bluuebunny/crossref_metadata_2025 huggingface dataset:
Almost free actions on manifolds
Authors: Church Philip T. , Lamotke Klaus
Published: April 2009

Let X be a compact, connected, oriented topological G-manifold, where G is a compact connected Lie group. Assume that the fixed point set is finite but nonempty, the action is otherwise free, and the orbit space is a manifold. It follows that either G = U(1) = S1 and dimX =4 or G = Sp(1) = S3 and dimX = 8, and the number of fixed points is even. The authors prove that these ∪(1)-manifolds (respectively, Sp

(1)-manifolds) are classified up to orientation-preserving equivariant homeomorphism by (1) the orientation-preserving homeomorphism type of their orbit 3-manifolds (respectively, 5-manifolds), and

(2) the (even) number of fixed points.

Both the homeomorphism type in (1) and the even number in (2) are arbitrary, and all the examples are constructed. The smooth analog for U(1) is also proved.
On the coefficients of transformation polynomials for the modular function
Authors: Mahler Kurt
Published: April 2009

In a previous paper (Acta Arith. 21 (1972), 89–97), I had proved that the sum of the absolute values of the coefficients of the mth transformation polynomial Fm (u, v) of the Weber modular function j(ω) of level 1 is not greater than
2(36n+57)2n
when m = 2n is a power of 2. The aim of the present paper is to give an analogous bound for the case of general m. This upper bound is much less good and of the form

where c > 0 is an absolute constant which can be determined effectively. It seems probable that also in the general case an upper bound of the form
eO(m10gm)
should hold, but I have not so far succeeded in proving such a result.
Computation of the stationary distribution of an infinite stochastic matrix of special form
Authors: Golub G.H. , Seneta E.
Published: April 2009

An algorithm is presented for computing the unique stationary distribution of an infinite regular stochastic matrix of a structural form subsuming both upper-Hessenberg and generalized renewal matrices of this kind. Convergence is elementwise, monotone from above, from information within finite truncations, of increasing order.
Padé approximation and orthogonal polynomials
Authors: Allen G.D. , Chui C.K. , Madych W.R. , Narcowich F.J. , Smith P.W.
Published: April 2009

By using a variational method, we study the structure of the Padé table for a formal power series. For series of Stieltjes, this method is employed to study the relations of the Padé approximants with orthogonal polynomials and gaussian quadrature formulas. Hence, we can study convergence, precise locations of poles and zeros, monotonicity, and so on, of these approximants. Our methods have nothing to do with determinant theory and the theory of continued fractions which were used extensively in the past.
Some remarks on stable graphs
Authors: Yap H.P.
Published: April 2009

We introduce some methods of constructing stable graphs and characterize a few classes of stable graphs. We also give a counter example to disprove Holton's conjecture.
Ascoli theorems and the pseudocharacter of mapping spaces
Authors: Guthrie J.A.
Published: April 2009

The pseudocharacter of the space YX of continuous functions from X to Y is studied. For certain topologies on YX sufficient conditions, which are shown in some cases also to be necessary, for YX to have a specified pseudocharacter are given. An Ascoli theorem is a theorem which characterizes the compact subsets of YX. Several Ascoli theorems are obtained, including ones which utilize results on pseudocharacter.
Commutative Banach algebras with non-unique complete norm topology
Authors: Loy Richard J.
Published: April 2009

It has recently been shown that discontinuous functional calculi exist for certain commutative Banach algebras. Such an algebra thus possesses two distinct calculi so that there exist analytic functions whose action on the algebra is not uniquely determined. In this note a method is given for constructing commutative Banach algebras which admit two inequivalent complete norm topologies and the result is applied to show that the action of any non-algebraic analytic function may fail to be uniquely defined.
Path decompositions of digraphs
Authors: Alspach Brian R. , Pullman Norman J.
Published: April 2009

A path decomposition of a digraph G (having no loops or multiple arcs) is a family of simple paths such that every arc of G lies on precisely one of the paths of the family. The path number, pn(G) is the minimal number of paths necessary to form a path decomposition of G.

We show that pn(G) ≥ max{0, od(v)-id(v)} the sum taken over all vertices v of G, with equality holding if G is acyclic. If G is a subgraph of a tournament on n vertices we show that pn(G) ≤ with equality holding if G is transitive.

We conjecture that pn(G) ≤ for any digraph G on n vertices if n is sufficiently large, perhaps for all n ≥ 4.
On a relation between the Fitting length of a soluble group and the number of conjugacy classes of its maximal nilpotent subgroups
Authors: Lausch H. , Makan A.
Published: April 2009

In a finite soluble group G, the Fitting (or nilpotency) length h(G) can be considered as a measure for how strongly G deviates from being nilpotent. As another measure for this, the number v(G) of conjugacy classes of the maximal nilpotent subgroups of G may be taken. It is shown that there exists an integer-valued function f on the set of positive integers such that h(G) ≦ f(v(G)) for all finite (soluble) groups of odd order. Moreover, if all prime divisors of the order of G are greater than v(G)(v(G) - l)/2, then h(G) ≦3. The bound f(v(G)) is just of qualitative nature and by far not best possible. For v(G) = 2, h(G) = 3, some statements are made about the structure of G.
Free products and residual nilpotency
Authors: Dey I. M. S.
Published: April 2009

