bluuebunny/crossref_metadata_2025 huggingface dataset:
On functions of bounded variation relative to a set
Authors: Bhakta P. C.
Published: April 2009

The present paper on functions of bounded variation relative to a set has its point of departure in the work of R. L. Jeffery [10]. Below we recapitulate Jeffery's class U of functions of bounded variation relative to a set, we state and prove a number of preliminary lemmas and theorems, we introduce a suitable pseudo- metric space (X, d) of such functions, and the analogous space , and prove that (X, d) is separable, that every closed sphere in (X, d) is compact and that is complete. These results extend known results of C. R. Adams, and C. R. Adams and A. P. Morse for the space of usual BV functions.
On inverses of products of idempotents in regular semigroups
Authors: Fitz-Gerald D. G.
Published: April 2009

Let E be the set of idempotents of a regular semigroup; we prove that V(En) = En+1 (see below for the meaning of this notation). This generalizes a result of Miller and Clifford ([3], theorem 4, quoted as exercise 3(b), p. 61, of Clifford and Preston [1]) and the converse, proved by Howie and Lallement ([2], lemma 1.1), which together establish the case n = 1. As a corollary, we deduce that the subsemigroup generated by the idempotents of a regular semigroup is itself regular.
An example of an orthogonal transformation of Hilbert sequence space
Authors: Smith R. C. T.
Published: April 2009

We use the equivalence of the sequence space 12 and the function space L2 to obtain an orthogonal transformation related to two series inversion formulae due, respectively, to Titchmarsh [1, 2] and to Linfoot and Shepherd [3].
Negative curvature and combinatorial group theory
Authors: Wolf Joseph A.
Published: April 2009

Combinatorial group theory has roots in Poincaré's work on the topology of manifolds, which in turn was based on problems in differential equations and analytic number theory. Thus the Fuchsian groups, which are the fundamental (first homotopy) groups of oriented negatively curved compact surfaces, served as important models in their day. In the last few years there have been advances in the understanding of the structure of fundamental groups of negatively curved manifolds, some of them based on examples from analytic number theory. Here I describe one of these developments and pose a few difficult combinatorial questions.
Isomorphisms induced by automorphisms
Authors: Thompson John G.
Published: April 2009

The object of this note is to record a property of finite, perfect, centrally closed groups, where, by definition, G si centrally closed if and only if whenever E/Z ≅ G and Z ⊆ E' ∩ Z(E), then Z = 1.
Groups with an automorphism squaring many elements
Authors: Liebeck Hans
Published: April 2009

A universal power automorphism (Cooper [1]) of a group is an automorphism mapping every element x to a power xn for some fixed integer n. It is long known that a group admitting such an automorphism with n= −1, 2 or 3 must be Abelian. Miller [5] showed that for every other non-zero integral value of n there exist non-Abelian groups admitting a non-trivial universal power automorphism x→xn.
The path functor and faithful representability of banach lie algebras
Authors: van Est W. T. , Świerczkowski S.
Published: April 2009

In this note “vector space” will mean “Banach space” unless otherwise specified. Accordingly “Lie algebra” will stand for “Banach Lie algebra”. Morphisms between Lie algebras will be assumed continuous. A Banach algebra B will be always assumed associative, and it will be also viewed as a Lie algebra with product [X, YXY− YX. In particular, the Lie algebra gl(V) of endomorphisms of a vector space V will be equipped with the uniform norm. A morphism of Lie algebras L → gl(V) will b called a representation of L in gl(V). Also, if B is a Banach algebra, a morphism of Lie algebras L → B will be called a representation of L in B. From such one evidently obtains a representation of L in gl(B). A representation will be called faithful if it is injective.
Formal multiplication of trigonometric series
Authors: Izumi Masako , Izumi Shin-Ichi
Published: April 2009

Let be the given trigonometric series, then the formal product of them is defined by where the last series is supposed to be convergent for every n.
On minimal irregular p-Groups
Authors: Groves J. R. J.
Published: April 2009

The study of minimal irregular p-groups was initiated by Mann [9]. He defines such a group to be a finite irregular p-group in which every proper section is regular. He then deduces a large number of properties of such groups and shows, by construction, that there is no bound on their exponent. In this note methods involving varieties of groups are used to show that all minimal irregular p-groups can be ‘built up’ from cyclic groups, groups of exponent p and a relatively small class of metabelian minimal irregular p-groups. To phrase this more precisely, say that a group P is derivable from a family of groups {Qi} if P is a homomorphic image of a subdirect product of the Qi. Then we will prove as Theorem 2.2: a minimal irregular p-group of exponent pn and nilpotency class c is derivable from a family of three groups: i) a cyclic group of order pn, ii) a 2-generator monolithic group of exponent p and class c, iii) a minimal irregular p-group whi h is metabelian, has class p and has exponent p2.
Subgroups of finitely presented metabelian groups
Authors: Baumslag Gilbert
Published: April 2009

In 1961 Graham Higman [1] proved that a finitely generated group is a subgroup of a finitely presented group if, and only if, it is recursively presented. Therefore a finitely generated metabelian group can be embedded in a finitely presented group.
Alternating trilinear forms and groups of exponent 6
Authors: Atkinson M. D.
Published: April 2009

The theory of alternating bilinear forms on finite dimensional vector spaces V is well understood; two forms on V are equivalent if and only if they have equal ranks. The situation for alternating trilinear forms is much harder. This is partly because the number of forms of a given dimension is not independent of the underlying field and so there is no useful canonical description of an alternating trilinear form.
Operator algebras with a reduction property
Authors: Gifford James A.
Published: April 2009

Given a representation θ: A → B(H) of a Banach algebra A on a Hilbert space H, H is said to have the reduction property as an A—module if every closed invariant subspace of H is complemented by a closed invariant subspace; A has the total reduction property if for every representation θ: A → B(H), H has the reduction property.

