bluuebunny/crossref_metadata_2025 huggingface dataset:
Monotone Homomorphisms of Compact Semigroups
Authors: Clark C. E.
Published: April 2009

The problem of determining the class of homomorphic images of a given class of topological semigroups seems to have received little attention in the literature. In [4] Cohen and Krule determined the homomorphic images of a semigroup with zero on an interval. Anderson and Hunter in [1] proved several theorems in this direction. In general, the problem seems to be rather difficult. However, the difficulty is lessened somewhat if all of the homomorphisms of the semigroups in question must be monotone. Phillips, [7], showed that every homomorphism of a standard thread is monotone and hence every homomorphic image of a standard thread is either a standard thread or a point. In this paper a larger class of topological semigroups which admit only monotone homomorphisms is given. These results are used to determine the topological nature of the homomorphic images of certain classes of topological semigroups. These include products of standard threads with min threads, certain semilattices on a two-cell, and compact connected lattices in the plane.
Some Uniqueness Theorems for Functional Equations
Authors: Howroyd T. D.
Published: April 2009

The generalized Pexider equation where f and g are unknown and x, y, are real, has been discussed by J. Aczél [1] and J. Aczé and M. Hosszú [2]. In [2] it is shown that if F is continuous and F and H are strictly increasing in their first variables and strictly decreasing in their second variables, then two initial conditions suffice to determine at most one continuous solution f of (1). We extend these results to strictly increasing and strictly decreasing functions F and derive results for strictly monotonic F and H.
A Note on the Approximation of Continuous Functions by Riesz Typical Means of Their Fourier Series
Authors: Kwee B.
Published: April 2009

In [1], the following theorem is proved:THEOREM. If f ∈C2π, α is a positive integer, and then Where , .
The Completion of a Lattice Ordered Group
Authors: Conrad Paul , McAlister Donald
Published: April 2009

A lattice ordered group(‘l-group’) is calledcompleteif each set of elements that is bounded above has a least upper bound (and dually). A completel-group is archimedean and hence abelian, and each archimedeanl-group has a completion in the sense of the following theorem.
A Double-Infinity Configuration
Authors: Barnes D. W.
Published: April 2009

The double-six configuration in classical 3-dimensional projective geometry has been discussed by a number of authors. It consists of two sets a1,…, a6 and b1, …, b6 of six lines such that no two lines of the same set intersect, and ai meets bi if and only if i †j. The existence of a doublesix in the 3-dimensional projective geometry over a field F has been proved by Hirschfeld in [2] for all fields F except those of 2, 3 and 5 elements. For an arbitrary 3-dimensional projective geometry in which the number of points on a line is at least 5 but is not 6, the existence of a double-six follows from the fact that the geometry is a geometry over a division ring D with a subfield F satisfying the conditions of Hirschfeld's theorem.
Idempotent-Separating Extensions Of Inverse Semigroups
Authors: D'Alarcao H.
Published: April 2009

Extensions of semigroups have been studied from two points of view; ideal extensions and Schreier extension. In this paper another type of extension is considered for the class of inverse semigroups. The main result (Theorem 2) is stated in the form of the classical treatment of Schreier extensions (see e.g.[7]). The motivation for the definition of idempotentseparating extension comes primarily from G. B. Preston's concept of a normal set of subsets of a semigroup [6]. The characterization of such extensions is applied to give another description of bisimple inverse ω-semigroups, which were first described by N. R. Reilly [8]. The main tool used in the proof of Theorem 2 is Preston's characterization of congruences on an inverse semigroup [5]. For the standard terminology used, the reader is referred to [1].
A Note on Separation Axioms Weaker than T1
Authors: Robinson S. M. , Wu Y. C.
Published: April 2009

In [1], Aull and Thron introduce several separation axioms between T0 and T1. In particular, they define a space X to be TD if for each x ∈ X, {x}' is a closed set.
The Terminal Behaviour of a Uniformly Shortening Pendulum
Authors: Brearley M. N.
Published: April 2009

In this paper, necessary and sufficient conditions are established for the continuation of small oscillations of a simple pendulum as its string length is decreased uniformly to zero, and the accompanying terminal value of the angular displacement is determined.
On Finite Soluble Groups
Authors: Ward J. N.
Published: April 2009

Let p be a class of finite soluble groups which is closed under epimorphic images and let g be a saturated formation. Then if G is a group of minimal order belonging to p but not to g, F(G), the Fitting subgroup of G, is the unique minimal normal subgroup of G. It is to groups with this property that the following proposition is applicable.
Representing varieties of algebras by algebras
Authors: Neumann Walter D.
Published: April 2009

In this paper we describe a way of representing varieties of algebras by algebras. That is, to each variety of algebras we assign an algebra of a certain type, such that two varieties are rationallv equivalent if and only if the assigned algebras are isomorphic.
Second nilpotent BFC groups
Authors: Bride Iain. M.
Published: April 2009

The BFC number of a group G is defined to be the least upper bound n of the cardinals of the conjugacy classes of G, provided this is finite, and we then say that G is n-BFC. It was shown by B. H. Neumann [2] that the derived group G′ of such a group is finite, and J. Wiegold [5] proved that.This bound was sharpened by I. D. Macdonald [1] to, and P. M. Neumann has recently communicated the (unpublished) result that G′ ≦ nq(n) with q(n) a quadratic in log2w, an immense improvement on the above. J. A. H. Shepperd and J. Wiegold [4] improved the bound in two special cases, showing that if G is soluble, G′ ≦ np(n) with p(n) a quintic in Iog2n, and that if G is nilpotent of class 2, , It is conjectured that for any n-BFC group G, , Wiegold [5] having shown that this bound is attained by certain nilpotent groups of class 2.
An improved bound for BFCp-groups
Authors: Neumann Peter M.
Published: April 2009