Let G be the free product of groups Gα, [Gα] the cartesian subgroup of G and k [Gα] the intersection of [G] with the k–th term of the lower central series for G. Then the k[Gα] form a descending chain of subgroups of ] and it is shown that if the intersection of all the subgroups in this chain is trivial then G and hence each Gα, is residually nilpotent. This answers a question of S. Moran.
Surface and internal waves in a liquid of variable depth
Authors: Hurley D. G. , Imberger J.
Published: April 2009

Consider a stably stratified liquid, whose density varies exponentially with the vertical co-ordinate, that is bounded above by a free surface and below by a bed whose height depends on only one of the horizontal co-ordinates. Suppose that a gravity wave, that may be either a surface or an internal one, is travelling in a direction normal to the lines of constant depth. It is shown that if the frequency is below a certain value an infinite number of waves, all of the same frequency but having differing wave lengths, are generated and expressions for their amplitude are given in terms of the changes in depth which are assumed to be small.
Representing N-semigroups
Authors: Higgins John C.
Published: April 2009

AnN-semigroup is a commutative, cancellative, archimedean semigroup with no idempotent element. This paper obtains a representation of finitely generatedN-semigroups as the subdirect product of an abelian group and a subsemigroup of the additive positive integers.
Anti-uniform semilattices
Authors: Howie J. M. , Schein B. M.
Published: April 2009

An inverse semigroup which is a union of groups is called Cliffordian. A semilattice E is called universally Cliffordian if every inverse semigroup having E as semilattice of idempotents is Cliffordian. It is shown that E is universally Cliffordian if and only if it is anti-uniform, that is, if and only if no two distinct principal ideals of E are isomorphic.

A semilattice E satisfying the minimum condition is anti-uniform if and only if it is a well-ordered chain. Examples are given of anti-uniform semilattices of more complicated types.
The completion by cuts of an orthocomplemented modular lattice
Authors: Adams D. H.
Published: April 2009

In this note we give an example of an orthocomplemented modular lattice whose completion by cuts is not orthomodular. This solves negatively Problem 36, p. 131, in G. Birkhoff: Lattice theory (3rd edition).
On Lie algebra obstructions
Authors: Knopfmacher J.
Published: April 2009

A basic problem in the theory of Lie algebra extensions concerns a given homomorphism X of a Lie algebra L into the Lie algebra of outer derivations of a Lie algebra B. In analogy with the theory of group extensions, Mori and HochschiId developed the concept of an obstruction to X being the homomorphism defined by some Lie algebra extension of B by L. This note considers an alternative approach to this theory, which is particularly simple when applied to the problem of realizing arbitrary three-cohomology classes of L as obstructions. The approach is analogous to one for groups, which was given recently by Gruenberg.
A finite basis theorem for product varieties of groups
Authors: Brooks M.S. , Kovács L.G. , Newman M.F.
Published: April 2009

It is shown that, if U is a subvariety of the join of a nilpotent variety and a metabelian variety and if V is a variety with a finite basis for its laws, then UV also has a finite basis for its laws. The special cases U nilpotent and U metabelian have been established by Higman (1959) and Ivanjuta (1969) respectively. The proof here, which is independent of Ivanjuta's, depends on a rather general sufficient condition for a product variety to have a finite basis for its laws.
Hadamard designs
Authors: Wallis Jennifer
Published: April 2009

In this paper it is shown that an Hadamard design with each letter repeated once and only once can exist for 2, 4 and 8 letters only.

L.D. Baumert and Marshall Hall, Jr have found a design with four letters each repeated three times. Their design and the design on four letters each repeated once, found by J. Williamson, is the totality previously published.
On the extension of orders in ordered modules
Authors: Ribenboim P.
Published: April 2009

We introduce the notion of a positively independent set of elements in an ordered module. With this concept we determine a necessary and sufficient condition which insures that on a strictly ordered module over a strictly ordered ring there exists a strict total order refining the given order. This generalizes a previous result of Fuchs, concerning the case of ordered abelian groups.

As an application, let R be a strictly ordered totally ordered ring and let M be the R-module of all mappings from a set I into R, with pointwise order; then this order on M may be refined to a strict total order.
The weak closure of the set of singular elements in a Banach algebra
Authors: Gray J.D.
Published: April 2009

In this note it is proved that for a certain class of infinite dimensional Banach algebras the set of singular elements (the non-units) is dense in the weak topology.
A Hahn-Banach theorem for complex semifields
Authors: Anton Howard , Pervin W.J.
Published: April 2009

The following form of the Hahn-Banach theorem is proved: Let X be a linear space over the complex semifield E and let f: S → E be a linear functional defined on a subspace S of X. If p: X → RΔ is a seminorm with the property that ∣f(s)∣ ≪ p(s) for all s in S, then f has a linear extension F to X with the property that ∣F(x)∣ ≪ p(x) for all x in X.
A generalisation of Turán's main theorems to binomials and logarithms
Authors: van der Poorten A. J.
Published: April 2009

The Main Theorems of P. Turán's book Eine neue Methode in der Analysis und deren Anwendungen concern only sums of powers but are easily generalised to exponential sums with polynomial coefficients. It does not appear to have been observed however, that similar such theorems with analogous implication as to value distribution and arithmetical behaviour can be formulated for a wider class of functions. We prove a result for functions of the form subsuming identities which Mahler has shown to contain transcendence results on the exponential and logarithmic functions and diophantine results of the Thue-Siegel-Roth type.
Averaging and integral manifolds
Authors: Coppel W. A. , Palmer K. J.
Published: April 2009

