter. It extends an earlier result of the author and generalizes an oscillation theorem of M. Morse. Among others the proof of the theorem requires a formula on the index of the difference of symmetric solutions of a corresponding Riccati matrix differential equation. This index formula is the other new result presented.

We consider queueing systems where the workload process is assumed to have an associated large deviation principle with arbitrary scaling: there exist increasing scaling functions (at, vt, t∈R+) and a rate function I such that if (Wt, t∈R+) denotes the workload process, then

on the continuity set of I. In the case that at = vt = t it has been argued heuristically, and recently proved in a fairly general context (for discrete time models) by Glynn and Whitt[8], that the queue-length distribution (that is, the distribution of supremum of the workload process Q = supt≥0Wt) decays exponentially:

and the decay rate δ is directly related to the rate function I. We establish conditions for a more general result to hold, where the scaling functions are not necessarily linear in t: we find that the queue-length distribution has an exponential tail only if limt→∞at/vt is finite and strictly positive; otherwise, provided our conditions are satisfied, the tail probabilities decay like

Let X, X1, X2, …, be independent random variables with a common distribution function F(x) = P {X ≤ x}, x∈ℝ, and for each n∈ℕ, let X1, n ≤ … ≤ Xn, n denote the order statistics pertaining to the sample X1, …, Xn. We assume that 1–F(x) = x−1/cl(x), 0 < x < ∞, where l is some function slowly varying at infinity and c > 0 is any fixed number. The class of all such distribution functions will be denoted by .