The theme of this paper is a conjecture whose origins go back to work of B. H. Neumann ([5], Theorem 3·1) and his former students I. Wiegold and I. D. Macdonald ([7], [4]). B. H. Neumann proved that if there is a bound to the sizes of conjugacy classes in the group G. that is. if G is a BFC group. then the derived group G′ is finite; Wiegold. and later Macdonald. produced explicit upper bounds for G′ in terms of the maximum n of the sizes of the conjugacy classes in G. The coniecture is that where l(n) should be the arithmetic function1 λ(n) for a best possible bound, or l(n) may be interpreted as log2n for a smooth, monotonicorderof-magnitude estimate. However, the bounds produced in [7] and [4], and even the vastly better upper bounds proved bv Sheooerd and Wieeold [6] for soluble groups, are very much bigger than , and it is mv aim in this paper to take a first step towards closing the gap. This first, step is study of BFC p-groups: in a sequel I hope to show how the results proved here can be used to obtain improved bounds for arbitrary BFC groups.
Generating groups for nilpotent varieties
Authors: Levin Frank
Published: April 2009

Let ℜc denote the variety of all nilpotent groups of class ≦ c, that is, ℜc is the class of all groups satisfying the law, where we define, as usual, and, inductively, . Further, let Fk(ℜc) denote a free group of ℜe of rank k. In her book Hanna Neumann ([4], Problem 14) poses the following problem: Determine d(c), the least k such that Fk(ℜc) generates ℜc. Further, she suggests, incorrectly, that d(c) = [c/2] + l. However, as we shall prove here, the correct answer is d(c) = c—1, for c ≦ 3. 2 More generally, we shall prove the following result.
A note on extending partial automorphisms of abelian groups
Authors: Chehata C. G. , Shawky A.
Published: April 2009

Given a group G and a partial automorphism μ of G, i.e. an isomorphism mapping a subgroup A of G onto another subgroup B of G, then it is known [3] that μ can always be extended to a total automorphism, in fact an inner one, of a supergroup of G; that is there exists a group G* ⊇ G with an inner automorphism μ* whose effect on the elements of A is the same as that of μ. Also any number of partial automorphisms μσ, where a ranges over some index set Σ can be simultaneously extended to inner automorphisms of one and the same group [3, Theorem II].
Optimal quadrature formulae and minimal monosplines in Lq
Authors: Kautsky J.
Published: April 2009

The quadrature formula of order m using values of derivatives up to the m — 1st order with the best possible bound in is derived. Using certain properties of the polynomials minimal in Lq norm, it is proved that the optimal formula not use the derivatives of m — 1st order if m is even.
On orthomodular posets
Authors: Finch P. D.
Published: April 2009

Let S be a poset with a greatest element 1. We denote order in S by ‘≦’ and, whenever they exist in S, l.u.b and g.l.b by ‘∨’ and ‘∧’ respectively. An orthocomplementation of S is a bijection w: S → S such that x ∨ xω exists for each x in S and (i) xωω = x, (ii) x ≦ y implies yω ≦ xω and (iii) x ∨xω = 1. If a poset S admits an orthocomplementation ω we call the pair (S, ω) an orthoposet.
A remark about a certain class of distribution spaces
Authors: Harasymiv S. R.
Published: April 2009

The object of this note is to exhibit a certain class of distribution spaces as being c-admissible in the sense of [1] and [2]. Throughout the terminology and notation are the same as in [1] and [2]. The only addition to this is that if x ∈ Rn then τx will denote the translation operator which carries each distribution u onto the distribution ux.
A method for solving problems of elastic plates of arbitrary shape
Authors: Mazumdar J.
Published: April 2009

The methods developed up to now in the theory of plates do not lend themselves easily to the solution of problems in the case of plates of arbitrary shape. Such methods lend themselves only to the solution of problems for plates of certain special simple shapes. Moreover, because of the complicated mathematical tools involved in these methods, the methods are not suitable for engineers. Hence there arises the need for the formulation of an approximate theory which may appear quite suitable for attacking problems on plates of arbitrary shape. In the present paper the author develops a new method which aims at the solution of the problem quite independently of the shape of the plate.
On the continuity of projections
Authors: Gemignani M. C.
Published: April 2009

Throughout this note X will be a topological space with geometry G of length m–1 with . The terminology will be that of [1].
On the behaviour of the characteristic function of a probability distribution in the neighbourhood of the origin
Authors: Pitman E. J. G.
Published: April 2009

Let X be a real valued random variable with probability measure P and distribution function F. It will be convenient to take F as the intermediate distribution function defined by . In mathematical analysis it is a little more convenient to use this function rather than , which arise more naturally in probability theory. In all cases we shall consider With this definition, if the distribution function of X is F(x), then the distribution function of −X is 1−F(−x). The distribution of X is symmetrical about 0 if F(x) = 1 − F(−x).
On the number of varieties of groups
Authors: Kovács L. G.
Published: April 2009