An integral manifold for a system of differential equations is a manifold such that any solution of the equations which has a point on it is entirely contained on it. The method of averaging establishes the existence of such a manifold for a system which is a perturbation of an autonomous system with a periodic orbit. The existence of the manifold is established here under more general hypotheses, namely for perturbations which are ‘integrally small’. The method differs from the original method of Bogolyubov and Mitropolskii and operates directly with the individual solutions. This is made possibly by the use of an appropriate norm, and is equivalent to solving the partial differential equation which occurs in work by Moser and Sacker by the method of characteristics rather than by the introduction of an artificial viscosity term. Moreover, detailed smoothness properties of the manifold are obtained. For periodic perturbations the integral manifold is a torus and these smoothness properties are just sufficient to permit the application of Denjoy's theorem.
Extensions of finite nilpotent groups
Authors: Poland John
Published: April 2009

If G is a finite group and P is a group-theoretic property, G will be called P-max-core if for every maximal subgroup M of G, M/MG has property P where MG = ∩ is the core of M in G. In a joint paper with John D. Dixon and A.H. Rhemtulla, we showed that if p is an odd prime and G is (p-nilpotent)-max-core, then G is p-solvable, and then using the techniques of the theory of solvable groups, we characterized nilpotent-max-core groups as finite nilpotent-by-nilpotent groups. The proof of the first result used John G. Thompson's p-nilpotency criterion and hence required p > 2. In this paper I show that supersolvable-max-core groups (and hence (2-nilpotent)-max-core groups) need not be 2-solvable (that is, solvable). Also I generalize the second result, among others, and characterize (p-nilpotent)-max-core groups (for p an odd prime) as finite nilpotent-by-(p-nilpotent) groups.
The formation generated by a finite group
Authors: Bryant R. M. , Bryce R. A. , Hartley B.
Published: April 2009

We prove here that the (saturated) formation generated by a finite soluble group has only finitely many (saturated) subformations. This answers a question asked by Professor W. Gaschütz. Some partial results are also given in the case of a formation generated by an arbitrary finite group.
On invariant measures for simple branching processes (Summary)
Authors: Seneta E.
Published: April 2009

Problems pertaining to invariant measures of a non-critical Galton-Watson process, whether with or without Immigration, may be discussed in terms of measures of a subcritical process with a possibly defective immigration distribution. There is in fact only one such measure satisfying a regular variation condition. This result provides a unifying principle for several contexts of Galton-Watson theory. A full discussion with analytical details will appear elsewhere.
Near-rings in which each element is a power of itself
Authors: Bell Howard E.
Published: April 2009

Let R denote a near-ring such that for each x ∈ R, there exists an integer n(x) > 1 for which xn(x) = x. We show that the additive group of R is commutative if 0.x; = 0 for all x ∈ R and every non-trivial homomorphic image R¯ of R contains a non-zero idempotent e commuting multiplicatively with all elements of R¯. As the major consequence, we obtain the result that if R is distributively-generated, then R is a ring – a generalization of a recent theorem of Ligh on boolean near-rings.
On the structure of Stone lattices
Authors: Finch P. D.
Published: April 2009

C.C. Chen and G. Grätzer have shown that a Stone lattice is determined by a triple (C, D, ø) where C is a boolean algebra, D is a distributive lattice with 1 and ø is an e-homomorphism from C into D(D), the lattice of dual ideals of D.

It is shown here that any Stone lattice is, up to an isomorphism, a subdirect product of its centre C(L) and a special Stone lattice M(L). Special Stone lattices are characterised, in the terminology of the Chen-Grätzer triple, by the fact that the e-homomorphism Φ is one to one.

In this paper we characterise a special Stone lattice L as a triple (H, C, Do) where H is a distributive lattice with 0 and 1, C is a boolean e-subalgebra of the centre of H and Do is a sublattice of H with o such that
d ∈ Do & c ∈ C = d ∧ c ∈ Do, and which separates the elements of C in the sense that for any c1 ≠c2 in C there is a d in Do with d ≤ c1 but d ≰ C2. It then turns out that C is C(L) and Do is the dual of D(L).
Some arithmetical identities for Ramanujan's and divisor functions
Authors: Lahiri D.B.
Published: April 2009

A new linear expression in σ(ν), ν = l, 2, …, n, which vanishes identically is established. A linear expression in σ(ν)'s has been found for σ3(n). A similar expression in σ3(ν)'s has been proved for σ7(n) also, Ramanujan's τ(n) = P24(n-1) is given in three different ways as linear expressions in σ2k+1(n) and σK(ν)'s with k = 1, 3, 5 respectively. Again, the coefficient p48(n-2) is expressed as a linear expression in σ11(v)'S and σ5(ν)'s. In establishing these results advantage is taken of the general theorem, also established, that the coefficients of the square of a power series whose coefficients satisfy a certain functional equation are expressible as linear functions of the latter coefficients.
Topologies on finite groups
Authors: Morris Sidney A. , Thompson H.B.
Published: April 2009

It has been shown by D. Stephen that the number N of open sets in a non-discrete topology on a finite set with n elements is not greater than 3 × 2n-2.We show that for admissable topologies on a finite group N ≦ 2n/r, where r is the least order of its non-trivial normal subgroups. This is clearly a sharper bound.
On the lattice structure of quantum logic
Authors: Finch P.D.
Published: April 2009

A weak logical structure is defined as a set of boolean propositional logics in which one can define common operations of negation and implication. The set union of the boolean components of a weak logical structure is a logic of propositions which is an orthocomplemented poset, where orthocomplementation is interpreted as negation and the partial order as implication. It is shown that if one can define on this logic an operation of logical conjunction which has certain plausible properties, then the logic has the structure of an orthomodular lattice. Conversely, if the logic is an orthomodular lattice then the conjunction operation may be defined on it.
Ext(Q,Z) is the additive group of real numbers
Authors: Wiegold James
Published: April 2009

Standard homological methods and a theorem of Harrison on cotorsion groups are used to prove the result mentioned.

