Introduction. In his article (22), de Possel laid the foundations for an abstract theory of differentiation of set functions, the term “abstract” being meant in the sense of Fréchet-Nikodym, that is, without reference to a euclidean, metric, or topological background. In 1.1, we adopt, substantially, his notion of derivation basis. De Possel considered two Vitali properties for a derivation basis.

In this paper we are going to use a result of H. Harada and Y. Sai concerning composition of nonisomorphisms between indecomposable modules and the theory of almost split sequences introduced in the representation theory of Artin algebras by M. Auslander and I. Reiten to obtain the inductive step in the second Brauer-Thrall conjecture.

Section 1 is devoted to giving the necessary background in the theory of almost split sequences.

As an application we get the first Brauer-Thrall conjecture for Artin algebras. This conjecture says that there is no bound on the length of the finitely generated indecomposable modules over an Artin algebra of infinite type, i.e., an Artin algebra such that there are infinitely many nonisomorphic indecomposable finitely generated modules. This result was first proved by A. V. Roiter [8] and later in general for Artin rings by M. Auslander [2] using categorical methods.

Let G be a group, ZG its integral group ring and Δ = Δ(G) the augmentation ideal of ZG. Denote by Gi the ith term of the lower central series of G. Following Passi [3], we set . It is well-known that (see, for example [1]). In [3] Passi shows that if G is an abelian group then , the second symmetric power of G.

The genus series for maps is the generating series for the number of rooted maps with a given number of vertices and faces of each degree, and a given number of edges. It captures topological information about surfaces, and appears in questions arising in statistical mechanics, topology, group rings, and certain aspects of free probability theory. An expression has been given previously for the genus series for maps in locally orientable surfaces in terms of zonal polynomials. The purpose of this paper is to derive an integral representation for the genus series. We then show how this can be used in conjunction with integration techniques to determine the genus series for monopoles in locally orientable surfaces. This complements the analogous result for monopoles in orientable surfaces previously obtained by Harer and Zagier. A conjecture, subsequently proved by Okounkov, is given for the evaluation of an expectation operator acting on the Jack symmetric function. It specialises to known results for Schur functions and zonal polynomials.

The nonnegative weight function pairs u, v for which the operator maps the nonnegative nonincreasing functions in LP(v) boundedly into weak Lq(u) are characterized. This result is used, in particular, both to generalize and to provide an alternate proof of certain strong type inequalities recently obtained by Ariño and Muckenhouptfor the Hardy averaging operator restricted to nonnegative nonincreasing functions.

A ring R (with identity element) is called a right PCI-ring if and only if every proper cyclic right R-module is injective; that is, if C is a cyclic right R-module then either C ≌ R or C is injective. Faith [3, Theorems 14 and 17] (or see [2, Proposition 6.12 and Theorem 6.17]) proved that if a ring R is a right PCI-ring then R is semiprime Artinian or R is a simple right semihereditary right Ore domain. These latter rings we shall call simple rightPCI-domains. Examples of non-Artinian simple right PCI-domains were produced by Cozzens [1]. The object of this paper is to examine rings with similar properties and thus extend Faith's results.

We call a digraph “antisymmetrical” if there is an automorphism θ of its graph, of period 2, which reverses the direction of every edge and maps no edge or vertex onto itself. We construct a theory of flows invariant under θ for such a diagraph. This theory is analogous to the Max Flow Min Cut theory for ordinary flows in digraphs. It is found to include that part of the theory of undirected graphs which discusses the existence of spanning subgraphs with a specified valency at each vertex.

If (L, ≥) is a lattice or partial order we may think of its Hesse diagram as a directed graph, G, containing the single edge E(c, d) if and only if c covers d in (L, ≥). This graph we shall call the graph of (L, ≥). Strictly speaking it is the basis graph of (L, ≥) with the loops at each vertex removed; see (3, p. 170).

We shall say that an undirected graph Gu can be realized as the graph of a (modular) (distributive) lattice if and only if there is some (modular) (distributive) lattice whose graph has Gu as its associated undirected graph.

In this paper we are chiefly concerned with the chromatic sums we have called l and h, with colour-number 3. In this case h can be interpreted as enumerating the rooted Eulerian triangulations with a given number of faces, and l as enumerating such triangulations with a given number of faces and a given valency for the root-vertex. The series h has been determined already, by summation from the formula enumerating even slicings [3]. However our formula for l does not seem to have been published before, though it could presumably be derived in a similar way.

σ-ideals (collections of sets which are closed under subset and countable union) are certainly important mathematically—consider first category sets, sets of measure zero, nonstationary sets, etc.—but aside from the observation that in certain spaces the first category σ-ideal is proper, cr-ideals have not been extensively studied by topologists. In this note we study a natural topology determined by a d-ideal, exploiting the interplay between the set-theoretic properties of the σ-ideal and the topological properties of the associated space.

1. Introduction. Let F, G (and all capital letters to be used below) denote n by n matrices the elements of which are real-valued continuous functions on an interval a ≤t ≤ b. Correspondingly, by a solution x = x(t) of a differential system

1,

with det G(t) ≠ 0, or of a differential system

2,

will be meant a (vector) solution all n components of which are real-valued.