There are infinitely, but at most continuously, many varieties of groups; the precise cardinal is unknown. It is easy to see that if there is no infinite properly descending chain of varieties (equivalently, if the laws of every variety have a finite basis), then the number of varieties is countable infinity; the converse implication does not seem to have been proved. This note presents an argument which implies that if the locally finite or the locally nilpotent varieties fail to satisfy the minimum condition, then there are continuously many such varieties. Alternatively, one can conclude that if a locally finite or locally nilpotent variety has a finite basis for its laws but some subvariety of has none, then there are continuously many varieties between and . This points again to the interesting question: is every locally finite [locally nilpotent] variety contained in a suitable locally finite [locally nilpotent] variety which has a finite basis for its laws? (That is, must be locally finite [locally nilpotent] for some finite n?) For, if the answer were affirmative, it would follow that the number of locally finite [locally nilpotent] varieties is either countable or the cardinal of the continuum, depending exactly on the existence of finite bases for the laws of such varieties.
A formula for the resolvent of a Reynolds operator
Authors: Miller J. B.
Published: April 2009

Let be a complex Banach algebra, possibly non-commutative, with identity e. By a Reynolds operator we mean here a bounded linear operator T: → satisfying the Reynolds identity for all x, y ∈ . We prove that under certain conditions the resolvent of T, R(p, T) = (pI−T)−1, has the form where s = −log(e−Te) and exp y = e+y+y2/2!+….
Abstract Kernels and Cohomology
Authors: Swierczkowski S.
Published: April 2009

Let G, N be groups, let A(N) be the automorphism group of N and let I(N) be the subgroup of inner automorphisms. A homomorphism θ: G → A(N)/I(N) will be denoted by (G, N, θ) and called an abstract kernel. (G, N, θ) induces in an obvious manner a structure of a (left) G-module on the centre C of N. A well known construction of Eilenberg and MacLane [1, § 7–9] assigns to (G, N, θ) its obstruction Obs (G, N, θ) ∈H3(G, C). This assignment is such that if C is an arbitrary G-module then every element of H3(G, C) is of the form Obs (G, N, θ) for a suitable abstract kernel (G, N, θ).
Solution of Schrödinger equation by Laplace transform
Authors: Englefield M. J.
Published: April 2009

Laplace transform techniques for solving differential equations do not seem to have been directly applied to the Schrödinger equation in quantum mechanics. This may be because the Laplace transform of a wave function, in contrast to the Fourier transform, has no direct physical significance. However, this paper will show that scattering phase shifts and bound state energies can be determined from the singularities of the Laplace transform of the wave function. The Laplace transform method can thereby simplify calculations if the potential allows a straightforward solution of the transformed Schrödinger equation. Suitable cases are the Coulomb, oscillator and exponential potentials and the Yamaguchi separable non-local potential.
The automata theory of semigroup embeddings
Authors: Arbib Michael
Published: April 2009

The theorem of Trevor Evans [1] that every countable semigroup can be embedded in a two-generator semigroup becomes obvious in automata theory as the statement that every countable automaton can be embedded in one with binary inputs. Standard techniques of automata theory [1], [3] yield a proof of the Evans Theorem using wreath products, as in Neumann [4].
Fundamental solutions for a class of equations with several singular coefficients
Authors: Weinacht R. J.
Published: April 2009

This paper is concerned with a class of singular elliptic partial differential equations related to the operator of Weinstein's generalized axially symmetric potential theory (GASPT) [1, 2] which has numerous applications.
A class of multipliers
Authors: Edwards R. E.
Published: April 2009

Throughout this paper G denotes an infinite compact connected Hausdorff Abelian group with character group X. Given a map of α X into itself, we are concerned with the set of a, ∈ G such that the function ϕa ∈ l∞ (X) defined by is a multiplier of type (þ, þ), where it can be assumed without loss of generality that 1 ≤ þ < ∞.
Commutative semigroup amalgams
Authors: Howie J. M.
Published: April 2009

In the terminology of J. R. Isbell [5], an element d of a semigroup S is dominated by a subsemigroup U of S if, for an arbitrary semigroup X and arbitrary homomorphisms α β, from S into X, α(u) = β(u) for every u in U implies α(d) = β(d). The set of elements of S dominated by U is a subsemigroup of S containing U and is called the dominion of U. It was shown by Isbell that if one takes two disjoint isomorphic copies S+, S− of S and forms their amalgamated free product S+ * US− that is to say, the quotient of the free product S+ * S− by the congruence p generated by , (u+ and u− being the images of u in S+, S− respectively) then the homomorphisms: μ+: S→ S+ * uS−, μ−: S→ S+ * uS− defined by are one-one. Moreover, μ+(s) = μ−(t) only if s = t, and μ+(s)= μ−(s) if and only if s is in the dominion of U. In other words, the two natural copies of S in S+ * uS− intersect precisely in the dominion of U. Thus, in particular, and in the terminology of [3], the amalgam [S+, S−; U] is embeddable if and only if U is self-dominating (that is to say, its own dominion) in S.
The H-function transform
Authors: Gupta K. C. , Mittal P. K.
Published: April 2009

Here we introduce a new integral transform whose kernel is the H-function. Since most of the important functions occurring in Applied Mathematics and Physics are special cases of the H-function, various integral transforms involving these functions as kernels follow as special cases of our transform. We mention some of them here and observe that a study of this transform gives general and useful results which serve as key formulae for several important integral transforms viz. Laplace transform, Hankel transform. Stieltjes transform and the various generalizations of these transforms. In the end we establish an inversion formula for the new transform and point out its special cases which are generalizations of results found recently.
The laminar free-convection boundary layer on a vertical heated plate in the neighbourhood of a discontinuity in plate temperature
Authors: Smith R. K.
Published: April 2009