In this note Z denotes an infinite cyclic group, Q the additive group of rational numbers, Zp ∞ a p–quasicyclie group, and Ip the group of p–adic integers.

Pascual Llorente proves in [3] that Ext(Q,z) is an uncountable group, and gives explicitly a countably infinite subset. Very little extra effort produces the result embodied in the title, as follows.
A note on certain subsets of algebraic integers
Authors: Atterton T.W.
Published: April 2009

This paper is concerned with certain subsets of a finite extension K of the quotient field of an integral domain R. These subsets are contained in the integral closure of R in K and when R is integrally closed they are identical with it, but generally they need not even be rings. Various inclusion relations are studied and examples are given to show that these inclusions may be strict (with one exception which is still undecided).
Sums of finite-dimensional spaces
Authors: Wenner B.R.
Published: April 2009

Analogues are developed to the sum theorems in the dimension theory of metric spaces. It is shown that, within the class of metric spaces, any locally countable, σ-locally finite, or closure-preserving sum of finite-dimensional sets is countable-dimensional. Similar results are obtained under the more general hypothesis of countable-dimensional rather than finite-dimensional sets.
A class of densely invertible parabolic operator equations
Authors: Anderssen R.S.
Published: April 2009

Before variational methods can be applied to the solution of an initial boundary value problem for a parabolic differential equation, it is first necessary to derive an appropriate variational formulation for the problem. The required solution is then the function which minimises this variational formulation, and can be constructed using variational methods. Formulations for K-p.d. operators have been given by Petryshyn. Here, we show that a wide class of initial boundary value problems for parabolic differential equations can be related to operators which are densely invertible, and hence, K-p.d.; and develop a method which can be used to prove dense invertibility for an even wider class. In this way, the result of Adler on the non-existence of a functional for which the Euler-Lagrange equation is the simple parabolic is circumvented.
On boolean near-rings
Authors: Ligh Steve
Published: April 2009

It is well-known that a boolean ring is commutative. In this note we show that a distributively generated boolean near-ring is multiplicatively commutative, and therefore a ring. This is accomplished by using subdirect sum representations of near-rings.
The approximation problem for compact operators
Authors: Caradus S.R.
Published: April 2009

The following sufficient condition is obtained for the uniform approximability of compact operators on a reflexive Banach space by operators of finite rank: if S is the unit ball of X and ø: X* → C(S) is the imbedding ø(x*)x = x*(x) then we require ø(X*) to be complemented in C(S).
On certain abelian-by-nilpotent varieties
Authors: Brady J.M. , Bryce R.A. , Cossey John
Published: April 2009

We show that, whenever m, n are coprime, each subvariety of the abelian-by-nilpotent variety has a finite basis for its laws. We further Show that the just non-Cross subvarieties of are precisely those already known.
Some varieties without the amalgam embedding property
Authors: Meskin Stephen
Published: April 2009

A variety of groups has the amalgam embedding property if every amalgam of two -groups can be embedded in a -group. In this note the author proves that if is a variety of exponent O which satisfies a law W(x1, X2,…, xt) but not W(x1, x2,… xt) then does not have the amalgam embedding property.
Congruences on distributive pseudocomplemented lattices
Authors: Cornish William H.
Published: April 2009

Let B0 ⊂ B1 ⊂ Bn ⊂ … ⊂: Bw be all the non-trivial varieties of distributive pseudocomplemented lattices (L; ∨, ∧, *, 0, 1) considered as algebras of type (2, 2, 1, 0, 0). A subset J of such an algebra L is a congruence-kernel if and only if it is a lattice-ideal and x** ∈ J for each x ∈ J. The smallest congruence having J as its kernel is Θ(J), where a ≡ b (Θ(J)), (a, b ∈ L) if and only if a ∧ c* = b ∧ c* for some c ∈ J. For given 0 ≤ n ≤ w, let Σn (J) be the smallest congruence having J as its kernel and such that the associated quotient algebra is in Bn. Of course, Σw (J) = Θ(J) and the main result of this paper shows that for 1 ≤ n < w, Σn(J) = ∩{Θ(P1 ∩ P2 ∩ … ∩ Pn): J ⊆ P1, P2, …, Pn ⊂ L are minimal prime ideals}.
The classification of some spherically symmetric space-time metrics
Authors: Foyster J.M. , McIntosh C.B.G.
Published: April 2009

It is shown that the Petrov-Plebański classification of the trace-free Ricci tensors of some spherically symmetric metrics is invariant, contrary to an assertion by Takeno and Kitamura concerning these metrics.
The product of vector-valued measures
Authors: Swartz Charles
Published: April 2009

