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The closest infinite dimensional relatives of compact Lie algebras are Hilbert-Lie algebras, i.e. real Hilbert spaces with a Lie algebra structure for which the scalar product is invariant. Locally affine Lie algebras (LALAs) correspond to double extensions of (twisted) loop algebras over simple Hilbert-Lie algebras $\mathfrak{k}$, also called affinisations of $\mathfrak{k}$. They possess a root space decomposition whose corresponding root system is a locally affine root system of one of the $7$ families $A_J^{(1)}$, $B_J^{(1)}$, $C_J^{(1)}$, $D_J^{(1)}$, $B_J^{(2)}$, $C_J^{(2)}$ and $BC_J^{(2)}$ for some infinite set $J$. To each of these types corresponds a "minimal" affinisation of some simple Hilbert-Lie algebra $\mathfrak{k}$, which we call standard. In this paper, we give for each affinisation $\mathfrak{g}$ of a simple Hilbert-Lie algebra $\mathfrak{k}$ an explicit isomorphism from $\mathfrak{g}$ to one of the standard affinisations of $\mathfrak{k}$. The existence of such an isomorphism could also be derived from the classification of locally affine root systems, but for representation theoretic purposes it is crucial to obtain it explicitely as a deformation between two twists which is compatible with the root decompositions. We illustrate this by applying our isomorphism theorem to the study of positive energy highest weight representations of $\mathfrak{g}$. In subsequent work, the present paper will be used to obtain a complete classification of the positive energy highest weight representations of affinisations of $\mathfrak{k}$., Comment: 22 pages; Minor corrections

In this paper we first generalize to the case of partial flags a result proved both by Spaltenstein and by Steinberg that relates the relative position of two complete flags and the irreducible components of the flag variety in which they lie, using the Robinson-Schensted-Knuth correspondence. Then we use this result to generalize the mirabolic Robinson-Schensted-Knuth correspondence defined by Travkin, to the case of two partial flags and a line., Comment: 27 pages, slightly rewritten to combine two papers into one and clarify some sections

Let f be a normalized Hecke eigenform with rational integer Fourier coefficients. It is an interesting question to know how often an integer n has a factor common with the n-th Fourier coefficient of f. It has been shown in previous papers that this happens very often. In this paper, we give an asymptotic formula for the number of integers n for which (n, a(n)) = 1, where a(n) is the n-th Fourier coefficient of a normalized Hecke eigenform f of weight 2 with rational integer Fourier coefficients and having complex multiplication.

This paper is concerned with the residual spectrum of the hermitian quaternionic classical groups G′n and H′n defined as algebraic groups for a quaternion algebra over an algebraic number field. Groups G′n and H′n are not quasi-split. They are inner forms of the split groups SO4n and Sp4n. Hence, the parts of the residual spectrum of G′n and H′n obtained in this paper are compared to the corresponding parts for the split groups SO4n and Sp4n.