If velocity and temperature profiles are known at a particular distance along a vertical heated plate, the equations of motion determine the velocity and temperature at points downstream, for a given varition of plate temperature. The problem of continuing the boundary layer solution for given initial conditions was investigated by Goldstein [2], for the isothermal case of the laminar, incompressible flow past plate, with a given streamwise varitain of pressure gradient outside the boundary layer. He showed that the solution is not always free from singularities and developed an expansion procedure to calculate the flow downstream when these occurred. Typical singularities occur, for instance, near the leading edge of the plate where the no-slip condition is imposed on the plate surface and near the trailing edge, where this condition is relaxed to one of zero stress along the axis of symmetry of the wake.
The uniqueness of bounded or measurable solutions of some functional equations
Authors: Howroyd T. D.
Published: April 2009

In this paper we are concerned with the uniqueness of solutions of functional equations of the form Some conditions for (1) or (2) to have at most one real continuous solution f which satisfies two given initial conditions are contained in [2], [3], [4] and [7]. Conditions sufficient for the equation to determine at most one continuous solution f with values in a Banach algebra are contained in [5]. It is well known (see [1] ch. 2) that one initial condition suffices for Cauchy's equation or two for Jensen's equation to uniquely determine a real solution f which is bounded on an interval or majorized on a set of positive measure by a measurable function. We place conditions on F and H so that similar statements can be made about solutions of (1) or (2). The corresponding results for solutions which are functions of many real variables follow as for Cauchy's and Jensen's equations (see [1] ch. 5).
On a result of Cassels
Authors: Worley R. T.
Published: April 2009

Let α be an irrational algebraic number of degree k over the rationals. Let K denote the field generated by α over the rationals and let a denote the ideal denominator of α. Then Cassels [3] has shown that for sufficiently large integral N > 0 distinctly more than half the integers n, are such that (n+α)a is divisible by a prime ideal pn which does not divide (m+α)a for any integer m ≠ n satisfying . The purpose of this note is to point out that minor modification of Cassel's proof enables the extension of the interval for n from to , and to derive results on the proportion of values n, for which the values f(n) of a given integral polynomial in n are divisible by a prime p > N.
A characterization of the group Aut(PGL(3, 4))
Authors: Taylor D. E.
Published: April 2009

In a recent paper, Z. Janko [5] announced the discovery of two new finite non-abelian simple groups and characterized these groups in terms of the centralizer of an involution. In fact, he proved the following result.
Cubic identity graphs and planar graphs derived from trees
Authors: Balaban A. T. , Davies Roy O. , Harary Frank , Hill Anthony , Westwick Roy
Published: April 2009

The smallest (nontrivial) identity graph is known to have six points and the smallest identity tree seven. It is now shown that the smallest cubic identity graphs have 12 points and that exactly two of them are planar, namely those constructed by Frucht in his proof that every finite group is isomorphic to the automorphism group of some cubic graph. Both of these graphs can be obtained from plane trees by joining consecutive endpoints; it is shown that when applied to identity trees this construction leads to identity graphs except in certain specified cases. In appendices all connected cubic graphs with 10 points or fewer, and all cubic nonseparable planar graphs with 12 points, are displayed.
Uniform O-estimates of certain error functions connected with k-free integers
Authors: Suryanarayana D.
Published: April 2009

Let k be a fixed integer ≧ 2. A positive integer m is called k-free if m is not divisible by the k'th power of any integer > 1. Let qk(m) be the characteristic function of the set of k-free integers; that is, qk(m) = 1 or 0 according as m is k-free or not. It can be easily shown that where μ(n) is the Mobius function. Let x ≧ 1 denote a real variable and n be a positive integer. Let Qk(x, n) and be the number and the sum of the reciprocals of the k-free integers ≦ x which are prime to n respectively.
The serial correlation coefficients of waiting times in a stationary single server queue
Authors: Daley D. J.
Published: April 2009

In a stationaryGI/G/1 queueing system in which the waiting time variance is finite, it can be shown that the serial correlation coefficients {ρn} of a (stationary) sequence of waiting times are non-negative and decrease monotonically to zero. By means of renewal theory we find a representation for Σ∞0ρnfrom which necessary and sufficient condition for its finiteness can be found. InM/G/1 rather more can be said: {ρn} is convex sequence, the asymptotic form of ρncan be given in a nearly saturated queue, and a simple explicit expression for Σ∞0ρnexists. For the stationaryM/M/1queue we find the ρn's explicitly, illustrate them numerically, and derive a representation which shows that {ρn} is completely monotonic.
Maximal connected topologies
Authors: Thomas J. Pelham
Published: April 2009

If (X, ) is a set X with topology we shall say that is connected if (X, ) is a connected topological space. We shall investigate the existence of and the properties of maximal connected topologies.
A note on commutative semigroups
Authors: Speed T. P.
Published: April 2009

In 1962, O. Frink [2] showed that in a pseudo-complemented semilattice 〈P; ∧, *, 0〉, the closed elements form a Boolean algebra. We shall consider an extension of this result to arbitrary commutative semigroups with zero.
Non-linear weak shock diffraction
Authors: De Mestre N. J.
Published: April 2009