We show that a C*—algebra has the total reduction property if and only if all its representations are similar to *—representations. The question of whether all C*-algebras have this property is the famous ‘similarity problem’ of Kadison.

We conjecture that non-self-adjoint operator algebras with the total reduction property are always isomorphic to C*-algebras, and prove this result for operator algebras consisting of compact operators.
Class preserving automorphisms of Blackburn groups
Authors: Herman Allen , Li Yuanlin
Published: April 2009

In this article, a Blackburn group refers to a finite non-Dedekind group for which the intersection of all nonnormal subgroups is not the trivial subgroup. By completing the arguments of M. Hertweck, we show that all conjugacy class preserving automorphisms of Blackburn groups are inner automorphisms.
Elements of rings and Banach algebras with related spectral idempotents
Authors: Castro-González N. , Vélez-Cerrada J. Y.
Published: April 2009

Let aπ denote the spectral idempotent of a generalized Drazin invertible element a of a ring. We characterize elements b such that 1 − (bπ − aπ)2 is invertible. We also apply this result in rings with involution to obtain a characterization of the perturbation of EP elements. In Banach algebras we obtain a characterization in terms of matrix representations and derive error bounds for the perturbation of the Drazin Inverse. This work extends recent results for matrices given by the same authors to the setting of rings and Banach algebras. Finally, we characterize generalized Drazin invertible operators A, B ∈ (X) such that pr(Bπ) = pr(Aπ + S), where pr is the natural homomorphism of (X) onto the Calkin algebra and S ∈(X) is given.

2000 Mathematics subject classification: primary 16A32, 16A28, 15A09.
Growth properties and sequences of zeros of analytic functions in spaces of Dirichlet type
Authors: Girela Daniel , Peláez José Ángel
Published: April 2009

For 0 < p < ∞, we let pp−1 denote the space of those functions f that are analytic in the unit disc Δ = {z ∈ C: |z| < 1} and satisfy ∫Δ(1 – |z|)p−1|f′(z)|pdx dy < ∞ The spaces pp−1 are closely related to Hardy spaces. We have, p−1p ⊂ Hp, if 0 < p ≦ 2, and Hp ⊂ pp−1, if 2 ≦ p < ∞. In this paper we obtain a number of results about the Taylor coefficients of pp-1 -functions and sharp estimates on the growth of the integral means and the radial growth of these functions as well as information on their zero sets.

2000 Mathematics subject classification: primary 30D35, 30D55, 46E15.
I-Subsemigroups and α-monomorphisms
Authors: Magill Kenneth D.
Published: April 2009

To each idempotent v of a semigroup T, there is associated, in a natural way, a subsemigroup Tv of T. The subsemigroup Tv is simply the collection of all elements of T for which v acts as a two-sided identity. We refer to such a subsemigroup as an I-subsemigroup of T. We first establish some elementary properties of these subsemigroups with no restrictions on the semigroup in which they are contained. Then we turn our attention to the semigroup of all continuous selfmaps of a topological space. The I-subsemigroups of these semigroups are investigated in some detail and so are the a-monomorphisms [3, p. 518] from one such semigroup into another. Among other things, a relationship is established between I-subsemigroups and α-monomorphisms. An analogous theory exists for semigroups of closed selfmaps on topological spaces. A number of results are listed for these semigroups with the proofs often deleted since, in many cases, the situation is much the same as for semigroups of continuous functions.
On rings of sets II. Zero-sets
Authors: Speed T. P.
Published: April 2009

In an earlier paper [11] we discussed the problem of when an (m, n)-complete lattice L is isomorphic to an (m, n)-ring of sets. The condition obtained was simply that there should exist sufficiently many prime ideals of a certain kind, and illustrations were given from topology and elsewhere. However, in these illustrations the prime ideals in question were all principal, and it is desirable to find and study examples where this simplification does not occur. Such an example is the lattice Z(X) of all zero-sets of a topological space X; we refer to Gillman and Jerison [5] for the simple proof that Z(X) is a (2, σ)-ring of subsets of X, where we denote aleph-zero by σ.
On the second cohomology of GL (n, 2)
Authors: Dempwolff U.
Published: April 2009

The purpose of this note is to prove the following result. Theorem A. Let G be a finite group with the following properties (i) V▵ G, &V& = 2n and V is elementary abelian, (ii) G/V ≃ GL(n, 2), (iii) CG(V) ⊆ V. If n ≧ 6 then G splits over V.
Invariant subspaces of operators related to the unilateral shift
Authors: Caradus S. R.
Published: April 2009

Among all non-self-adjoint operators, the shift has a special place in questions relating to invariant subspaces. It is therefore natural to attempt to use this fact to study other operators related in some way to the shift. Examples of this procedure are available in the work of Freeman [3] and Lim [5] where perturbations of the shift are studied.
A note on near-rings of mappings
Authors: Yamamuro Sadayuki
Published: April 2009