We study the cardinal invariants of analytic P-ideals, concentrating on the ideal Zof asymptotic density zero. Among other results we prove min{b, cov (N)} ≤ cov∗(Z) ≤ max{b, non(N)}. Introduction Analytic P-ideals and their quotients have been extensively studied in recent years. The first step to better understanding the structure of the quotient forcings P(ω)/I is to understand the structure of the ideal itself. Significant progress in understanding the way in which the structure of an ideal affects the structure of its quotient has been done by I. Farah [Fa1, Fa2, Fa3, Fa4, Fa5]. Typically (but not always) the quotients P(ω)/I, where I is an analytic P-ideal, are proper and weakly distributive. For some special ideals these quotients have been identified: P(ω)/Z is as forcing notion equivalent to P(ω)/fin ∗ B(2) [Fa5], and P(ω)/ tr(N) [HZ1] is as forcing notion equivalent to the iteration of B(ω) followed by an א0-distributive forcing (see the definitions below). A secondary motivation comes from the problem of which ideals can be destroyed by a weakly distributive forcing. Even for the class of analytic P-ideals only partial results are known (see Section 3). In this note we contribute to this line of research by investigating cardinal invariants of analytic P-ideals, comparing them to other, standard, cardinal invariants of the continuum. In the first section we introduce cardinal invariants of ideals on ω, along the lines of the cardinal invariants contained in the Cichon’s diagram. We also recall the definitions of standard orderings on ideals on ω (Rudin–Keisler, Tukey, Katětov) and their impact on the cardinal invariants of the ideals. Basic theory of analytic P-ideals on ω and examples are also reviewed here. Known results on additivity and cofinality of analytic P-ideals are summarized in the second section. The main part of the paper is contained in the third section. There we study the order of Katětov restricted to analytic P-ideals, giving a detailed description of how the summable and density ideals are placed in the Katětov order. For the rest of the section, we focus on the ideal of asymptotic density zero and compare its covering number to standard cardinal invariants of the continuum. We prove that min{b, cov(N)} ≤ cov(Z) ≤ max{b, non(N)} and mention some consistency results. We introduce the notion of a totally bounded analytic P-ideal and show that Received by the editors October 5, 2004; revised September 19, 2005. The authors gratefully acknowledge support from PAPIIT grant IN106705. The second author’s research was also partially supported by GA CR grant 201-03-0933 and CONACYT grant 40057-F AMS subject classification: 03E17, 03E40. c ©Canadian Mathematical Society 2007. 575 576 F. Hernandez-Hernandez and M. Hrusak all analytic P-ideals which are not totally bounded can be destroyed by a weakly distributive forcing. In the last section we study the separating number of analytic P-ideals, an invariant closely related to the Laver and Mathias–Prikry type forcings associated with the ideal. Two major problems remain open here: (1) Is add(I) = add(N) for every tall analytic P-ideal I? (2) Can every analytic P-ideal be destroyed by a weakly distributive forcing? What about Z? We assume knowledge of the method of forcing as well as the basic theory of cardinal invariants of the continuum as covered in [BJ]. Our notation is standard and follows [Ku, Je, BJ]. In particular, c0, l1 and l∞ denote the standard Banach spaces of sequences of reals. For A,B infinite subsets of ω, we say that A is almost contained in B (A ⊆ B) if A \ B is finite. The symbol A = Bmeans that A ⊆ B and B ⊆ A. For functions f , g ∈ ω we write f ≤ g to mean that there is some m ∈ ω such that f (n) ≤ g(n) for all n ≥ m. The bounding number b is the least cardinal of an ≤-unbounded family of functions in ω . The dominating number d is the least cardinal of a ≤-cofinal family of functions in ω . Recall that a family of subsets of ω has the strong finite intersection property if any finite subfamily has infinite intersection. The pseudointersection number p is the minimal size of a family of subsets of ω with the strong finite intersection property but without an infinite pseudointersection (i.e., without a common lower bound in the ⊆ order). A family S ⊆ P(ω) is a splitting family if for every infinite A ⊆ ω there is an S ∈ S such that S ∩ A and A \ S are infinite. The splitting number s is the minimal size of a splitting family in P(ω). The set 2 is equipped with the product topology, that is, the topology with basic open sets of the form [s] = {x ∈ 2 : s ⊆ x}, where s ∈ 2 . The topology of P(ω) is that obtained via the identification of each subset of ω with its characteristic function. An ideal on X is a family of subsets of X closed under taking finite unions and subsets of its members. We assume throughout the paper that all ideals contain all singletons {x} for x ∈ X. An ideal I on ω is called P-ideal if for any sequence Xn ∈ I, n ∈ ω, there exists X ∈ I such that Xn ⊆ ∗ X for all n ∈ ω. An ideal I on ω is analytic if it is analytic as a subspace of P(ω) with the above topology. Recall that an ideal on ω is tall (or dense) if every infinite set of ω contains an infinite set from the ideal. If I is an ideal on ω and Y ⊆ ω is an infinite set, then we denote by I↾Y the ideal {I ∩ Y : I ∈ I}; note that the underlying set of the ideal I↾Y is not the underlying set of I but Y . For an ideal I on ω, I denotes the dual filter, M denotes the ideal of meager subsets of R, and N the ideal of Lebesgue null subsets of R. Given an ideal I on a set X, the following are standard cardinal invariants associated with I: add(I) = min{|A| : A ⊆ I ∧ ⋃ A / ∈ I}, cov(I) = min{|A| : A ⊆ I ∧ ⋃ A = X}, cof(I) = min{|A| : A ⊆ I ∧ (∀I ∈ I)(∃A ∈ A)(I ⊆ A)}, non(I) = min{|Y | : Y ⊆ X ∧Y / ∈ I}. The provable relationships between the cardinal invariants of M and N are Cardinal Invariants of Analytic P-Ideals 577 summed up in the following diagram: cov(N) // non(M) // cof(M) // cof(N)

The main problem that is solved in this paper has the following simple formulation (which is not used in its solution). The group K = Op(C) × Oq(C) acts on the space Mp, q of p × q complex matrices by (a; b) · x = axb–1, and so does its identity component K0 = SOp(C)×SOq(C). A K-orbit (or K0-orbit) in Mp,q is said to be nilpotent if its closure contains the zero matrix. The closure, , of a nilpotent K-orbit (resp. K0-orbit) in Mp,q is a union of and some nilpotent K-orbits (resp. K0-orbits) of smaller dimensions. The description of the closure of nilpotent K-orbits has been known for some time, but not so for the nilpotent K0-orbits. A conjecture describing the closure of nilpotent K0-orbits was proposed in [11] and verièd when min(p, q) ≤ 7. In this paper we prove the conjecture. The proof is based on a study of two prehomogeneous vector spaces attached to and determination of the basic relative invariants of these spaces.The above problem is equivalent to the problem of describing the closure of nilpotent orbits in the real Lie algebra so(p, q) under the adjoint action of the identity component of the real orthogonal group O(p, q).