Perturbation expansions are sought for the flow variables associated with the diffraction of a plane weak shock wave around convex-angled corners in a polytropic, inviscid, thermally-nonconducting gas. Lighthill's method of strained co-ordinates [4] produces a uniformly valid expansion for most of the diffracted front, while the remainder of this front is treated by a modification of the shock-ray theory of Whitham [6]. The solutions from these approaches are patched just inside the ‘shadow’ region yielding a plausible description of the entire diffracted shock front.
The Morse inequalities for line bundles
Authors: Khabbaz Samir
Published: April 2009

In place of a real valued differentiable (C2) function on a closedn-dimensional differentiable manifoldM, we may more generally consider a differentiable section s in any line bundleLonM, assumed to have structural groupZ2, the group of integers modulo two. Since the usual definitions of a critical point and of a non-degenerate critical point are local in nature, and since composing a differentiable real valued function with the functiont→—t does not change its set of critical points or its set of non-degenerate critical point, it is clear that we may speak of critical points and nondegenerate critical points of the section s. Unless the bundle has a fixed trivialization however, the index of a non-degenerate critical point must be thought of as a set of two numbers {k, n—k), corresponding to the two indices arising from the two trivializations possible forLrestricted to a small enough neighborhood of the point, i.e. corresponding to the two possible ways of reading the index. With this understanding we extend the usual definitions, and call a differentiable (C2) section s of L a Morse section if each of its critical points is non-degenerate.
Composable topological properties and semigroups of relations
Authors: Magill Kenneth D.
Published: April 2009

It is assumed throughout this paper that all topological spaces under consideration are Hausdorff. Since the notion of topological property is fundamental in this paper, we begin by making it precise. For our purposes here, it is sufficient to think of a topological property Q as being a class of space such that if X ∈ Q and Y is homeomorphic to X, then Y∈Q. To say that a space X has property Q would then be equivalent to saying that X ∈ Q.
Positive linear operators and summability
Authors: King J. P. , Swetits J. J.
Published: April 2009

Let {Ln} be a sequence of positive linear operators defined on C[a, b] of the form where xnk ∈ [a, b] for each k = 0, 1,…, n = 1, 2,…. The convergence properties of the sequences {Ln(f)} to for each f ∈ C[a, b] have been the object of much recent research (see e.g. [4], [8], [11], [13]). In many cases positive linear operators of the form (1) give rise to interesting summability matrices A = (ank(x)) and vice- versa.
Some results on finiteness of radical algebras
Authors: Mirbagheri Ahmad
Published: April 2009

R denotes always a radical algebra over a field φ. A left ring ideal of R which is also a subvector space over φ is called a left algebra ideal of R. R is said to be left algebra noetherian if it satisfies the ascending chain condition for left algebra ideals. If dim R < ∞, then
(i) R is finitely generated
(ii) R is left alehra noetherian
(iii) R is algebraic.
Since the radical of an algebraic algebra is nil ([4] P. 19), conditions (i), (ii), (iii) are also sufficient for R to be finite-dimensional.
Cutting down random trees
Authors: Meir A. , Moon J. W.
Published: April 2009

Let Tn denote a tree with n(≧ 2) labelled points: we assume Tn is rooted at a given point x, say the point labelled 1 (see [3] for definitions not given here). If we remove some edge e of Tn, then Tn falls into two subtrees one of which, say Tk, contains the root x. If k ≧ 2 we can remove some edge of Tk and obtain an even smaller subtree of Tn that contains x. If we repeat this process we will eventually obtain the subtree consisting of x itself. Let λ = λ(Tn) denote the number of edges removed from Tn before the root x is isolated. Our main object here is to determine the expected value μ(n) and variance σ2(n) of λ(Tn) under the assumptions (1) Tn is chosen at random from the set of nn−2 trees with n labelled points that are rooted at point x, and (2) at each stage the edge removed is chosen at random from the edges of the remaining subree containing x. It follows from our results that μ(n) ~ (½πn)½ and (2−½π)n ~ (2−½π)n as n tends to infinity. We also consider the corresponding problem for forests of rooted trees and for trees in which the degree of the root is specified. We are indebted to Professor Alistair Lachlan for suggesting the original problem to us.
Quasi-normal matrices and products
Authors: Wiegmann N. A.
Published: April 2009

A normal matrix A = (aij) with complex elements is a matrix such that AACT = ACTA where ACT denotes the (complex) conjugate transpose of A. In an article by K. Morita [2] a quasi-normal matrix is defined to be a complex matrix A which is such that AACT = ATAC, where T denotes the transpose of A and AC the matrix in which each element is replaced by its conjugate, and certain basic properties of such a matrix are developed there. (Some doubt might exist concerning the use of ‘quasi’ since this class of matrices does not contain normal matrices as a sub-class; however, in deference to the original paper and the normal canonical form of Theorem 1 below, the terminology in [2] is used.)
On decomposable varieties of groups
Authors: Cossey John
Published: April 2009

We shall be concerned with problem 7 of Hanna Neumann [7], which asks: Prove or disprove that if U and B are varieties, and neither of U and B is contained in the other, then U ∪ B and [U, B] are decomposable if and only if U and B have a common non-trivial right hand factor.
The radical of the algebra of any finite semigroup over any field
Authors: Hall T. E.
Published: April 2009