Hanna Neumann was one of the pioneers of the algebraic theory of nearrings. In the functional analysis, the near-rings appear as the sets of non-linear mappings of a space into itself. The relations between the algebraic properties of these near-rings and the analytic structures of the spaces and mappings involved have not been fully investigated. The purpose of this note is to consider one of such problems.
Annihilators of relation modules
Authors: Mital J. N. , Passi I. B. S.
Published: April 2009

Let be a non-cyclic free presentation of a group G, the lower central series of R. Then Rn/Rn+1, n ≧ 1, can be regarded as a right G-module by defining the action of G via conjugation in F. We wish to investigate the annihilators of these modules which we call higher relation modules.
Radial distributions of Julia sets of meromorphic functions
Authors: Qiu Ling , Wu Shengjian
Published: April 2009

We consider a meromorphic function of finite lower order that has ∞ as its deficient value or as its Borel exceptional value. We prove that the set of limiting directions of its Julia set must have a definite range of measure.
Resolution in hölder spaces of an elliptic problem in an unbounded domain
Authors: Berroug Tarik , Labbas Rabah , Sadallah Boubaker-Khaled
Published: April 2009

In this paper we give new results concerning the maximal regularity of the strict solution of an abstract second-order differential equation, with non-homogeneous boundary conditions of Dirichlet type, and set in an unbounded interval. The right-hand term of the equation is a Hölder continuous function.
The factorial moments of additive functions with rational argument
Authors: Šiaulys J. , Stepanauskas G.
Published: April 2009

We consider the weak convergence of the set of strongly additive functions f(q) with rational argument q. It is assumed that f(p) and f(1/p) ∈ {0, 1} for all primes. We obtain necessary and sufficient conditions of the convergence to the limit distribution. The proof is based on the method of factorial moments. Sieve results, and Halász's and Ruzsa's inequalities are used. We present a few examples of application of the given results to some sets of fractions.
Fixed Point Free Actions of Groups of Exponent 5
Authors: Jabara Enrico
Published: April 2009

In this paper we prove that if V is a vector space over a field of positive characteristric p ≠ 5 then any regular subgroup A of exponent 5 of GL(V) is cyclic. As a consequence a conjecture of Gupta and Mazurov is proved to be true.
The unreasonable effectualness of continued function expansions
Authors: Martin Greg
Published: April 2009

Many generalizations of continued fractions, where the reciprocal function has been replaced by a more general function, have been studied, and it is often asked whether such generalized expansions can have nice properties. For instance, we might ask that algebraic numbers of a given degree have periodic expansions, just as quadratic irrationals have periodic continued fractions; or we might ask that familiar transcendental constants such as e or π have periodic or terminating expansions. In this paper, we show that there exist such generalized continued function expansions with essentially any desired behaviour.
The strong closure of Boolean algebras of projections in Banach spaces
Authors: Diestel J. , Ricker W. J.
Published: April 2009

This note improves two previous results of the second author. They turn out to be special cases of our main theorem which states: A Banach space X has the property that the strong closure of every abstractly σ-complete Boolean algebra of projections in X is Bade complete if and only if X does not contain a copy of the sequence space ℓ∞.
Certain relations between p-regular class sizes and the p-structure of p-solvable groups
Authors: Beltrán Antonio , Felipe María José
Published: April 2009

Let G be a finite p-solvable group for a fixed prime p. We study how certain arithmetical conditions on the set of p-regular conjugacy class sizes of G influence the p-structure of G. In particular, the structure of the p-complements of G is described when this set is {1, m, n} for arbitrary coprime integers m, n > 1. The structure of G is determined when the noncentral p-regular class lengths are consecutive numbers and when all of them are prime powers.
On the pseudorandomness of the signs of Kloosterman sums
Authors: Fouvry Étienne , Michel Philippe , Rivat Joël , Sárközy András
Published: April 2009

In this paper we study the pseudorandom properties of the signs of Kloosterman sums.
Kazhdan constants for compact groups
Authors: Neuhauser Markus
Published: April 2009

It is shown that for the computation of the Kazhdan constant for a compact group only the regular representation restricted to the orthogonal complement of the constant functions needs to be taken into account.
Strict feasibility of generalized complementarity problems
Authors: He Y. R. , Ng K. F.
Published: April 2009

The existence of strictly feasible points is shown to be equivalent to the boundedness of solution sets of generalized complementarity problems with stably pseudomonotone mappings. This generalizes some known results in the literature established for complementarity problems with monotone mappings.
The structure of groups whose subgroups are permutable-by-finite
Authors: De Falco M. , De Giovanni F. , Musella C. , Sysak Y. P.
Published: April 2009

A subgroup H of a group G is said to be permutable if HX = XH for each subgroup X of G, and the group G is called quasihamiltonian if all its subgroups are permutable. We shall say that G is a Q F-group if every subgroup H of G contains a subgroup K of finite index which is permutable in G. It is proved that every locally finite Q F-group contains a quasihamiltonian subgroup of finite index. In the proof of this result we use a theorem by Buckley, Lennox, Neumann, Smith and Wiegold concerning the corresponding problem when permutable subgroups are replaced by normal subgroups: if G is a locally finite group such that H/HG is finite for every subgroup H, then G contains an abelian subgroup of finite index.
Polyhedral convex cones and the equational theory of the bicyclic semigroup
Authors: Pastijn F.
Published: April 2009