In an earlier paper [10], we studied a generalized Rao bound for ordered orthogonal arrays and (T, M, S)-nets. In this paper, we extend this to a coding-theoretic approach to ordered orthogonal arrays. Using a certain association scheme, we prove a MacWilliams-type theorem for linear ordered orthogonal arrays and linear ordered codes as well as a linear programming bound for the general case. We include some tables which compare this bound against two previously known bounds for ordered orthogonal arrays. Finally we show that, for even strength, the LP bound is always at least as strong as the generalized Rao bound.

In this paper we consider the following problem: Given a unital C*- algebra A and a collection of elements S in the identity component of the invertible group of A, denoted inv0(A), characterize the group of finite products of elements of S. The particular C*-algebras studied in this paper are either unital purely infinite simple or of the form (A ⊗ K)+, where A is any C*-algebra and K is the compact operators on an infinite dimensional separable Hilbert space. The types of elements used in the factorizations are unipotents (1+ nilpotent), positive invertibles and symmetries (s2 = 1). First we determine the groups of finite products for each collection of elements in (A ⊗ K)+. Then we give upper bounds on the number of factors needed in these cases. The main result, which uses results for (A ⊗ K)+, is that for A unital purely infinite and simple, inv0(A) is generated by each of these collections of elements.

We formulate a conjecture analogous to Gross' refinement of the Stark conjectures on special values of abelian L-series at s = 0. Some evidence for the conjecture can be obtained, thanks to the fundamental ideas of F. Thaine. 1. Introduction. This paper formulates a refined analogue of the usual class number formula for a real quadratic extension of Q, using circular units. The statement of this conjecture is inspired by an analogous conjecture of Gross (Gr). Strong evidence for this conjecture can be given thanks to F. Thaine's powerful method (Th) for generating relations in ideal class groups using circular units. The first two sections briefly recall Dirichlet's analytic class number formula and Gross's refinement of it; they are there mainly to fix notations and provide motivation. Section 4 states the new conjecture. The remaining sections are devoted to proving various results that support it. ACKNOWLEDGEMENTS. I wish to thank Massimo Bertolini and Benedict Gross for many stimulating conversations on the topics of this paper. NOTATIONS. If K is a number field and w is a place of K lying above a prime v of Q, we denote by Kw the localization of K at w, and let Nw be the order of its residue field. The w-adic norm || || w is normalized so that it is equal to Nw" 1 on uniformizing elements.

Introduction. The concept of a mitotic group was introduced in [3] by Baumslag, Dyer and Heller who showed that mitotic groups were acyclic. In [8] one of the authors introduced the concept of a pseudo-mitotic group, a concept weaker than that of a mitotic group, and showed that pseudo-mitotic groups were acyclic and that the group Gn of homeomorphisms of R n with compact support is pseudo-mitotic. In our present paper we develop techniques to prove pseudomitoticity of certain other homeomorphism groups. In [5] Kan and Thurston observed that the group of set theoretic bijections of Q with bounded support is acyclic. A natural question is to decide whether the group of homeomorphisms of Q (resp. the irrationals / ) with bounded support is acyclic or not. In the present paper we develop techniques to answer this question in the affirmative. Also the techniques developed here enable us to show that the group of homeomorphisms of the Cantor set which are identity in a neighbourhood of 0 and 1 is pseudo-mitotic and hence acyclic. It is worth noticing the contrast between our results and the results of R. D. Anderson in [1] and [2]. Anderson, using his techniques shows that the group of all homeomorphisms of Q, / or the Cantor set is a simple group. He also shows that the group of orientation preserving homeomorphisms of S or S is simple.

The aim of this paper is to study circular units in the compositum K of t cyclic extensions of ${\mathbb {Q}}$ ( $t\ge 2$ ) of the same odd prime degree $\ell $ . If these fields are pairwise arithmetically orthogonal and the number s of primes ramifying in $K/{\mathbb {Q}}$ is larger than $t,$ then a nontrivial root $\varepsilon $ of the top generator $\eta $ of the group of circular units of K is constructed. This explicit unit $\varepsilon $ is used to define an enlarged group of circular units of K, to show that $\ell ^{(s-t)\ell ^{t-1}}$ divides the class number of K, and to prove an annihilation statement for the ideal class group of K.

In the classical setting, a convex polytope is said to be semiregular if its facets are regular and its symmetry group is transitive on vertices. This paper continues our study of alternating semiregular abstract polytopes, which have abstract regular facets, still with combinatorial automorphism group transitive on vertices and with two kinds of regular facets occurring in an alternating fashion.Our main concern here is the universal polytope ${\mathcal{U}}_{{\mathcal{P}},{\mathcal{Q}}}$, an alternating semiregular $(n+1)$-polytope defined for any pair of regular $n$-polytopes ${\mathcal{P}},{\mathcal{Q}}$ with isomorphic facets. After a careful look at the local structure of these objects, we develop the combinatorial machinery needed to explain how ${\mathcal{U}}_{{\mathcal{P}},{\mathcal{Q}}}$ can be constructed by “freely assembling” unlimited copies of ${\mathcal{P}}$, ${\mathcal{Q}}$ along their facets in alternating fashion. We then examine the connection group of ${\mathcal{U}}_{{\mathcal{P}},{\mathcal{Q}}}$, and from that prove that ${\mathcal{U}}_{{\mathcal{P}},{\mathcal{Q}}}$ covers any $(n+1)$-polytope ${\mathcal{B}}$ whose facets alternate in any way between various quotients of ${\mathcal{P}}$ or ${\mathcal{Q}}$.