For a finite semigroup S and a field Φ, denote by Φ [S] the semigroup algebra of S over Φ and when S has a zero element, denote by Φ0[S] the contracted semigroup algebra of S over Φ (see § 5.2 [1]). Then theorem 5.31 [1], due to E.Hewitt and H. S. Zuckerman, gives a determination of the radical of Φ0[S] when S is commutative and the characteristic of Φ does not divide the order of any subgroup of S; and on page 168 [1] some results concerning the radical of Φ0[S] when S is completely 0-simple are given, the determination of the radical in the general case remaining open.
The convolutional approach to the estimation of spectral maxima directly from the autocovariance function
Authors: Hutchens Helen , Keats R. G.
Published: April 2009

It is common practice for research workers in a great number of widely differing fields, to gather vast amounts of experimental data (see [1 – 3], for example). These data are then analysed, using various statistical or other techniques, in an attempt to obtain information concerning the phenomenon being studied. One important source of such information is evidence of oscillations in the data collected. Various techniques for revealing frequencies of oscillations are available (see [4], for example); they usually involve some form of data processing. In many cases the collected data are records of one or more realisations of a stationary stochastic process, and the frequencies of interest appear as local maxima or peaks in the spectrum of the process [5]. In this paper a method for determining the position of the maximum spectral value directly from the autocovariance function is presented and discussed.
Zero divisors and finite near-rings
Authors: Ligh S. , Malone J. J.
Published: April 2009

A near-ring is a triple (R, +, · such that (R, +) is a group, (R, ·) is a semigroup, and is left distributive work on near-rings is [1]. A near-ring R is distributively generated if there exists S ⊂ R such that (S, ·) is sub-semigroup of (R, ·), each element of S is right distributive, and S is an additive generating set for (R, +). Distributively generated near-rings, first treated in [3], arise out of consideration of the system generated by the endomorphisms of an (not necessarily commutative) additive group. A near-field is a near-ring such that the nozero elements form a group under multiplication. Near fields are discussed in [9]. An element x ≠ 0 in R is a left (right) zero divisor if there is a ≠ 0 in R such that xa = 0 (ax = 0). A zero divisor is an element that is either a left or a right zero divisor. In a near-ring R it will be assumed that Ox = 0 for each x ∈ R.
A non-existence theorem for (v, k,λ)-graphs
Authors: Wallis W. D.
Published: April 2009

A (ν, κ, λ)-graph is defined in [3] as a graph on ν points, each of valency κ, and such that for any two points P and Q there are exactly λ points which are joined to both. In other words, if Si is the set of points joined to Pi, then

Si has k elements for any i

SiSj has λ elements if i≠j

The sets Si are the blocks of a (v, k, λ)-configuration, so a necessary condition on v, k, and λ that a graph should exist is that a (v, k, λ)- configuration should exist. Another necessary condition, reported by Bose (see [1]) and others, is that there should be an integer m satisfying have equal parity. We shall prove that these conditions are not sufficient.
The descending chain condition in join-continuous modular lattices
Authors: Kovács L. G.
Published: April 2009

If L is a distributive lattice in which every element is the join of finitely many join-irreducible elements, and if the set of join-irreducible elements of L satisfies the descending chain condition, then L satisfies the descending chain condition: this follows easily from the results of Chapter VIII, Section 2, in the Third (New) Edition of Garrett Birkhoff's ‘Lattice Theory’ (Amer. Math. Soc., Providence, 1967). Certain investigations (M. S. Brooks, R. A. Bryce, unpublished) on the lattice of all subvarieties of some variety of algebraic systems require a similar result without the assumption of distributivity. Such a lattice is always join-continuous: that is, it is complete and (∧X) ∨ y = ∧ {x ∨ y: x ∈ X} whenever X is a chain in the lattice (for, the dual of such a lattice is complete and ‘algebraic’, in Birkhoff's terminology). The purpose of this note is to present the result:
Varieties and finite groups
Authors: Kovács L. G.
Published: April 2009

The theory of discrete, abstract groups, as presented in current texts, consists of investigations of various special classes of groups: it has very few completely general results. For some classes (say, for finite groups) the investigations have been extensive and successful; in a few cases (say, for finitely generated abelian groups) they have even reached a sense of completeness. The choice of some of these classes was dictated by the needs of other branches of mathematics. Many more were introduced with the view of extending the scope of certain powerful but special results, and a large part of the literature is taken up by elaborate counterexamples which mark the limits of these generalizations. In so far as one is looking for some kind of classification theory, it is immediately evident that the classes investigated were chosen by historical accident rather than by any master plan, and so far do not appear to form the initial part of a pattern which could be enlarged and completed in the future.
A Hahn-Banach theorem for semifields
Authors: Kleiber Martin , Pervin W. J.
Published: April 2009

Iseki and Kasahara (see [3]) have given a Hahn-Banach type theorem for semifield-valued linear functionals on real linear spaces. We shall generalize their result by considering linear spaces over semifields.
A Characterization of the finite simple group U4(3)
Authors: Phan Kok-Wee
Published: April 2009

The aim of this paper is to give a characterization of the finite simple group U4(3) i.e. the 4-dimensional projective special unitary group over the field of 9 elements. More precisely, we shall prove the following result.
On the finite similarity groups
Authors: Lorimer Peter
Published: April 2009

A result contained in a previous paper [1] of the author is Theorem 1. If (G, K, H) is a T3-triple, G is finite, a is an involution contained in G–N(K) and H ∩ Ka=1, then the factor group of G over its centre is isomorphic to a group of similarities over a finite field.
On permutational products of groups
Authors: Gregorac R. J.
Published: April 2009

