Let K be a nonarchimedean local field, let L be a separable quadratic extension of K, and let h denote a nondegenerate sesquilinear formk on L3. The Bruhat-Tits building associated with SU3(h) is a tree. This is applied to the study of certain groups acting simply transitively on vertices of the building associated with SL(3, F), F = Q3 or F3((X)).

We obtain a complete structural characterization of Cohn–Leavitt algebras over no-exit objects as graded involutive algebras. Corollaries of this result include graph-theoretic conditions characterizing when a Leavitt path algebra is a directed union of (graded) matricial algebras over the underlying field and over the algebra of Laurent polynomials and when the monoid of isomorphism classes of finitely generated projective modules is atomic and cancelative. We introduce the nonunital generalizations of graded analogs of noetherian and artinian rings, graded locally noetherian and graded locally artinian rings, and characterize graded locally noetherian and graded locally artinian Leavitt path algebras without any restriction on the cardinality of the graph. As a consequence, we relax the assumptions of the Abrams–Aranda–Perera–Siles characterization of locally noetherian and locally artinian Leavitt path algebras.

We describe the group of all reflection-preserving automorphisms of an imprimitive complex reflection group. We also study some properties of this automorphism group.

In this paper, we prove that two homogeneous quasi-invariant subspaces are similar only if they are equal. Moreover, we exhibit an example to show how to determine the similarity orbits of quasi-invariant subspaces.

The order topology is compact and T2 in both the scale and retracted scale of any uniform space (S, U). if (S, U) is T2 and totally bounded, the Samuel compactification associated with (S, U) can be obtained by uniformly embedding (S, U) in its order retracted scale (that is, the retracted scale with its order topology). This implies that every compact T2 space is both a closed subspace of a complete, infinitely distributive lattice in its order topology, and also a continuous, closed image of a closed subspace of a complete atomic Boolean algebra in its order topology.

In the past a number of papers have appeared which give representations of abstract lattices as rings of sets of various kinds. We refer particularly to authors who have given necessary and sufficient conditions for an abstract lattice to be lattice isomorphic to a complete ring of sets, to the lattice of all closed sets of a topological space, or to the lattice of all open sets of a topological space. Most papers on these subjects give the conditions in terms of special elements of the lattice. We thus have completely join-irreducible elements — G. N. Raney [7]; join prime, completely join prime, and supercompact elements — V. K. Balachandran [1], [2]; N-sub-irreducible elements — J. R. Büchi [5]; and lattice bisectors — P. D. Finch [6]. Also meet-irreducible and completely meet-irreducible dual ideals play a part in some representations of G. Birkhoff & 0. Frink [4].

In his Topics in Number Theory, vol. 2, chapter 2 (Reading, Mass., 1956) W. J. LeVeque proved an important generalisation of Roth's theorem (K. F. Roth, Mathematika 2,1955, 1—20).

Let ξ be a fixed algebraic number, σ a positive constant, and K an algebraic number field of degree n. For κ∈K denote by κ(1), …, κ(n) the conjugates of κ relative to K, by h(κ) the smallest positive integer such that the polynomial has rational integral coefficients, and by q(κ) the quantity

Let H be a group, let {G1: i ∈ I} be a set of groups and, for each i, let θi be a a monomorphism: H → G1, with Hθ1 = H1. We call such a system of groups and monomorphisms an amalgam and denote it by [G1; H; θi; Hi]. By an embedding of the amalgam into a group G is meant a set of monomorphisms ϕi: G1 → G such that θiϕi= θjϕjfor all i, j and G1ϕi = Hθϕk, for all i, j, k.

We establish the uniform almost sure convergence of the partitioning estimate, which is a histogram-like mean regression function estimate, under ergodic conditions for a stationary and unbounded process. The main application of our results concerns time series analysis and prediction in the Markov processes case.

We determine which varieties of commutative semigroups have the weak or strong amalgamation property. These are precisely the varieties of inflations of semilattices of abelian groups.

Let p,m be the modular representation algebra of the cyclic group of order pm over the prime field Zp. The characters of p, m are derived. For p = 2, this provides an alternative proof of a result due to Carlson (1975), tha 2,m is a local ring. It is shown that for p>2, p, m is a direct sum of 2m local rings. Their dimensions and primitive idempotents are derived.

Subject classification (Amer. Math. Soc. (MOS) 1970): 20 C 20, 12 C 05, 12 C 30, 33 A 65.