To any given balanced semigroup identityU≈Wa number of polyhedral convex cones are associated. In this setting an algorithm is proposed which determines whether the given identity is satisfied in the bicylic semigroupor in the semigroup. The semigroupsBCandEdeserve our attention because a semigroup variety contains a simple semigroup which is not completely simple (respectively, which is idempotent free) if and only if this variety containsBC(respectively,E). Therefore, for a given identityU≈Wit is decidable whether or not the variety determined byU≈Wcontains a simple semigroup which is not completely simple (respectively, which is idempotent free).
Expansion and dimension
Authors: Mancuso Vincent J.
Published: April 2009

If (X, τ) is a topological space, is a family of subsets of X, then the expansion of τ by is the topology τ() on X having τ ∪ as a subbase. If (X, τ) has property , when does (X, τ()) have property ? In this paper we answer this question for the property of dimension. We apply our techniques to first countability and related properties.
Representation of permutations as products of cycles of fixed length
Authors: Herzog Marcel , Reid K. B.
Published: April 2009

We study the problem of representing a permutation C as a product of a minimum number, fk(C), of cycles of length k. Upper and lower bounds on fk(C) are obtained and exact results are derived for k = 2, 3, 4.
On lattice-ordered rings in which the square of every element is positive
Authors: Steinberg Stuart A.
Published: April 2009

It is shown that a unital lattice-ordered ring in which the square of every element is positive is embeddable in a product of totally ordered rings provided it is archimedean, semiperfect, or π-regular. Also, some canonical examples of unital l-domains with squares positive that are not totally ordered are discussed.
Corrigendum
Authors: Hall P.
Published: April 2009

My attention has been called to the fact that Theorem E1, quoted on p. 445 of my paper “On the embedding of a group in a join of given groups’, J. Austral. Math. Soc. 17 (1974), 434–495, is practically identical with Théoréme I (p. 47) of the second part of the three-part paper “Produit complet des groupes de permutations et probléme d'extension des groupes” by M. Krasner and L. Kaloujnine, Acta Sci. Math. Szeged 14 (1951), 39–66. My oversight in referring only to the third part of this important work, which contains the special case often quoted as the Kaloujnine-Krasner Theorem, is regrettable: it is the much more general and definitive Théoréme I which should be equally or better known by that name.
A maximum principle
Authors: Ng Kung-Fu
Published: April 2009

Let K be a nonempty compact set in a Hausdorff locally convex space, and F a nonempty family of upper semicontinuous convex-like functions from K into [–∞, ∞). K is partially ordered by F in a natural manner. It is shown among other things that each isotone, upper semicontinuous and convex-like function g: K → [ – ∞, ∞) attains its K-maximum at some extreme point of K which is also a maximal element of K.

Subject classification (Amer. Math. Soc. (MOS) 1970): primary 46 A 40.
On groups of finite height
Authors: Pride Stephen J.
Published: April 2009

In a previous paper, ‘The concept of “largeness” in group theory’, a partial order was defined on the class of infinite groups, and this partial order was seen to give some precision to our intuitive notions of what it means for one infinite group to be ‘larger’ than another. The aim of this paper is to look more closely at groups which are ‘low down’ in this partial order, and to examine the interplay between properties of groups and finiteness conditions in group theory.

Subject classification (Amer. Math. Soc. (MOS) 1970): primary 20 E 15, 20 F 15, 20 K 10, 20 K 25; secondary 20 E 10.
The nullity of a wild knot in a compact 3-manifold
Authors: McPherson James M.
Published: April 2009

The nullity of the Alexander module of the fundamental group of the complement of a knot in S3 was one of the invariants of wild knot type defined and investigated by E. J. Brody in [1], in which he developed a generalised elementary divisor theory applicable to infinitely generated modules over a unique factorisation domain. Brody asked whether the nullity of a knot with one wild point was bounded above by its enclosure genus; for knots in S3, the present author showed in [6] that this was indeed the case. In [7], it was (prematurely) stated by the author that this was also the case for knots k embedded in a 3-manifold M so that H,(M — k) was torsion-free.
The soluble subgroups of a one-relator group with torsion
Authors: Newman B. B.
Published: April 2009

In a survey article [1] Baumslag posed the problem of determining the abelian subgroups of a one-relator group. The solution of this problem was stated but not proved in [5], and partly solved by Moldavanskii [4]. In this paper it will be proved that the centralizer of every non-trivial element in a one-relator group with torsion is cyclic, and that the soluble subgroups of a one-relator groups with torsion are cyclic groups or the infinite dihedral group. That both types of groups may occur as subgroups is easily seen by considering
On some infinitely presented associative algebras
Authors: Lewin Jacques
Published: April 2009

We prove here that if F is a finitely generated free associative algebra over the field and R is an ideal of F, then F/R2 is finitely presented if and only if F/R has finite dimension. Amitsur, [1, p. 136] asked whether a finitely generated algebra which is embeddable in matrices over a commutative f algebra is necessarily finitely presented. Let R = F′, the commutator ideal of F, then [4, theorem 6], F/F′2 is embeddable and thus provides a negative answer to his question. Another such example can be found in Small [6]. We also show that there are uncountably many two generator I algebras which satisfy a polynomial identity yet are not embeddable in any algebra of n xn matrices over a commutative algebra.
Closure operations for Schunck classes
Authors: Hawkes Trevor
Published: April 2009