We here study permutational products of groups, a study which was initiated by B. H. Neumann [8], [9].
On conformally recurrent spaces of second order
Authors: Chaki M. C. , Chowdhury A. N. Roy
Published: April 2009

In a recent paper [1] Adati and Miyazawa studied conformally recurrent spaces, that is, Riemannian spaces defined by where is the conformal curvature tensor: λi is a non-zero vector and comma denotes covariant differentiation with respect to the metric tensor gij.
An elementary proof of Minkowski's second inequality
Authors: Danicic I.
Published: April 2009

Let K be an open convex domain in n-dimensional Euclidean space, symmetric about the origin O, and of finite Jordan content (volume) V. With K are associated n positive constants λ1, λ2,…,λn, the ‘successive minima of K’ and n linearly independent lattice points (points with integer coordinates) P1, P2, …, Pn (not necessarily unique) such that all lattice points in the body λ,K are linearly dependent on P1, P2, …, Pj-1. The points P1,…, Pj lie in λK provided that λ > λj. For j = 1 this means that λ1K contains no lattice point other than the origin. Obviously
A note on Baker's method
Authors: Ramachandra K.
Published: April 2009

Let …, αn, (n ≧ 2) be (fixed) multiplicatively independent non zero algebraic numbers and set M(H) = min |β1log α1+…+βn log αn| the minimum taken over all algebraic numbers bgr;1,…βn not all equal to zero, of degrees not exceeding a fixed natural number d0, and heights not exceeding an arbitrary natural number H. Then an important result [1] of Baker states that for every fixed ε > 0 and an explicit constant . It may be remarked that Baker deduces his general result from the special case where βn is fixed to be —1. The following straight forward generalization might be of some interest since it shows that the exponent n+1+ε need not be the best, and that the best exponent obtainable by his method has some chance of being 1 + ε (see the corollary to the Theorem).
A note on the intersection of free maximal ideals
Authors: Robinson Stewart M.
Published: April 2009

In [2] we proved that if X admits a complete uniform structure, the intersection of the free maximal ideals in C(x) is precisely Ck(X), the ring of functions with compact support. In the present paper we are able to sharpen this result somewhat and give necessary and sufficient conditions on a space X so that this conclusion holds. Both our previous result and that of Kohls for p-spaces follow as special cases of our theorem.
On Koenigs' ratios for iterates of real functions
Authors: Seneta E.
Published: April 2009

In a recent note, M. Kuczama [5] has obtained a general result concerning real solutions φ(x) on the interval 0 ≦x<a≦∞ of the Schröder functional equationproviding the known real function satisfies the following (quite weak) conditions:f(x)is continuous and strictly increasing in ([0 a);(0)= 0 and 0 <f(x)<x for x ∈ (0, a); limx→0+{f(x)/x} = s; and f(x)/x is monotonic in (0, a).
An expansion related to the central limit theorem
Authors: Heathcote C. R.
Published: April 2009

Let X1, X2,…be independent and identically distributed non-lattice random variables with zero, varianceσ2<∞, and partial sums Sn = X1+X2+…+X.
Bounds for the Asymptotic Growth Rate of an Age-Dependent Branching Process
Authors: Brockwell P. J.
Published: April 2009

Let M(t) denote the mean population size at time t (conditional on a single ancestor of age zero at time zero) of a branching process in which the distribution of the lifetime T of an individual is given by Pr {T≦t} =G(t), and in which each individual gives rise (at death) to an expected number A of offspring (1λ A λ ∞). expected number A of offspring (1 < A ∞). Then it is well-known (Harris [1], p. 143) that, provided G(O+)-G(O-) 0 and G is not a lattice distribution, M(t) is given asymptotically by where c is the unique positive value of p satisfying the equation .
H-extension of ring
Authors: Lin Bertrand I-Peng
Published: April 2009

A ring R is called an H-ring if for every x ∈ R there exists an interger n=n(x)> 1 such that xn-x ∈C, where C is center of R. I. N. Herstein proved that H-rings must be commutative [See 3 pp. 220–221]. We now introduce the following definition.
Formations of Π-soluble groups
Authors: Lausch H.
Published: April 2009

The theory of formations of soluble groups, developed by Gaschütz [4], Carter and Hawkes[1], provides fairly general methods for investigating canonical full conjugate sets of subgroups in finite, soluble groups. Those methods, however, cannot be applied to the class of all finite groups, since strong use was made of the Theorem of Galois on primitive soluble groups. Nevertheless, there is a possiblity to extend the results of the above mentioned papers to the case of Π-soluble groups as defined by Čunihin [2]. A finite group G is called Π-soluble, if, for a given set it of primes, the indices of a composition series of G are either primes belonging to It or they are not divisible by any prime of Π In this paper, we shall frequently use the following result of Čunihin [2]: Ift is a non-empty set of primes, Π′ its complement in the set of all primes, and G is a Π-soluble group, then there always exist Hall Π-subgroups and Hall ′-subgroups, constituting single conjugate sets of subgroups of G respectively, each It-subgroup of G contained in a Hall Π-subgroup of G where each ′-subgroup of G is contained in a Hall Π′-subgroup of G. All groups considered in this paper are assumed to be finite and Π-soluble. A Hall Π-subgroup of a group G will be denoted by G.
The Riemann surface of a ring
Authors: Stanley M. G.
Published: April 2009