Archdeacon and Grable (1995) proved that the genus of the random graph $G\in\mathcal{G}_{n,p}$ is almost surely close to $pn^2/12$ if $p=p(n)\geq3(\ln n)^2n^{-1/2}$. In this paper we prove an analogous result for random bipartite graphs in $\mathcal{G}_{n_1,n_2,p}$. If $n_1\ge n_2 \gg 1$, phase transitions occur for every positive integer $i$ when $p=\Theta((n_1n_2)^{-\frac{i}{2i+1}})$. A different behaviour is exhibited when one of the bipartite parts has constant size, $n_1\gg1$ and $n_2$ is a constant. In that case, phase transitions occur when $p=\Theta(n_1^{-1/2})$ and when $p=\Theta(n_1^{-1/3})$., Comment: 19 pages

Factorization structures on a category form a useful categorical tool. As is known, any , satisfying suitable completeness—and smallness—conditions, has a sufficient supply of factorization structures; in fact, there is a bijection between the class of all epireflective (full and isomorphism- closed) subcategories of and the class of all so called perfect factorizationstructures of In this paper, for an arbitrary category supplied with a fixed factorization structure (E, M), a similar bijection between the class of all E-reflective (full and isomorphism-closed) subcategories of and the class of all (E, M)-dispersed factorization structures on , introduced in this paper, will be established.

Throughout this paper L will be an orthomodular lattice and the set of all maximal Boolean subalgebras, also called blocks [4], of L. For every x ∈ L, C(x) will be the set of all elements of L which commute with x. Let n ≧ 1 be a natural number. In this paper we consider the following conditions for L:An: L has at most n blocks,Bn: there exists a covering of L by at most n blocks,Cn: the set ﹛C(x)| x ∈ L﹜ has at most n elements,Dn: out of any n + 1 elements of L at least two commute.We also consider quantified versions of these statements, namely the statements A, B, C, D defined by: A ⇔ ∃ nAn, B ⇔ ∃ nBn, C ⇔ ∃ nCn and D ⇔ ∃ nDn.

Erdös [1] has shown that, if the square-free numbers in ascending order be denoted by s1, s2, … , sn, … , then for 0 ≦ γ ≦ 2as x → ∞. In this paper we shall extend this result by proving that the asymptotic formula in fact holds for the wider range 0 ≦ γ ≦ 3.Similar results have been obtained previously by the author in respect of both the sequence of numbers expressible as the sum of two squares and also sequences of numbers relatively prime to given large integers, although the method used here differs from that of the earlier papers [2; 3].

In this paper we study the question “which generalized metric spaces can be written as the union of k (closed) metrizable subspaces, where k is a cardinal number with k ≤ c?“ Questions of this type first arose in [16] where J. Nagata asked for examples of certain generalized metric spaces which could not be written as the union of countably many closed metrizable subspaces. Using Baire Category arguments, Fitzpatrick provided the required examples in [12]. We begin this paper by sharpening Fitzpatrick's examples, showing in Section 2 that there is a Moore space which is not the union of countably many metrizable subspaces of any kind. Then in Section 3 we present a positive result, proving that any cr-space, and a fortiori any Moore space, can be written as the union of c = 2ω0 closed metrizable subspaces.

In the recent paper [8] Phillips has shown that for each prime p there are 2 ℵ0 non-isomorphic 2-generatecl p-groups. This same result was obtained independently by S. Jeanes and J. S, Wilson (unpublished) who show that the groups constructed in [1] have 2 ℵ0 non-isomorphic images. The groups in both of these proofs all have infinite exponent. In this paper we show that, for large enough primes p, there are 2 ℵ0 non-isomorphic 2-generated groups of exponent p2.

1. Introduction. In this paper we give several general constructions for lattice packings of spheres in real n-dimensional space Rn and complex space Cn. These lead to denser lattice packings than any previously known in R36, R64, R80, …, R128, …. A sequence of lattices is constructed in Rn for n = 24m ≦ 98328 (where m is an integer) for which the density Δ satisfies log2 Δ ≈ – (1.25 …)n, and another sequence in Rn for n = 2m (m any integer) withThe latter appear to be the densest lattices known in very high dimensional space. (See, however, the Remark at the end of this paper.) In dimensions around 216 the best lattices found are about 2131000 times as dense as any previously known.Minkowski proved in 1905 (see [20] and Eq. (23) below) that lattices exist with log2 Δ > –n as n → ∞, but no infinite family of lattices with this density has yet been constructed.

The topic of homotopy pull-backs and push-outs has recently been discussed by a number of authors; Boardman and Vogt [5], Bousfield and Kan [6], Fantham [7], Mather [11], and Vogt [16]. Mather develops the theory with an eye to applications and of particular interest is his cube theorem which appears in this paper as Theorem (1.10); the significance of this theorem to applications is shown in [11]. As often occurs in homotopy theory the dual is not true. The purpose of this paper is to examine approximations to the dual in order to obtain new information concerning classical problems of duality.