Kent [4] showed that each uniform space (S, u) could be embedded in a complete, uniform lattice, called the scale of (S, u). The scale was first introduced by Bushaw [3] for studying stability in topological dynamics. In [5], the notions of connectedness and local connectedness were studied. This note is a follow-up of [5]; the purpose being to characterise the uniform subspaces of the reals, R, which have connected (locally connected) scales. The reader is asked to refer to [5] for definitions and notation not given here.

The classical 2-dimensional Laguerre plane is obtained as the geometry of non-trivial plane sections of a cylinder in R3 with a circle in R2 as base. Points and lines in R3 define subsets of the circle set of this geometry via the affine non-vertical planes that contain them. Furthemore, vertical lines and planes define partitions of the circle set via the points and affine non-vertical lines, respectively, contained in them.

We investigate abstract counterparts of such sets of circles and partitions in arbitrary 2-dimensional Laguerre planes. We also prove a number of related results for generalized quadrangles associated with 2-dimensional Laguerre planes.

Equations describing a possible mode of propagation of plane waves in isotropic plastic solids are solved numerically.

It has been shown by one of the authors that the system of idempotents of monoids on a group G of Lie type with Dynkin diagram Γ can be classified by the following data: a partially ordered set U with maximum element 1 and a map λ: U → 2Γ with λ(1) = Γ and with the property that for all J1, J2, J3 ∈ U with J1 > J2 > J3, any connected component of λ(J2) is contained in either λ(J1) or λ(J3). In this paper we show that λ comes from a regular monoid if and only if the following conditions are satisfied: (1) U is a ∧-semilattice; (2) If J1, J2 ∈ U, then λ(J1)∧ λ(J2) λ(J1 ∧ J2); (3) If θ ∈ Γ, J ∈ U, then max{J1 ∈ U|J1 > J, θ ∈ λ (J1)} exists; (4) If J1, J2 ∈ U with J1 > J2 and if X is a two element discrete subset of λ(J1) ∪ λ(J2), then X λ(J) for some J ∈ UJ with J1 > J > J2.

Fellowship of the Australian Academy of Science and Fellowship of the Australian College of Education are formal recognition of Hanna Neumann's impact on a country she had first set foot in only in August 1963. But then`Hanna Neumann was a remarkable person. Throughout her life she had won the love and respect of many people. The extent of this can not really be measured, however some indications can be given. A memorial meeting held in Canberra overflowed a large lecture theatre even though it was virtually vacation time. A collection of papers dedicated to her memory and a fund to provide some form of memorial to her have both drawn quite overwhelming support from many parts of the world. One finds a tremendous list of words describing her memorable qualities: warm, enthusiastic, inspiring, energetic, firm, tactful, sympathetic, efficient, patient, shrewd, humble, peace-loving, courageous, gracious, … No words can hope to evoke more than a pale shadow of such a person; this story must be read in such a light.

Let N be a finitely generated normal subgroup of a finitely generated group G. We show that if the trivial subgroup is tame in the factor group G/N, then N is that in G. We also give a short new proof of the fact that quasiconvex subgroups of negatively curved groups are tame. The proof utilizes the concept of the geodesic core of the subgroup and is related to the Dehn algorithm.

Let P be the algebra of polynomials in one inderminate x over the complex field C. Suppose ∥ · ∥ is a norm on P such that the coefficient functionals cj: ∑αix1 → αj (j = 0,1,2,…) are all continuous with respect to ∥·∥, and Let K ⊂ C be the set of characters on P which are ∥·∥-continuous. then K is compact, C\K is connected, and 0∈K. K. Let A be the completion of P with respect to ∥·∥. Then A is a singly generated Banach algebra, with space of characters (homeomorphic with) K. The functionals cj have unique extensions to bounded linear functionals on A, and the map a →∑Ci(a)xi (a ∈ A) is a homomorphism from A onto an algebra of formal power series with coefficients in C. We say that A is an algebra of power series if this homomorphism is one-to-one, that is if a ∈ A and a≠O imply cj(a)≠ 0 for some j.

We give a constructive proof for the theorem of Lovász and Plummer which asserts the existence of an ear decomposition of a 1-factor covered graph.

The objective of this paper is to prove the following generalization of the main result in [2]: THeorem. Let G be a finite simple group which possesses an involution t such that the centralizer of t in G is isomorphic to the centralizer of an involution in H. Then G is isomorphic to L5(2), M24, or H.