Let M (N) be a σ–algebra of subsets of a set S (T) and let X, Y be Banach spaces with (,) a continuous bilinear map from X × Y into the scalar field. If μ: M → X (v: N → Y) is a vector measure and λ is the scalar measure defined on the measurable rectangles A × B, A ∈ M, B ∈ N, by λ(A×B) = 〈μ(A), v(B)〉, it is known that λ is generally not countably additive on the algebra generated by the measurable rectangles and therefore has no countably additive extension to the σ-algebra generated by the measurable rectangles. If μ (v) is an indefinite Pettis integral it is shown that a necessary and sufficient condition that λ have a countable additive extension to the σ-algebra generated by the measurable rectangles is that the function F: (s, t) → 〈f(s), g(t)〉 is integrable with respect to α × β.
Twisted group rings and a problem of Faith
Authors: Cozzens John H.
Published: April 2009

A homological characterization is given of when a twisted group ring relative to an automorphism of an arbitrary field has all of its simple right modules injective (= a right V-ring). This answers a question raised by Osofsky. A “Hilbert Theorem 90” type theorem determines the cardinality of the isomorphism classes of one-dimensional simple modules.
Embedding near domains
Authors: Graves J.A. , Malone J.J.
Published: April 2009

A class of near rings which generalizes the class of integral domains is defined. The definition of near domain is derived from the desirability of embedding near domains in near fields. The near domains presented here are shown to contain the D-rings of Berman and Siverman.
The zero divisor question for supersolvable groups
Authors: Formanek Edward
Published: April 2009

Using a theorem of Jacques Lewin, the zero divisor question is solved for group rings of supersolvable groups.
The algebra of functions with Fourier transforms in a given function space
Authors: Tewari U.B. , Gupta A.K.
Published: April 2009

Let G be a locally compact abelian group and Ĝ be its dual group. For 1 ≤ p < ∞, let Ap (G) denote the set of all those functions in L1(G) whose Fourier transforms belong to Lp (Ĝ). Let M(Ap (G)) denote the set of all functions φ belonging to L∞(Ĝ) such that is Fourier transform of an L1-function on G whenever f belongs to Ap (G). For 1 ≤ p < q < ∞, we prove that Ap (G) Aq(G) provided G is nondiscrete. As an application of this result we prove that if G is an infinite compact abelian group and 1 ≤ p ≤ 4 then lp (Ĝ) M(Ap(G)), and if p > 4 then there exists ψ є lp (Ĝ) such that ψ does not belong to M(Ap (G)).
A useful relationship in the conformal mapping of quadrilaterals
Authors: Hughes O.F.
Published: April 2009

A relationship is presented which greatly simplifies the application of the Schwarz-Christoffel formula in mapping a quadrilateral having one axis of symmetry. The relationship has been verified numerically, and it is hoped that its evident simplicity and usefulness will stimulate the development of an analytical proof.
A generalization of a theorem of Aquaro
Authors: Aull C.E.
Published: April 2009

In this paper we introduce the concept of a δθ-cover to generalize Aquaro's Theorem that every point countable open cover of a topological space such that every discrete closed family of sets is countable has a countable subcover. A δθ-cover of a space X is defined to be a family of open sets where each Vn covers X and for x є X there exists n such that Vn is of countable order at x. We replace point countable open cover by a δθ-cover in Aquaro's Theorem and also generalize the result of Worrell and Wicke that a θ-refinable countably compact space is compact and Jones′ result that ℵ1-compact Moore space is Lindelöf which was used to prove his classic result that a normal separable Moore space is metrizable, using the continuum hypothesis.
Uniqueness and representation of a function in terms of its translated averages
Authors: Ching Chin-Hung
Published: April 2009

We give the representation formula of a periodic function with period one in terms of its translated means on the unit interval. We also give an application of this formula to a boundary value problem.
Metrization of spaces having Čech dimension zero
Authors: Broughan K.A.
Published: April 2009

A metrizable topological space has a metric taking values in a closed subset of the real numbers having Čech dimension zero if and only if the space itself has Čech dimension zero. We call a development D = {Dn} for a topological space (X, T) a sieve for X if the sets in each Dn are pairwise disjoint. Then a Hausdorff topological space (X, T) has a compatible metric taking values in a closed subset of the real numbers having Čech dimension zero if and only if there exists a sieve for X.
On rings with trivial torsion parts
Authors: Bican L. , Jambor P. , Kepka T. , Němec P.
Published: April 2009

In this paper, we exhibit the necessary and sufficient conditions for a ringRto have only the trivial torsion parts with respect to any (hereditary) radical on the category of leftR-modules.
Divergence of Fourier series: Corrigenda
Authors: Izumi Masako , Izumi Shin-ichi
Published: April 2009

The authors regret that the following corrections need to be made tc their paper [1].

p. 293, 1. 2 delete the factor ½.

p. 294, 1. 6 replace the denominator 2πl by πl.
Laws in finite strictly simple loops
Authors: Macdonald Sheila Oates
Published: April 2009

It is shown that a finite loop with no proper nontrivial subloops has a finite basis for its laws.
Completely semi-stable trees
Authors: Holton D.A.
Published: April 2009

Completely semi-stable trees are characterised by the complete absence of three types of subtrees.
Macroscopic descriptions of microscopic phenomena
Authors: Finch Peter D.
Published: April 2009