In an earlier paper [6] we considered the application of Lagrangian methods to the enumeration of plane rooted trees with given colour partition. We obtained an expression which generalised Tutte’s result [9], and a correspondence, which, when specialised, gives the de Bruijn-van Aardenne Ehrenfest-Smith-Tutte Theorem [1]. A corollary of these results is a one-to-one correspondence [4], between trees and generalised derangements, for which no combinatorial description has yet been found.In this paper we extend these methods to the enumeration of rooted labelled trees to demonstrate how another pair of well-known and apparently unrelated theorems may be obtained as the result of a single enumerative approach. In particular, we show that a generalisation of Good's result [3], also considered by Knuth [7], and the matrix tree theorem [8] have a common origin in a single system of functional equations, and that they correspond to different coefficients in the power series solution.

A dissectible polyhedron is a natural extension of a concept whose history dates back to at least 1758 and Euler [7]—the concept of a dissection of a polygon. An interesting historical survey of dissections of a polygon is given by Brown [4]. Some approaches to the classical problem have been given by Moon and Moser [9] and by Guy [8] ; the latter provides an approach which is the basis of the work in this paper. A summary of enumeration results on dissections of polygons and polyhedra by automorphism groups has been given by the authors [2].Recent extensions of the problem have been investigated in a series of papers by Brown and Tutte [3; 5; 14; 15] and b y Takeo [10; 11; 12; 13].

In this paper we consider a method for enumerating trees with respect to colour and degree information. The method makes use of elementary decompositions of trees, and the functional equations which are induced. A number of new results are obtained by this means. More specifically, we consider (Section 3) the enumeration of rooted plane X-coloured trees with given colour and edge partitions. Remarkably, the number of such trees which are planted is a multiple of the number of spanning arbores­ cences on a graph with the edge partition as its adjacency matrix. This result is used (Section 4) to obtain the number of rooted plane i£-chromatic trees with fixed chromatic partition, a number given by Tutte [5] for planted trees in the case K = 2. Finally, a new combinatorial correspon­ dence between two sets of trees is given (Section 5) which yields the de Bruijn-van Aardenne Ehrenfest-Smith-Tutte (BEST) theorem as a special case. We use a familiar decomposition of rooted trees. Associated with this is a system of functional equations which may be solved by a specialisa­ tion, given in Section 2, of the Lagrange theorem in many variables. The specialisation accounts for the persistence, in combinatorial enumeration, of determinants of matrices with row and column constraints (see, for example, [6]). Throughout this paper we use the notation: the number of non-root vertices of colour i is nt = ]C^i/

This paper extends several graph-theoretic results to matroids. The main result of Tutte's paper [10] which introduced the theory of n-connection for matroids was a generalization of an earlier result of his [9] for 3-connected graphs. The latter has since been strengthened by Halin [3] and in Section 3 of this paper we prove a matroid analogue of Halin's result. Tutte used his result for 3-connected graphs to deduce a recursive construction of all simple 3-connected graphs having at least four vertices. In Section 4 we generalize this by giving a recursive construction of all 3-connected matroids of rank at least three. Section 2 contains a generalization to minimally n-connected matroids of a result of Dirac [2] for minimally 2-connected graphs.

All rings in this paper are assumed to be commutative with identity, and the undefined terminology is the same as that in [3]. In 1956, in an important paper [2], M. Nagata constructed an example which showed (among other things): (i) a maximal chain of prime ideals in an integral extension domain R' of a local domain (R, M) need not contract in R to a maximal chain of prime ideals; and, (ii) a prime ideal P in R' may be such that height P < height P ∩ R. In his example, Rf was the integral closure of R and had two maximal ideals. In this paper, by using Nagata's example, we show that there exists a finite local integral extension domain of D = R[X](M,X) for which (i) and (ii) hold (see (2.8.1) and (2.10)).

Let S denote the unit sphere in Cn, B the (open) unit ball in Cn and H∞(B) the collection of all bounded holomorphic functions on B. For f ∈ H∞(B) the limitsexist for almost every ζ in S, and the map ƒ → ƒ* defines an isometric isomorphism from H∞(B) onto a closed subalgebra of L∞(S), denoted H∞(S). (The only measure on S we will refer to in this paper is the Lebesgue measure, dσ, generated by Euclidean surface area.) Rudin has shown in [4] that the spaces H∞(B) + C(B) and H∞(S) + C(S) are Banach algebras in the sup norm. In this paper we will show that the maximal ideal space of H∞(B) + C(B), Σ (H∞(B) + C(B)), is naturally homeomorphic to Σ (H∞(B)) and that Z (H∞(S) + C(S)) is naturally homeomorphic to Σ (H∞(S))\B.