In his Canberra lectures on finite soluble groups, [3], Gaschütz observed that a Schunck class (sometimes called a saturated homomorph) is {Q, Eφ, D0}-closed but not necessarily R0closed(*). In Problem 7·8 of the notes he then asks whether every {Q, Eφ, D0}-closed class is a Schunck class. We show below with an example † that this is not the case, and then we construct a closure operation R0 satisfying Do < ro < Ro such that is a Schunck class if and only if = {QEφ, Ro}. In what follows the class of finite soluble groups is universal. Let B denote the class of primitive groups. We recall that a Schunck class is one which satisfies: (a) = Q, and (b) contains all groups G such that Q(G) ∩ B ⊆.
On the characterization of Shannon's entropy by Shannon's inequality
Authors: Aczél J. , Ostrowski A. M.
Published: April 2009

1. In [2,5,6,7] a.o. several interpretations of the inequalityfor allsuch thatwere given and the following was proved.
An analogue of lubeseder's theorem for formations of finite rings
Authors: Christensen Carlton
Published: April 2009

For formations of finite soluble groups, the properties of Frattini closure and local defineability are known to be equivalent (see [2]). The investigations of Barnes and Gastineau-Hills [1] on the other hand reveal that although every Frattini closed formation of finite-dimensional soluble Lie algebras over an algebraically closed field of zero characteristic is local, without the algebraic closure condition the relationship between the two properties breaks down even for supersoluble Lie algebras. We are concerned here with the analogous problem for rings. The main results are;
Artinian quotient rings of group rings
Authors: Hughes Ian
Published: April 2009

Smith [6, Theorem 2.18] proved that if A is a ring which has a right artinian right quotient ring and G is a poly- (cyclic-or-finite) group, then the group ring AG has a right artinian right quotient ring. We give here a different proof (and a generalization) of this result using methods developed by Jategaonkar [3,4]. THEOREM. Let A be a ring which has a right artinian right quotient ring, and let G be a group which has a (transfinite) ascending normal series with each factor either finite or cyclic, but only a finite number of finite factors. Then AG has a right artinian right quotient ring.
Finite stability domains for difference equations
Authors: Diamond Phil
Published: April 2009

A technique for the estimation of domains of local stability for difference equations is discussed. A Liapunov function is used in the estimation. Sharper results are possible if there is only one type of nonlinearity. when open Liapunov surfaces are possible. An example of the technique is given.
Some covering and embedding theorems for inverse semigroups
Authors: McAllister D. B.
Published: April 2009

An inverse semigroupSis calledE-unitary if the equationsea = e = e2together implya2=a. In a previous paper the author showed that every inverse semigroup is an idempotent separating homomorphic image of anE-unitary inverse semigroup. The main question considered in this paper is the following. Given an inverse semigroupSgive a method for constructingE-unitary inverse semigroupsPand idempotent separating homomorphisms φP→Sin such a way that the structure ofPas aP-semigroup is evident.
Graphs With Stability Index One
Authors: Holton D. A. , Sims J. A.
Published: April 2009

We consider the effect on the stability properties of a graph G, of the presence in the automorphism group of G of automorphisms (uv)h, where u and v are vertices of G, and h is a permutation of vertices of G excluding u and v. We find sufficient conditions for an arbitrary graph and a cartesian product to have stability index one, and conjecture in the latter case that they are necessary. Finally we exhibit explicitly a large class of graphs which have stability index one.
The representation of d. g. near-rings
Authors: Meldrum J. D. P.
Published: April 2009

In this paper we study the problem of the representation of d.g. near-rings, and in particular the problem of a faithful representation, which is equivalent to the adjoining of an identity. This problem has been considered by Malone [5] and Malone and Heatherly [6] and [7]. They have shown that a finite near-ring with two sided zero can be embedded in the d.g. near-ring generated by the inner automorphisms of a suitable group, and that an identity can always be adjoined to a near-ring with two sided zero. They have also given some special conditions under which a faithful representation of a d.g. near-ring exists.
Some varieties of groups
Authors: Newman M. F.
Published: April 2009

For positive integers n, c the class of groups all of whose n-generator subgroups are nilpotent of class (at most) c is a variety, here denoted [n → c]. Hanna Neumann in her book ([15] pp. 93–98) reported on the first stage of the investigation of these varieties. The main result was that [n → c] is nilpotent if and only if c ≦ n ≧ 2 ([15] 34.33 and 34.54).
Groups with trivial Schur multiplicator
Authors: Wamsley J. W.
Published: April 2009

LetGbe a finite group generated by n elements and defined by m relators, thenGhas a presentationwhereFfree on generatorsx1, …,xn, andRis the normal closure inFofR1, …,Rm. The deficiency of this presentation isn–mand the deficiency ofGis the maximum over all presentations forG.
Multiplicities in Hayman's alternative
Authors: Bergwelier Walter , Langley J. K.
Published: April 2009

In 1959 Hayman proved an inequality from which it follows that if f is transcendental and meromorphic in the plane then either f takes every finite complex value infinitely often or each derivative f(k), k ≥1, takes every finite non-zero value infinitely often. We investigate the extent to which these values may be ramified, and we establish a generalization of Hayman's inequality in which multiplicities are not taken into account.
Heath g-functions and metrization
Authors: Mohamad Abdul M.
Published: April 2009

In this paper, we present some new metrization theorems in terms of Heath g–functions.
Characterizations of collectively precompact and semi-precompact opterators on topological vector spaces
Authors: Deshpande M. V. , Padhye S. M.
Published: April 2009

Characterizations of collectively precompact and collectively semi-precompact sets of operators on topological vector spaces are obtained. These lead to the characterization of totally bounded sets of semi-precompact operators on locally convex spaces.