The contravariant functor F from the category of Riemann surfaces and analytic mappings to the category of complex algebras and homomorphisms which takes each surface Ω to the algebra of analytic functions on Ω does not have an adjoint on the right; but it nearly does. To each algebra A there is associated a surface Σ1 (A) and a homomorphism A from A into FΣ1 (A), indeed onto an algebra of functions not all of which are constant on any component of Σ1 (A), such that every such non-trivial representation A A → F(Ω) is induced by a unique analytic mapping Ω → Σ1(A)
On Stone Lattices
Authors: Speed T. P.
Published: April 2009

M. H. Stone raised the problem ([1] Problem 70) of characterising the class of distributive pseudo-complemented lattices ℒ = 〈L; ∨, ∧, 0, 1〉 in which a* ∨ a** = 1 holds identically. Several solutions to this problem have now been offered — the first being by G. Grätzer and E. T. Schmidt [6], who gave this class of lattices the name Stone lattices. Later solutions were given by J. Varlet [11], O. Frink [4] and G. Grätzer [5]; see also G. Bruns [2].
Some Remarks on Complete Integral Closure
Authors: Heinzer William
Published: April 2009

This paper continues an investigation of the complete integral closure of an integral domain which was begun in [2]. We recall that if D is an integral domain with quotient field K then an element x of K is said to be almost integral over D if there exists a nonzero element y of D such that yxn is an element of D for each positive integer n. The set D* of elements of K almost integral over D is called the complete integral closure of D and D is said to be completely integrally closed if D* = D.
The Lengthening Pendulum
Authors: Werner A. , Eliezer C. J.
Published: April 2009

Some recent papers have revived interest in some questions concerning the motion of a simple pendulum which is oscillating with small angular amplitude under gravity, when the length of the pendulum changes with time in some prescribed manner.
Some Results on ω-Derivatives and BV-ω Functions
Authors: Chakrabarty M. C.
Published: April 2009

Let ω(x) be non-decreasing on the closed interval [a, b]. Outside the interval ω(x) is defined by ω(x) = ω(a) for x < a and ω(x) = ω(b) for x > b. Let S denote the set of points of continuity of ω(x) and D denote the set of points of discontinuity of ω(x). R. L. Jeffery [5] has defined the class U, of functions f(x) as follows:
A Theorem on Partially Ordered Sets and its Application
Authors: Devidé Vladimir
Published: April 2009

Let (S, ≦) be a (non-void) partially ordered set with the property that for every (non-void) chain C (i.e., every totally ordered subset) of S, there exists in S the element sup C. Let SM be the set of all maximal elements s of S. ƒ:S/SM→S be a slowly increasing mapping in the sense that
A Restricted Inhomogeneous Minimum for Forms
Authors: Blanksby P. E.
Published: April 2009

Let us suppose that ƒ(x, y) is an indefinite binary quadratic form that does not represent zero. If P is the real point (x0, y0) then we may define where the infimum is taken over all integral x, y. The inhomogeneous minimum of the form ƒ is defined where the supremum taken over all real points P, need only extend over some complete set of points, incongruent mod 1.
On an Inequality of Bombieri
Authors: Peaslee D. C. , Coppel W. A.
Published: April 2009

Bieberbach's conjecture, proposed in 1916 and still unsolved, states that if ƒ(z) = z+a2z2+… is holomorphic and univalent in the disc ∣z∣ < 1 then ∣an∣ ≦ n for each n ≧ 2, with equality for some n only if ƒ(z) is the Koebe function of is obtained from this function by a rotation. Very recently Bombieri has succeeded in showing that if ƒ(z) is sufficiently close to the Koebe function, then with equality only if ƒ(z) = k(z). This had previously been proved by Garabedian, Ross and Schiffer [3] for even values of n.
Witt's Theorem for Symplectic Modular Forms
Authors: James D. G.
Published: April 2009

LetLdenote a freeZ-module of rank 2nand Φ an alternating bilinear mapping fromL×Linto the rational integersZ. Writing α · β for Φ(α, β), where α, β ∈L, we haveWe shall assume that Φ is non-singular and unimodular (see Bourbaki [1]).Lis now a (symplectic)lattice.
Multipliers between Some Spaces of Distributions
Authors: Price J. F.
Published: April 2009

By the systematic use of Fourier transforms and suitable weight functions L. R. Volevich and B. P. Paneyakh brought many classes of spaces of distributions (including the Sobolev spaces) and their topological duals under the one unifying definition. The main purpose here is to demonstrate that the representation of multipliers between pairs of these spaces (that is, continuous linear operators from one space into another which commute with translations) may be related to the representation of multipliers betweenLpandLq.
Some Restricted Partition Functions: Congruences Modulo 5
Authors: Lahiri D. B.
Published: April 2009

Ramanujan was the first mathematician to discover some of the arithmetical properties of p(n), the number of unrestricted partitions of n. His congruence,

for example, is famous [2; 3]. Some progress has been made since then; it is known that the congruence,

has an infinitude of solutions for any arbitrary value of r [4]. This is a somewhat weak relation, and one would have liked to obtain, if possible, stronger results of the type,

for ‘almost all’ values of n, which in its turn is derivable from another stronger relation, viz.,

also established by Ramanujan [2], where r(n) is Ramanujan's function defined by
On the Complete Continuity of Differentiable Mappings
Authors: Palmer K. J.
Published: April 2009

Let E1 and E2 be real Banach spaces. We denote by E12 the Banach space of all bounded linear operators mapping E1 into E2 with the usual operator norm. By Dr we denote the open ball of radius r, centred at the origin.