Introduction. Until recently none of the numerous papers on the distribution of quadratic and higher power residues was concerned with questions of the following sort: Let k and m be positive integers. According to a theorem of Brauer (1), for every sufficiently large prime p there exist m consecutive positive integers r, r + l , . . . , r + m — 1, each of which is a &th power residue of p. Let r(k, m, p) denote the least such r. What can be said about the behaviour of r(k, m, p) as p varies? In particular, for what values of k and m is r(k, m, p) bounded, and for these values what is its maximum? For a fixed k and m we call a prime exceptional and denote it by p* if there do not exist m consecutive integers each of which is a &th power residue of p*. Set A(k,m) = Max r (k, m, p), where the maximum is taken over all nonexceptional primes p. In a previous paper (4) Lehmer and Lehmer showed that

By an Euler polyhedron of valence three or a trivalent convex polyhedron in Euclidean 3-space (4) we mean in the present paper an Euler polyhedron in the sense of Steinitz (8, p. 113), such that each vertex is incident with exactly three edges.In the present paper we establish a theorem concerning the colouring of trivalent polyhedra. A specialization of this theorem solves the following problem implicit in Eberhard (1, p. 84): Does there exist a trivalent Euler polyhedron with an odd number of faces such that the number of edges incident with any face is divisible by three?

If G is a finite group which has a faithful complex representation of degree nit is said to be a linear group of degree n. It is convenient to consider only unimodular irreducible representations. For n ≦ 4 these groups have been known for a long time. An account may be found in Blichfeldt's book (1). For n= 5 they were determined by Brauer in (4). In (4), many properties of linear groups of prime degree pwere determined for pa prime greater than or equal to 5.In a forthcoming series of papers these results will be extended and the linear groups of degree 7 determined. In the first paper, some general results on linear groups of degree p, p≧ 7, will be given. These results will later be applied to the prime p = 7.

We say a (plane) set A contains all sets of some type if, for each B of type , there is a subset of A that is congruent to B. Recently, Besicovitch and Rado [3] and independently, Kinney [5] have constructed sets of plane measure zero containing all circles. In these papers it is pointed out that the set of all similar rectangles, some sets of confocal conies and other such classes of sets can be contained in sets of plane measure zero, but all these generalizations rely in some way on the symmetry, or similarity of the sets within the given type.In this paper we construct a set of plane measure zero containing all finite polygonal arcs (i.e., the one-dimensional boundaries of all polygons with a finite number of sides) with slightly stronger results if we restrict our attention to k-gons for some fixed k.

In a recent paper by Douglas Underwood [8] several definitions of “associated prime divisors” were discussed and shown to be unique. In this note we produce a fifth type, which is due to W. Krull, and is found in his classical paper [2] and further discussed by B. Banaschewski in[1].Historically this characterization considerably predates the other four definitions.Throughout this note,Rdenotes a commutative ring with unity, and all ideals and elements are assumed to be in such a ring. We shall let upper case letters, most frequently the beginning of the alphabet, denote ideals and lower case letters, elements ofR.On the whole, our terminology will be that of [9]. We do, however, take exception with [9] in two instances, viz.

The existence of certain formulae analogous to Poisson's summation formula (9, pp. 60-64),where αβ = 2π, α > 0, and Fc(x) is the Fourier cosine transform of f(x), but involving number-theoretic functions as coefficients, was first demonstrated by Voronoï (10) in 1904. He proved thatwhere r(n) is an arithmetic function,/(x) is continuous in (a, b) and a(x) and i?(x) are analytic functions dependent on τ(n). Later, numerous papers were published by various authors giving formulae of this type involving d(n), the number of divisors of n (3), and rp(n), the number of ways of expressing n as the sum of p squares of integers (8).In 1937, Ferrar (4) developed a general theory of summation formulae, using complex analysis. Around that time, Guinand (5) also published papers where he developed the general theory from a different point of view. He applied the theory of mean convergence for the transforms of class L2(0, ∞ ). Later in 1950, Bochner (1) gave a general summation formula.

In two previous papers (see4;5) O. T. O'Meara and I investigated the problem of generating the integral orthogonal group of a quadratic form by symmetries in the case where the underlying ring of integers was the integers of a dyadic local field of characteristic not 2. In this paper, the investigation is concerned with a local field of characteristic 2. As in (5), only the unimodular case is treated. As in (4) and (5), groupsS(L), Xh(L), andO(L) are introduced for a unimodular latticeLand the relationship betweenS(L) andO(L) studied. As in the previously cited papers, generation by symmetries means thatS(L) =O(L). The following result is obtained.