1980 Mathematics subject classification (Amer. Math. Soc): primary 47 B 05, 47 D 15; secondary 46 A 05, 46 A 15.
Differentiability and local rotundity
Authors: Yorke A. C.
Published: April 2009

The concept of very weak local uniform rotundity (very WLUR) is introduced. It is shown that if a Banach space E is WLUR, very WLUR, or LUR then its dual E* is smooth, very smooth, or Fréchet differentiable, respectively, on a norm dense subset. This leads to examples of nonreflexive spaces which are Fréchet differentiable at every nonzero point of a (relatively) norm dense subset of the embedding of E** in the fourth dual. When E is reflexive, necessary and sufficient conditions for E and E* to be WLUR and LUR are given.

Subject classification (Amer. Math. Soc. (MOS) 1970): 46 B 99.
Inverse semigroups determined by their partial automorphism monoids
Authors: Goberstein Simon M.
Published: April 2009

The partial automorphism monoid of an inverse semigroup is an inverse monoid consisting of all isomorphisms between its inverse subsemigroups. We prove that a tightly connected fundamental inverse semigroup S with no isolated nontrivial subgroups is lattice determined ‘modulo semilattices’ and if T is an inverse semigroup whose partial automorphism monoid is isomorphic to that of S, then either S and T are isomorphic or they are dually isomorphic chains relative to the natural partial order; a similar result holds if T is any semigroup and the inverse monoids consisting of all isomorphisms between subsemigroups of S and T, respectively, are isomorphic. Moreover, for these results to hold, the conditions that S be tightly connected and have no isolated nontrivial subgroups are essential.
The palindromic width of a free product of groups
Authors: Bardakov Valery , Tolstykh Vladimir
Published: April 2009

Palindromes are those reduced words of free products of groups that coincide with their reverse words. We prove that a free product of groups G has infinite palindromic width, provided that G is not the free product of two cyclic groups of order two (Theorem 2.4). This means that there is no uniform bound k such that every element of G is a product of at most k palindromes. Earlier, the similar fact was established for non-abelian free groups. The proof of Theorem 2.4 makes use of the ideas by Rhemtulla developed for the study of the widths of verbal subgroups of free products.
Maximal compatible extensions of partial orders
Authors: Foldes Stephan , Szigeti Jenő
Published: April 2009

We give a complete description of maximal compatible partial orders on the mono-unary algebra (A, f), wheref: A → Ais an arbitrary unary operation.
Finitely valued commutator sequences
Authors: Baer Reinhold
Published: April 2009

If x and y are elements in the group G, then we denote their commutator by x o y = x-1y-1 = x-1xy; and x o G is the set of all commutators x o g with g ∈ G. A G-commutator sequence is a series of elements ci ∈ G with c1 + 1 ∈ ci O G. Slightly generalizing well known results one proves that the hypercenter of the group G is exacly the set of all elements h ∈ G with the property: every G-commutator sequence, containing h, contains 1 [Proposition 1.1].
Bernstein's inequality for locally compact Abelian groups
Authors: Bloom Walter R.
Published: April 2009

This paper is concerned with version of Bernstein's inequality for Hausdroff locally compact Abelian groups. The ideas used are suggested by Exercise 12, p. 17 of Katznelson's book [4].
On the square root of the centre of the Hecke algebra of type A
Authors: Francis Andrew , Jones Lenny
Published: April 2009

In this paper we investigate non-central elements of the Iwahori-Hecke algebra of the symmetric group whose squares are central. In particular, we describe a commutative subalgebra generated by certain non-central square roots of central elements, and the generic existence of a rank-three submodule of the Hecke algebra contained in the square root of the centre, but not in the centre. The generators for this commutative subalgebra include the longest word and elements related to trivial and sign characters of the Hecke algebra. We find elegant expressions for the squares of such generators in terms of both the minimal basis of the centre and the elementary symmetric functions of Murphy elements.
On the module structure of a group action on a Lie algebra
Authors: Papistas Athanassios I.
Published: April 2009

Let G be a finite group, K a field, and V a finite-dimensional K G-module. Write L(V) for the free Lie algebra on V; similarly, let M ( V) be the free metabelian Lie algebra. The action of G extends naturally to these algebras, so they become KG-modules, which are direct sums of finite-dimensional submodules. This paper explores whether indecomposable direct summands of such a KG-module (for some specific choices of G, K and V) must fall into finitely many isomorphism classes. Of course this is not a question unless there exist infinitely many isomorphism classes of indecomposable KG-modules (that is, K has positive characteristic p and the Sylow p-subgroups of G are non-cyclic) and dim V > 1.