Some problems in the behavioural and physical sciences arise in the context of an incomplete knowledge of the fine detail of underlying practical situations. This paper presents a general mathematical framework for the discussion of such problems. This framework provides an algebraic language for the discussion of ecological analysis in the social sciences, aggregation in economics and macroscopic descriptions in statistical physics. Here, however, only the mathematical framework is presented; detailed applications will be presented elsewhere.
Characters for summary functions associated with cartesian products
Authors: Finch Peter D. , Heathcote Elizabeth
Published: April 2009

In deriving macroscopic descriptions of microscopic phenomena one often uses a vector valued summary function which is defined on a cartesian product in terms of component summary functions. We show that any character of such a summary function is the product of characters of its component summary functions.
An application of combinatorial techniques to a topological problem
Authors: Janos Ludvik
Published: April 2009

The following statement is proved: Let X be a set having at most continuously many elements and f: X → X a mapping such that each iteration fn (n = 1, 2, …) has a unique fixed point. Then for every number c ∈ (0, 1) there exists a metric p on X such that the metric space (X, p) is separable and the mapping f is a.contraction with the Lipschitz constant c.
Inverse subsemigroups of Rees matrix semigroups
Authors: Zitarelli David E.
Published: April 2009

According to the Sees Theorem, every completely 0-simple semigroup can be represented by a Rees matrix semigroup over a group with zero. A characterization of all subsemigroups of the latter is given in terms of the structure group, structure sets, and two mappings. Next all congruences on such subsemigroups are described, along with conditions for comparability. Finally, an algorithm for computing the number of nonisomorphic inverse subsemigroups is constructed.
On representations of selfmappings by contractions
Authors: Janos Ludvik
Published: April 2009

It is shown that any selfmapping (X, f) can be equivariantly and naturally embedded in the selfmapping of the form (X1, f1) ∪ (x2, f2) × (Y, g) where f1 and f2 are contractive relative to suitably chosen metrics and g is a bijection with all points periodic.
Approximative compactness and continuity of metric projections
Authors: Panda B.B. , Kapoor O.P.
Published: April 2009

In the paper “Some remarks on approximative compactness”, Rev. Roumaine Math. Pures Appl. 9 (1964), Ivan Singer proved that if K is an approximatively compact Chebyshev set in a metric space, then the metric projection onto K is continuous. The object of this paper is to show that though, in general, the continuity of the metric projection supported by a Chebyshev set does not imply that the set is approximatively compact, it is indeed so in a large class of Banach spaces, including the locally uniformly convex spaces. It is also proved that in such a space X the metric projection onto a Chebyshev set is continuous on a set dense in X.
Matrices of rational functions
Authors: Coppel W.A.
Published: April 2009

The properties of the degree of a matrix of rational functions are obtained in a simplified way, which enables them to be generalised to matrices whose elements are not necessarily rational functions. On the basis of these results a theory of realisations is developed, which similarly generalises the theory of state space realisations of a matrix of rational functions.
Soluble Lie algebras having finite-dimensional maximal subalgebras
Authors: Stewart Ian N.
Published: April 2009

Infinite-dimensional soluble Lie algebras can possess maximal subalgebras which are finite-dimensional. We give a fairly complete description of such algebras: over a field of prime characteristic they do not exist; over a field of zero characteristic then, modulo the core of the aforesaid maximal subalgebra, they are split extensions of an abelian minimal ideal by the maximal subalgebra. If the field is algebraically closed, or if the maximal subalgebra is supersoluble, then all finite-dimensional maximal subalgebras are conjugate under the group of automorphisms generated by exponentials of inner derivations by elements of the Fitting radical. An example is given to indicate the differences encountered in the insoluble case, and the nonexistence of group-theoretic analogues is briefly discussed.
Duals of Banach spaces which admit nontrivial smooth functions
Authors: John K. , Zizler V.
Published: April 2009

If a Banach space X admits a continuously Fréchet differentiable function with bounded nonempty support, then X* admits a projectional resolution of identity and a continuous linear one-to-one map into co(Γ).
Polynomials biorthogonal to Appell's polynomials
Authors: Fackerell Edward D. , Littler R.A.
Published: April 2009

We present the solution of a long-standing problem, namely, the determination of a set of polynomials in two independent variables which are biorthogonal over a triangular region to a set of polynomials previously introduced by Appell. Some elementary properties of our polynomials are investigated.
Collineation groups which are sharply transitive on an oval
Authors: Kirkpatrick P.B.
Published: April 2009

Let G be a group of collineations in a protective plane Π of order n. Suppose that one of the point orbits of G is an oval of Π, and that G acts regularly on this orbit. We prove that G fixes a non-incident point-line pair if either n is even, or n is odd and G is abelian, or n ≠ 11, 23, 59 is odd and is a pseudo-conic. It is then easy to deduce information about the lengths of the other orbits of G, and about the structure of G as an abstract group.
A Fritz John type sufficient optimality theorem in complex space
Authors: Gulati T.R.
Published: April 2009

A Fritz John type sufficient optimality theorem is proved for nonlinear programming problems in finite dimensional complex space over polyhedral cones, which may include equality as well as inequality constraints.
Concerning collectionwise Hausdorff spaces
Authors: Boone James R.
Published: April 2009