Let $F$ be a divisor on the blow-up $X$ of ${{\mathbf{P}}^{2}}$ at $r$ general points ${{p}_{1}},...,{{p}_{r}}$ and let $L$ be the total transform of a line on ${{\mathbf{P}}^{2}}$. An approach is presented for reducing the computation of the dimension of the cokernel of the natural map ${{\mu }_{F}}:\Gamma ({{\mathcal{O}}_{_{X}}}(F))\otimes \Gamma ({{\mathcal{O}}_{_{X}}}(L))\to \Gamma ({{\mathcal{O}}_{_{X}}}(F)\otimes {{\mathcal{O}}_{_{X}}}(L))$ to the case that $F$ is ample. As an application, a formula for the dimension of the cokernel of ${{\mu }_{_{F}}}$ is obtained when $r\,=\,7$, completely solving the problem of determining the modules in minimal free resolutions of fat point subschemes ${{m}_{1}}\,{{p}_{1}}\,+\,\cdot \cdot \cdot \,+\,{{m}_{7}}\,{{p}_{7}}\,\subset \,{{\mathbf{P}}^{2}}$. All results hold for an arbitrary algebraically closed ground field $k$.

1. Introduction. An extension E* of a topological space E (that is, a space containing E as a dense subspace) determines a family of filters (u) on E, given by the traces U ∩ E of the neighbourhoods U ⊆ E* of each u ∈ E* − E. Many topological properties of an extension E* of a given space E can be related to properties of these trace filters (as we shall call them) belonging to E*.

In this paper we shall derive a concise formula for the number of Euler graphs on n labelled nodes and k edges. An Euler graph is a connected graph in which every node has even valency, where by the valency of a node is meant the number of edges which are incident with that node. Throughout most of the paper we shall be dealing with graphs whose nodes have even valencies but which may or may not be connected. For convenience we shall refer to these graphs as Euler graphs, although the usage is not, strictly speaking, correct. We shall impose the condition of connectedness in § 4.

Two elements A, B of a lattice are said to form a modular pair when (X ∨ A) Λ B = X ∨ (A Λ B) holds for all X ≦ B, and are said to form a dual-modular pair when (X Λ A) ∨ B = X Λ (A ∨ B) holds for all X ≧ B.

We are concerned here with a particular lattice, the lattice of closed subspaces of a normed linear space, and with a question posed by Mackey in 1945 (6, p. 206, problem 2), namely:

“Are there any incomplete normed linear spaces in whose lattices of closed subspaces modularity and d-modularity are equivalent?”.

The principal result of this paper is the following.

We recall that the ideal I in an Archimedean Riesz space L is called a d-ideal whenever it follows from ƒ ∊ I that {ƒ}dd ⊂ I. Several authors (see [4], [5], [6], [12], [13], [15] and [18]) have considered the class of all d-ideals in L, but the set ℐd of all maximal d-ideals in L has not been studied in detail in the literature. In [12] and [13] the present authors paid some attention to certain aspects of the theory of maximal d-ideals, however neglecting the fact thatℐd, equipped with its hull-kernel topology, is a structure space of the underlying Riesz space L.

The main purpose of the present paper is to investigate the topological properties of ℐd and to compare ℐd to other structure spaces of L, such as the space of minimal prime ideals and the space of all e-maximal ideals in L (where e > 0 is a weak order unit).

In this paper we study the existence problem for topological actions of finite groups on euclidean spaces Rn+k which are free outside a fixed point set Rk (embedded as a vector subspace). We refer to such an action as a semi-free action on (Rn+k, Rn) and note that all our actions will be assumed orientation-preserving.

Suppose the finite group π acts semi-freely on (Rn+k, Rn), then it acts freely on (Rn+k – Rn) = Sn–l × Rk+1. Since this space is homotopy equivalent to Sn–l, π will have periodic integral cohomology and n will be a multiple of the period. In fact the orbit space

is a finitely-dominated Poincaré complex of formal dimension n – 1 with π1W = π and as considered by Swan [41].

Let $X$ be a locally finite, connected graph without vertices of degree 1. Non-backtracking random walk moves at each step with equal probability to one of the “forward” neighbours of the actual state, i.e., it does not go back along the preceding edge to the preceding state. This is not a Markov chain, but can be turned into a Markov chain whose state space is the set of oriented edges of $X$. Thus we obtain for infinite $X$ that the $n$-step non-backtracking transition probabilities tend to zero, and we can also compute their limit when $X$ is finite. This provides a short proof of an old result concerning cogrowth of groups, and makes the extension of that result to arbitrary regular graphs rigorous. Even when $X$ is non-regular, but small cycles are dense in$X$, we show that the graph $X$ is non-amenable if and only if the non-backtracking $n$-step transition probabilities decay exponentially fast. This is a partial generalization of the cogrowth criterion for regular graphs which comprises the original cogrowth criterion for finitely generated groups of Grigorchuk and Cohen.

The theory of associated prime ideals is one of the most basic notions in the study of modules over commutative Noetherian rings. For modules over non-Noetherian rings however, the classical associated primes are not so useful and in fact do not exist for some modules M. In [4] [22] a prime ideal P of a ring R is said to be attached to an R-module M if for each finite subset I of P there exists m ∊ M such that I ⊆ annR(m) ⊆ P. In [4] the attached primes were compared to the associated primes and the results of [4], [22], [23], [24] show that the attached primes are a useful alternative in non-Noetherian rings to associated primes. Several other methods of associating a set of prime ideals to a module M over a non-Noetherian ring have proven very useful in the past. The most common of these is the set Assf(M) of weak Bourbaki primes of M [2, pp. 289-290].

The Sierpiński carpets first considered by C.McMullen and later studied by Y. Peres are modified by insisting that the allowed digits in the expansions occur with prescribed frequencies. This paper (i) calculates the Hausdorff, box (or Minkowski), and packing dimensions of the modified Sierpiński carpets and (ii) shows that for these sets the Hausdorff and packing measures in their dimension are never zero and gives necessary and sufficient conditions for these measures to be infinite.