Error-correcting codes are used in several constructions for packings of equal spheres in ^-dimensional Euclidean spaces En. These include a systematic derivation of many of the best sphere packings known, and construction of new packings in dimensions 9-15, 36, 40, 48, 60, and 2m for m g 6. Most of the new packings are nonlattice packings. These new packings increase the previously greatest known numbers of spheres which one sphere may touch, and, except in dimensions 9, 12, 14, 15, they include denser packings than any previously known. The density A of the packings in En for n = 2m satisfies log A ~ — \n log log n as n —* oo. 1.1. Introduction. In this paper we make systematic use of error-correct ing codes to obtain sphere packings in En, including several of the densest packings known and several new packings. By use of cross-section s we then obtain packings in spaces of lower dimension, and by building up packings by layers we obtain packings in spaces of higher dimension. Collectively, these include all of the densest packings known, and further new packings are also con­ structed. Part 1 of the paper is devoted to groundwork for the constructions. § 1.2 introduces sphere packings, and §§ 1.3-1.8 survey the error-correcting code theory used in the later Parts. Part 2 describes and exploits Construction A, which is of main value in up to 15 dimensions. Part 3 describes Construction B% of main value in 16-24 dimensions. Part 4 digresses to deal with packings built up from layers, while Part 5 gives some special constructions for dimen­ sions 36, 40, 48 and 60. Part 6 deals with Construction C, applicable to dimensions n = 2m and giving new denser packings for m ^ 6. We conclude with tables summarizing the results. Table I, for all n S 24, supersedes the tables of [18; 19], and Table II gives results for selected n > 24. The tables may be used as an index giving references to the sections of the paper in which the packings are discussed. Partial summaries of this work have appeared in [22; 23]. General references for sphere packing are [18; 19; 31] and for coding theory [4; 25].

I t is straightforward, but tedious, to write down the integers whose representations in a given base do not have particular digits in certain positions. In the first section of this paper we give a computational scheme that enables us to carry out such operations in a rapid and simple fashion.In the second section of the paper we derive a general identity involving the digits of integers in arbitrary Cantor systems of notation.In the third section we apply this identity and deduce a number of results concerned with the splitting of integers into classes with equal power sums. The computational scheme of the first section leads us to an algorithm for the determination of such splittings.

LetRbe the discrete group ring of the groupGover the ringA. In this paper we attempt to find necessary and sufficient conditions onGandAso thatRwill have some standard ring-theoretic property ; among the properties considered are those of being artinian, regular, self-injective, and semi-prime.The contents of this paper form essentially the author's doctoral thesis. The author would like to thank his supervisor Dr. J. Lambek for his generous encouragement and continued interest.

Let Mm be a closed PL manifold of dimension m. Then a concordance between two PL-homeomorphisms h0, h1:M → M is a PL-homeomorphismH: M × I → M × I such that H|M × 0 = h0 and H|M × 1 = h. Concordance is an equivalence relation and in his paper [2], M. Kato classifies PL-homeomorphisms of Sp × Sq up to concordance. To do this he treats first the problem of classifying those homeomorphisms that induce the identity in homology, and then describes the automorphisms of the cohomology ring that can arise from homeomorphisms of Sp × Sq. In this paper we show that for sufficiently connected PL-manifolds that embed in codimension 1, one can extend Kato's classification of the homeomorphisms that induce the identity in homology.

For a formal definition of “map” the reader is referred to (7, §2). The maps in this paper are rooted by specifying an orientation for one of the edges. This also specifies a root vertex, the negative end of the root, and a root face, the face on the left of the root edge. Counting is, as usual, defined on isomorphism classes.Regular maps of even valence have been enumerated in a recent paper by Tutte. In this paper we determine the average number of trees in such maps, and include similar results for regular tri valent maps, that is, maps with three edges incident on every vertex. In the development for the latter, a formula for the number of trivalent maps with 2t vertices is produced.

HNN groups have appeared in several papers, e.g., [3; 4; 5; 6; 8]. In this paper we use the results in [6] to obtain a structure theorem for the subgroups of an HNN group and give several applications.We shall use the terminology and notation of [6]. In particular, if K is a group and {φi} is a collection of isomorphisms of subgroups {Li} into K, then we call the group1the HNN group with base K, associated subgroups { Li,φi(Li)} and free part the group generated by t1, t2, …. (We usually denote φi(Li) by Mi or L–i.) The notion of a tree product as defined in [6] will also be needed.

Throughout this paper, Z is the ring of integers, ƒ*(t) (ƒ(t)) is an integer monic (co-monic) polynomial in the indeterminate t (i.e., each coefficient of ƒ* (ƒ) is in Z and its highest (lowest) coefficient is 1 (5, p. 121, Definition) and M* (M) is the multiplicative semigroup of all integer monic (co-monic) polynomials ƒ* (ƒ) having no constant term. In (3, Theorem 2), Herstein proved that if R is a division ring with centre C such that1then R = C. In this paper we seek a generalization of Herstein's result to semi-simple rings. We also study the following condition:(1)*Our results are quite complete for a semi-simple ring R in which there exists a bound for the codegree ofƒ (ƒ*) (i.e., the degree of the lowest monomial of ƒ(ƒ*)) appearing in the left-hand side of (1) ((1)*).

Soon after the publication of Tutte's paper [5] on m-cages, H. S. M. Coxeter asked in a letter to the author whether one-regular graphs of degree 3 exist. The purpose of the following paper is to show by an example that the answer is in the affirmative.