In this paper the notion of a space which has property (ω) pointwise is studied. The primary application of this concept is a reformulation of Tall's existence theorem for normal non-metrizable metacompact Moore spaces in terms of families which are finite on convergent sequences. Since the first countability in Tall's theorem yields an abundance of convergent sequences, this reformulation places the existence theorem in a natural setting.
A simple model for a weak system of arithmetic
Authors: Barnes D.W. , Monro G.P.
Published: April 2009

The natural, first order version of Peano's axioms (the theory T with 0, the successor function and an induction schema) is shown to possess the following nonstandard model: the natural numbers together with a collection of ‘infinite’ elements isomorphic to the integers. In fact, a complete list of the models of this theory is obtained by showing that T is equivalent to the apparently weaker theory with the induction axiom replaced by axioms stating that there are no finite cycles under the successor function and that 0 is the only non-successor.
Solution of a linear difference equation
Authors: Brown A.
Published: April 2009

A solution is given for un + 1 in terms of u1 and u0, where the elements of the sequence {un} satisfy the linear difference equation

Two linearly independent solutions of the equation are written as determinants and relations are given which can be used to check the evaluation of these determinants.
Convergence and applications of reproducing kernels for classes of discrete harmonic functions
Authors: Mastin C. Wayne
Published: April 2009

This paper gives convergence properties and applications of the discrete analogs of reproducing kernels for various families of harmonic functions. From these results information is obtained on the solution of interpolation problems, the convergence of the discrete Neumann's function, and the solution to problems involving the discrete biharmonic operator.
Torsionfree modules and classes of orders
Authors: Gustafson William H.
Published: April 2009

It is shown how torsionfree modules can be used to characterize certain important classes of orders over Dedekind rings. In particular, we show that an order is Gorenstein if and only if each of its lattices can be embedded as a pure sublattice of a free lattice. We also show that an order is hereditary if and only if the tensor product of any of its right lattices with any of its left lattices is torsionfree over the ground domain.
A characterization of left semiartinian rings
Authors: Golan Jonathan S.
Published: April 2009

In defining the torsion-theoretic Krull dimension of an associative ring R we make use of a function δ from the complete lattice of all subsets of the torsion-theoretic spectrum of R to the complete lattice of all hereditary torsion theories on R-mod. In this note we give necessary and sufficient conditions for δ to be injective, surjective, and bijective. In particular, δ is bijective if and only if R is a left semiartinian ring.
An algebraic characterization of finite symmetric tournaments
Authors: Berggren J.L.
Published: April 2009

A tournament T is called symmetric if its automorphism group is transitive on the points and arcs of T. The main result of this paper is that if T is a finite symmetric tournament then T is isomorphic to one of the quadratic residue tournaments formed on the points of a finite field GF(pn), pn ≡ 3 (4), by the following rule: If a, b ∈ GF(pn) then there is an are directed from a to b exactly when b – a is a non-zero quadratic residue in GF(pn).
A formula for accelerating the convergence of a general series
Authors: Drummond J.E.
Published: April 2009

A weighted average of the partial sums of a series provides a quick and moderately powerful sum for any series in which the ratio of successive terms varies slowly along the series and this ratio is not close to +1. Some properties of the sum are examined.
Schauder decompositions in non-separable Banach spaces
Authors: Chadwick J.J.M. , Cross R.W.
Published: April 2009

It is shown that Schauder decompositions exist in non-separable weakly compactly generated spaces and in certain non-separable conjugate spaces. Some results are obtained concerning shrinking and boundedly complete Schauder decompositions in non-separable spaces.
A non-cyclic, locally free, free-by-cyclic group all of whose finite factor groups are cyclic
Authors: Baumslag Gilbert
Published: April 2009

We construct a group G with the properties stated in the title.
On a problem in the theory of ordered groups
Authors: Fox Colin D.
Published: April 2009

The group G presented on two generators a, c with the single defining relation a−1c2a = c2a2c2 [proposed by B.H. Neumann in 1949 (unpublished), discussed by Gilbert Baumslag in Proc. Cambridge Philos. Soc. 55 (1959)] has been considered as a possible example of an orderable group which can not be embedded in a divisible orderable group, contrary to the conjecture that no such examples exist. It is known from Baumslag's discussion that G can not be embedded in any divisible orderable group. However, it is shown in this note that G is not orderable, and thus is not a counter-example to the conjecture.
On a cardinal equation in set theory
Authors: Hickman J.L.
Published: April 2009

We work in a Zermelo-Fraenkel set theory without the Axiom of Choice. In the appendix to his paper “Sur les ensembles finis”, Tarski proposed a finiteness criterion that we have called “C-finiteness”: a nonempty set is called “C-finite” if it cannot be partitioned into two blocks, each block being equivalent to the whole set. Despite the fact that this criterion can be shown to possess several features that are undesirable in a finiteness criterion, it has a fair amount of intrinsic interest. In Section 1 of this paper we look at a certain class of C-finite sets; in Section 2 we derive a few consequences from the negation of C-finiteness; and in Section 3 we show that not every C-infinite set necessarily possesses a linear ordering. Any unexplained notation is given in my paper, “Some definitions of finiteness”, Bull. Austral. Math. Soc. 5 (1971).
Error bounds in the approximation of functions: Corrigenda and an acknowledgement
Authors: Sahney Badri N. , Gopal Rao V. Venu
Published: April 2009

The following corrections should be made in the authors' paper [2]: Replace p. 12, line 4 by ‖f−Nn‖p, = 0(1/nα),