The first two results show that the answer is positive for M(V) when K is finite and dim V = 2, but negative when G is the Klein four-group, the characteristic of K is 2, and V is the unique 3-dimnsional submodule of the regular module D. In the third result, G is again the Klein four-group, K is any field of charateristic 2 with more than 2 elements, V is any faithfull module of dimension 2, and B is the unique 3-dimensional quotient of D; the answer is positive for L(V) if and only if it is positive for each of L(B), L(D), and L(V ⊗ V).
On the accuracy of bootstrapping sample quantiles of strongly mixing sequences
Authors: Sun Shuxia
Published: April 2009

In this paper, we examine the rate of convergence of moving block bootstrap (MBB) approximations to the distributions of normalized sample quantiles based on strongly mixing observations. Under suitable smoothness and regularity conditions on the one-dimensional marginal distribution function, the rate of convergence of the MBB approximations to distributions of centered and scaled sample quantiles is of order O(n−1¼ log logn).
Derivative-free characterizations of Qk spaces
Authors: Wulan Hasi , Zhu Kehe
Published: April 2009

We give two charaterizations of the Möbius invariant QK spaces, one in terms of a double integral and the other in terms of the mean oscillation in the Bergman metric. Both charaterizations avoid the use of derivatives. Our results are new even in the case of Qp.
Product structure in topological algebras
Authors: Robbie Desmond A.
Published: April 2009

It is shown that every compact nonconnected semigroup (semiring) which has commuting congruences, has a nontrivial continous homomorphic image which is iseomorphic to a direct product of finite congruence free semigroups (semirings). (This extends parts of earlier work by Kaplansky (1947) on compact rings.) It is also shown that there is a possibly finer representation but onto a product of congruence free semigroups (semirings) known only to be compact Hausdorff. A number of the techniques used evolve from work of Professor Wallace, who retired in mid-1973, and to whom this paper is dedicated.
Generalized Ramsey theory VI: Ramsey numbers for small plexes
Authors: Duke Richard A. , Harary Frandk
Published: April 2009

Generalized Ramsey theory for graphs was formulated and developed in the previous papers in this series. We extend the area here by introducing generalized Ramsey numbers for higher dimensional simplicial complexes. In particular we calculate explicitly the Ramsey numbers for several small “pure 2-complexes”, or more brieflyplexes, in which each edge is contained in some 2–call.
An extended inhomogeneous minimum
Authors: Barnes E. S.
Published: April 2009

A new arithmetic invariant E(f) is defined for integral binary quadratic forms f. It has the property that, denoting by fm the norm-form of a quadratic number field Q(m), E(fm) < 1 if and only if Q(m) has class number one.
A note on polyhedral cones
Authors: Tam Bit-Shun
Published: April 2009

In this short note, two results on a solid, pointed, closed cone C in Rn will be given: first, C is polyhedral iff it has a finite number of maximal faces; second, for any face F of C, C* ∩ F┴ is a face of its dual cone C* of dimension n – dim F.
Separation axioms and subcategories of top
Authors: Nakagawa Ryosuke
Published: April 2009

(Point, closed subset)-separation axioms and closed subsets separation axioms for topological spaces will be uniformly defined. Then it is shown that a subcategory of TOP is bireflective in TOP if and only if Ob consists of all separated spaces for some (point, closed subset)-separation axiom. A characterization theorem on subcategories of all separated spaces for closed subsets separation axioms is also given by using the category SEP of all separation spaces and the embedding functor G: TOP → SEP. As an application we have that a T1-space is normal if and only if it is embedded in a product space of the unit intervals in SEP.
Classification of totally umbilical submanifolds in symmetric spaces
Authors: Chen Bang-Yen
Published: April 2009

A submanifold of a Riemannian manifold is called a totally umbilical submanifold if the second fundamental form is proportional to the first fundamental form. In this paper, we shall prove that there is no totally umbilical submanifold of codimension less than rank M — 1 in any irreducible symmetric space M. Totally umbilical submanifolds of higher codimensions in a symmetric space are also studied. Some classification theorems of such submanifolds are obtained.
Completely semisimple ring semigroups
Authors: Hannah John , Richardson J. S. , Zeleznikow John
Published: April 2009

Let R be a ring in which the multiplicative semigroup is completely semisimple. If R has the maximum (respectively, minimum) condition on principal multiplicative ideals. then R is semiprime artinian (respectively, a direct sum of dense rings of finite-rank linear transformations of vector spaces over division rings).
Positive linear operators and the approximation of continuous functions on locally compact abelian groups
Authors: Bloom Walter R. , Sussich Joseph F.
Published: April 2009

In 1953 P. P. Korovkin proved that if (Tn) is a sequence of positive linear operators defined on the space C of continuous real 2π-periodic functions and limn→rTnf = f uniformly for f = 1, cos and sin. then limn→rTnf = f uniformly for all f∈C. We extend this result to spaces of continuous functions defined on a locally compact abelian group G, with the test family {1, cos, sin} replaced by a set of generators of the character group of G.
Slim trees
Authors: Evans Elliott
Published: April 2009

A semilattice tree T with 0 is slim if there is a chain C with 0 so that the lattices θ (T) and θ(C) of semilattice congruences are isomorphic. This paper establishes elementary consequences of slimness and uses simple constructive techniques to show certain small trees slim. If T is the union of at most countably many branches, each of which has a maximum or a countable cotinal subset, then T is slim. For trees with enough maximals slimness is equivalent with not having any uncountable anti-chains. If a tree T has a countable cofinal subset then T is slim. Thus finitary trees are slim.
On the uniform convergence of interpolatory polynomials
Authors: Prasad J.
Published: April 2009