Throughout this paper, the symbol P = [Pij] will represent the transition probability matrix of an irreducible, null-recurrent Markov process in discrete time. Explanation of this terminology and basic facts about such chains may be found in (6, ch. 15). It is known (3) that for each such matrix P there is a unique (except for a positive scalar multiple) positive vector Q = {qi} such that QP = Q, or1this vector is often called the "invariant measure" of the Markov chain.The first problem to be considered in this paper is that of determining for which vectors U(0) = {μi(0)} the vectors U(n) converge, or are summable, to the invariant measure Q, where U(n) = U(0)Pn has components2In § 2, this problem is attacked for general P. The main result is a negative one, and shows how to form U(0) for which U(n) will not be (termwise) Abel summable.

For various classesQof metric spaces, there are several well-known results characterizing the localn-connectivity of a metric space in terms ofn-ANR(Q)'s. Specifically, we have in mind the results of Kuratowski (13, p. 265) and Kodama (10, p. 79). The main purpose of this paper will be to obtain similar results along these lines for non-metric classesQ.In the last part of the paper we specifyQto be the class of totally normal spaces and characterize the localn-connectivity of ann-dimensional separable metric space in terms of ANR(Q)'s.

1. 1. This paper is the second in a series of papers discussing linear groups of prime degree, the first being (8). In this paper we discuss only linear groups of degree 7. Thus, G is a finite group with a faithful irreducible complex representation Xof degree 7 which is unimodular and primitive. The character of Xis x- The notation of (8) is used except here p= 7. Thus Pis a 7-Sylow group of G.In §§ 2 and 3 some general theorems about the 3-Sylow group and 5-Sylow group are given. In § 4 the statement of the results when Ghas a non-abelian 7-Sylow group is given. This corresponds to the case |P| =73 or |P|= 74. The proof is given in §§ 5 and 6. In a subsequent paper the results when Pis abelian will be given.

These notes are to supplement my paper (4), and should be read in conjunction with it. Both are divided into three parts, and in these notes the section numbers have a further digit added; thus §1.41 here supplements §1.4 of (4). References by section numbers are always to (4) or to the present notes, but references to other papers are numbered independently.The principal results of these notes are the following. New sphere packings are given in [2m], m ⩾ 6, and in [24], which are twice as dense as those of §§1.6, 2.3. Others are given in [2m], m ⩾ 5, with the same density as those of §1.6, but in which each sphere touches fewer other spheres than in the earlier packings.

In a previous paper (2), one of us has derived a formula for the counting series for bicoloured graphs.2 These are graphs each of whose points has been coloured with exactly one of two colours in such a way that every two adjacent points have different colours.In this paper we first enumerate bicoloured graphs without isolated points and connected bicoloured graphs. This leads us to corresponding problems for bicolourable graphs. Such a graph has the property that its points can be coloured with two colours so as to obtain a bicoloured graph.

Our aim is to present some global results about the set of maximal abelian subgroups of the symmetric group Sn. We shall show that certain properties are true for “almost all” subgroups of this set in the sense that the proportion of subgroups which have these properties tends to 1 as n → ∞. In this context we consider the order and the number of orbits of a maximal abelian subgroup and the number of generators which the group requires.Earlier results of this kind may be found in the papers [1; 2; 3; 4; 5]; the papers of Erdös and Turán deal with properties of the set of elements of Sn. The present work arose out of a conversation with Professor Turán who posed the general problem: given a specific class of subgroups (e.g., the abelian subgroups or the solvable subgroups) of Sn, what kind of properties hold for almost all subgroups of the class?

The main theorem of this paper names seven ring classifications which coincide with the class of rings named in the title. Three of them are (1) rings over which every module is rationally complete, (2) left and right perfect rings over which every module is corationally complete, and (3) right perfect rings over which every right module is rationally complete. Corational completeness (introduced in this paper) and rational completeness are generalizations of projectivity and injectivity, respectively. One recalls that a proper subclass of the rings investigated in this paper, the artinian rings with zero radical, is known to have one-sided characterizations in terms of the injectivity (projectivity) of modules. An example proves, however, that the class of rings over which every right module is rationally complete properly contains the rings of the title.

Let X be an n-square matrix with elements in a field F. The permanent of X is defined by1.1where σ runs over the symmetric group of permutations on 1, 2, … , n. This function makes its appearance in certain combinatorial applications (13), and is involved in a conjecture of van der Waerden (6; 9). Certain formal properties of per (X) are known (1), and an old paper of Pólya (12) shows that for n > 2 one cannot multiply the elements of X by constants in any uniform way so as to convert the permanent into the determinant. In a subsequent paper we intend to investigate this problem for more general operations on X.

In this paper we consider undirected graphs, with no edges joining a vertex to itself, but with possibly several edges joining pairs of vertices. The first part of the paper deals with the question of characterizing those sets of non-negative integers d1d2 . . . , dn and {cij}, 1 ≤ i < j ≤ n, such that there exists a graph G with n vertices whose valences (degrees) are the numbers di, and with the additional property that the number of edges joining i and j is at most cij. This problem has been studied extensively, in the general case (1, 2, 9, 11), in the case where the graph is bipartite (3, 5, 7, 10), and in the case where the Cij are all 1 (6).