The author's theorems are contained as special cases in [1], Theorem 1.
Three-dimensional Stokes flow
Authors: Mahony J.J.
Published: April 2009

A method is present for simplifying the calculation of steady Stokes flows. It is applicable for domains which are essentially half spaces and utilizes a special vector stream potential to decouple the boundary conditions. The results are applied to estimate the effect of no-shear zones on a shear flow.
Asymptotic expansion of integrals occurring in linear wave theory
Authors: van den Driessche P. , Braddock R.D.
Published: April 2009

The asymptotic expansion of an integral of the type , is derived in terms of the large parameter t. Functions Φ(k) and ψ(k) are assumed analytic, and ψ(k) may have zeros at a stationary phase point. The usual one dimensional stationary phase and Airy integral terms are found as special cases of a more general result. The result is used to find the leading term of the asymptotic expansion of the double integral. A particular two dimensional Φ(k) relevant to surface water wave problems is considered in detail, and the order of magnitude of the integral is shown to depend on the nature of ψ(k) at the stationary phase point.
Elastic waves in a prestressed Mooney material
Authors: Belward J.A.
Published: April 2009

The dynamic response of a prestressed incompressible Mooney material is studied by investigating plane wave propagation and the response of the material to impulsive lines of force. The choice of an initial deformation which is axially symmetric gives a particularly simple form for the secular equation for the plane wavefront velocities. The speeds of propagation and the amplitudes of the two permissible transverse waves are found and necessary and sufficient conditions for there to exist two real wave speeds in all directions are established. The simple form of the secular equation enables the response of the material to concentrated disturbances to be readily solved using Fourier transforms. The motions caused by a line of impulsive forces is examined in some detail.
On the units of a modular group ring
Authors: Pearson K.R.
Published: April 2009

It is shown that a finite group G is a normal subgroup of the group of units of the group ring of G over the ring of integers modulo n if and only if G is abelian or n = 2 and G is isomorphic to the symmetric group on 3 letters.
Some classes of Hadamard matrices with constant diagonal
Authors: Wallis Jennifer , Whiteman Albert Leon
Published: April 2009

The concepts of circulant and backcirculant matrices are generalized to obtain incidence matrices of subsets of finite additive abelian groups. These results are then used to show the existence of skew-Hadamard matrices of order 8(4f+1) when f is odd and 8f + 1 is a prime power. This shows the existence of skew-Hadamard matrices of orders 296, 592, 1184, 1640, 2280, 2368 which were previously unknown.

A construction is given for regular symmetric Hadamard matrices with constant diagonal of order 4(2m + 1)2 when a symmetric conference matrix of order 4m + 2 exists and there are Szekeres difference sets, X and Y, of size m satisfying x є X ⇒ −xє X, y є Y ⇒ −y єY.
Maximal sum-free sets in abelian groups of order divisible by three: Corrigendum
Authors: Street Anne Penfold
Published: April 2009

The last step of the proof in [2] was omitted. To complete the argument, we proceed in the following way. We had shown that H = H(S) = H(S+S) = H(S-S), that |S-S| = 2|S| - |H| and hence that in the factor group G* = G/H of order 3m, the maximal sum-free set S* = S/H and its set of differences S* - S* are aperiodic, with

so that

By (1) and Theorem 2.1 of [1], S* - S* is either quasiperiodic or in arithmetic progression.
On three-Engel groups
Authors: Kappe L.-C. , Kappe W.P.
Published: April 2009

The following conditions for a group G are investigated:

(i) maximal class n subgroups are normal,

(ii) normal closures of elements have nilpotency class n at most,

(iii) normal closures are n–Engel groups,

(iv) G is an (n+1 )-Engel group.

Each of these conditions is a consequence of the preceding one. The second author has shown previously that all conditions are equivalent for n = 1. Here the question is settled for n = 2 as follows: conditions (ii), (iii) and (iv) are equivalent. The class of groups defined by (i) is not closed under homomorphisms, and hence (i) does not follow from the other conditions.
On varieties of soluble groups II
Authors: Groves J.R.J.
Published: April 2009

It is shown that, in a variety which does not contain all metabelian groups and is contained in a product of (finitely many) varieties each of which is soluble or locally finite, every group is an extension of a group of finite exponent by a nilpotent group by a group of finite exponent.
Bounded approximate identities and tensor products
Authors: Holub J.R.
Published: April 2009

The question has been raised [R.J. Loy, Bull. Austral. Math. Soc. 5 (1970), 253–260] as to whether the existence of a bounded (left) approximate identity in the tensor product A ⊗αB of Banach algebras A and B (for a a crossnorm on A ⊗ B ) implies the existence of a bounded (left) approximate identity in A and B. This is known [David A. Robbins, Bull. Austral. Math. Soc. 6 (1972), 443–445] to be the case for α equal to the greatest crossnorm. This paper answers the general question affirmatively.
Presentations of metacyclic groups
Authors: King Bruce W.
Published: April 2009

Each metacyclic p–group has a natural canonical presentation which is easily derived from the usual presentation. Parameters in the canonical presentation measure how far the group is from splitting, and from being either commutative or dihedral. The structure of the groups is discussed.
A note on generalized Hall planes
Authors: Johnson N.L.
Published: April 2009

We prove that if π is a generalized Hall plane of odd order with associated Baer subplane π0 then π is a Hall plane if and only if there is a collineation σ of π such that π0σ ∩ π0 is an affine point.