Here we consider a problem on weighted (0, 2) interpolation. We choose the interpolatory conditions in such a way that we get the polynomial of degree ≤2n−1, satisfying those conditions. Moreover we prove that the sequence of these interpolatory polynomials under certain conditions converges uniformly to a function belonging to the Zygmund class.
An isoperimetric problem with lattice point constraints
Authors: Arkinstall J. R. , Scott P. R.
Published: April 2009

The isoperimetric problem in the Euclidean plane is completely solved for bounded, convex sets which are symmetric about the origin, and which contain no non-zero point of the integral lattice.
Semigroup ideals in rings
Authors: Hill David A.
Published: April 2009

A ringRis called anl-ring (r-ring) in caseRcontains an indentity and every left (right) semigroup ideal is a left (right) ring ideal. A number of structure theorems are obtained forl-rings whenRis left noetherian and left artinian. It is shown that left noetherianl-rings are local left principal ideal rings. WhenRis a finite dimensional algebra over a field, the property of being anl-ring is equivalent to being anr-ring. However, examples are given to show that these two concepts are in general not equivalent even in the artinian case.
Growth sequences of finite groups
Authors: Wiegold James
Published: April 2009

During his investigation of the possible non-Hopf kernels for finitely generated groups in [1], Dey proves that the minimum number of generators d(Gn) of the n-th direct power Gn of a non-trival finite group G tends to infinity with n. This has prompted me to ask the question: what are the ways in which the sequence {d(Gn)} can tend to infinity? Let us call this the growth sequence for G; it is evidently monotone non-decreasing, and is at least logarithmic (Theorem 2.1). This paper is devoted to a proof that, broadly speaking, there are two different types of behaviour. If G has non-trivial abelian images (the imperfect case, § 3), then the growth sequence of G is eventually an arithmetic progression with common difference d(G/G'). In special cases (Theorem 5.2) the initial behaviour can be quite nasty. Our arguments in § 3 are totally elementary. If G has only trivial abelian images (the perfect case,§ 4), then the growth sequence of G is eventually bounded above by a sequence that grows logarithmically. It is a simple consequence of this fact that there are arbitrarily long blocks of positive integers on which the growth sequence takes constant values. This is a characteristic property of perfect groups, and indeed it was this feature in the growth sequences of large alternating groups (which I found by using ad hoc permutational arguments) that attracted me to the problem in the first place. The discussion of the perfect case rests on the lovely paper of Hall [2], which was brought to my notice by M. D. Atkinson.
Formulae for well formed formulae and their enumeration
Authors: Tamari Dov
Published: April 2009

1. General Idea. In this paper formal operators and bracketings devices are essentially the same; so are well formed formulae and correct bracketings.

A well known theorem of symbolic logic (Post languages, etc.) and universal algebra characterizes well formed formulae among (linear) strings of symbols for variables and operators in terms of a system of numerical inequalities, one of them an equality.
Varieties and simple groups
Authors: Jones Gareth A.
Published: April 2009

In her book on varieties of groups, Hanna Neumann posed the following problem [13, p. 166]: “

Can a variety other than D contain an infinite number of non-isomorphic non-abelian finite simple groups?”

The answer to this question does not seem to be known at present. However, in [7], Heineken and Neumann described an algorithm for determining whether or not there are any non-abelian finite simple groups satisfying a given law. They also outlined a way in which their algorithm could be used to show that “only finitely many of the known non-abelian finite simple groups can satisfy a given non-trivial law”; in this paper, we shall follow their suggestions, and prove the

THEOREM. Let g be a set of mutually non-isomorphic non-abelian finite simple groups, each of which is either an alternating group or a group of Lie type, and let g generate a proper subvariety of D. Then y is finite.
Totally pure sequences and totally pure projection
Authors: Cook Donald
Published: April 2009

All groups in this paper are Abelian. The notation is similar to Fuchs (1970) and Maclane (1963).
A note on Galois groups of algebraic closures
Authors: Krakowski Don
Published: April 2009

A group will be called full if it is the Galois group of an algebraic closure of a field. In this paper we first investigate full Abelian groups and classify them. Then we examine full groups from the point of view of how we can operate on them and still maintain the property of being full. Of course, by the fundamental theorem of the infinite Galois theory, closed subgroups (with standard profinite topology) of full groups are full. In general, products of full groups are notfull (for example, Z2 × Z2 is not full, by a theorem of Artin and Schreier (1927)); however we produce a set of groups which can always be attached as direct factors to full groups and still retain full groups. For the definition and basic properties of profinite groups we refer the reader to Cassels and Frohlich (1967). It has been shown by Leptin (1955) and independently by the author that profinite groups are Galois groups. The author (1971) has shown that if G is profinite, then G = Gdl(k/L) where L may have any desired characteristic and contains all primitive nth roots of unity, for all n. We will denote by p the pro-p-group which is the inverse limit of cyclic p-groups. (This is the group of p-adic integers).
Mobs and semitrees
Authors: Muenzenberger T. B. , Smithson R. E.
Published: April 2009

In (1957) Ward characterized trees as being certain compact idempotent commutative monotone mobs. The purpose of the present note is to obtain a similar characterization for the semitrees as studied by Muenzenberger and Smithson (1973, 1975).
Constructions preserving the associative and the commutative laws
Authors: Shafaat A.
Published: April 2009

The associative and the commutative laws are characterized by preservation under the construction of powers and addition of a (new) unit element. This is used to generate the varieties defined by the two laws from two element groupoids.