bluuebunny/crossref_metadata_2025 huggingface dataset:
The Abstract Group G3,7,16: A Correction
Authors: Coxeter H. S. M.
Published: January 2009

Dr. A. D. Sands has pointed out that the following sentence, occurring in (2, p. 60, lines 2 to 4), should be deleted: “Hence Sinkov's group of order 1344 … is the holomorph of the Abelian group {pi}, of order 8 (1, pp. 111–117).” In fact, Sinkov's group and the holomorph of C2×C2×C2 are not isomorphic. For, the holomorph is representable on 8 letters by definition (1, p. 87), whereas Sinkov's group is not representable on 8 letters. To see this, we recall that Sinkov's group (3, p. 584) is generated by two elements of periods 2 and 3 (namely, QP3 and QP2) whose commutator is of period 8. If these two generators could be represented as permutations of 8 letters, their commutator would be an even permutation and thus could not be of period 8.
On bounded sequences satisfying a linear inequality
Authors: Russell Dennis C.
Published: January 2009

In a recent paper, E. T. Copson (2) proves the following result:

Theorem C.Letki >0 (i = 1, …, m), k1 +…+ km = 1, and the real sequence (an) satisfy the inequality

If (an) is bounded, then it must be convergent.
Passman-Zalesskii Radical of group algebras
Authors: Sinha I.
Published: January 2009

Recently Passman (attributing the origin of the idea to Zalesskii) has defined the following ideal in a ring, (2).

Definition. For a unitary ring R,

N * R = {α ∈ R | αS is nilpotent for all finitely generated subrings S of R}.

For a group algebra KG over a field K of characteristic p ≠ 0, he has proved the radical property:
Minimal cohesive basic sets
Authors: Goldsmith Donald L.
Published: January 2009

A basic set (formerly basic sequence) ℬ is a set of pairs (a, b) of positive integers satisfying

(1) if (a, b) ∈ ℬ, then (b, a) ∈ ℬ,

(2) (a, bc) ∈ ℬ if and only if (a, b) ∈ ℬ and (a, c) ∈ ℬ,

(3) (1, k ∈ ℬ, k = 1, 2, ….

Some familiar examples of basic sets are

, where Sk = {(1, k), (k, 1)},

= {(a, b)| a and b are relatively prime positive integers},

ℒ = {(a, b)| a and b are any positive integers}.
The word problem for lattice ordered groups
Authors: Glass A. M. W.
Published: January 2009

The purpose of this note is to show the existence of a lattice ordered group G with a finite set of generators and a recursively enumerable set of defining relations such that there is no decision procedure to determine whether or not an arbitrary word in the generators reduces to the identity in G. In addition to the usual group-theoretic words, we may also use the two lattice operations ∨ and ∧ ; for example, a−l(b∨c) is a word in the generators a, b and c. At first sight it might appear that since we have an even greater harvest of words than in group theory and there exist finitely presented groups H (H has a finite number of generators and defining relations) with an insoluble word problem (no decision procedure to determine whether an arbitrary word in the generators reduces to the identity)—see (1), (2), (4) or (6)—the same would be true of lattice ordered groups. Unfortunately, such a naïve approach overlooks two salient points. First, the class of lattice ordered groups is strictly smaller than the class of all groups; second, there are certain relations connecting the lattice operations with the group operations which hold true for all lattice ordered groups. For example, a(b∧c) = ab∨ac and (a∨b)−1 = a−1∧b−1.
Spectra of irreducible matrices
Authors: Minc Henryk
Published: January 2009

A real matrix is called non-negative (positive) if all its entries are non-negative (positive). Two matrices A and B are said to be cogredient if there exists a permutation matrix Q such that QAQT = B. A square non-negative matrix is called reducible if it is cogredient to a matrix of the form

where the blocks X and Y are square. Otherwise it is called irreducible.
The radical equation P(An) = (P(A))n
Authors: Propes R. E.
Published: January 2009

The purpose of this paper is to impose conditions on a radical class P so that the P-radical of the ring of n × n-matrices over a ring A is equal to the ring of n×n-matrices over the ring P(A). In (1), Amitsur gave such conditions, but with the stipulation that the radical class P contained all zero-rings (rings in which all products are zero). In what follows, we shall be working within the class of associative rings.
On the Stepanov-almost periodic solution of a second-order operator differential equation
Authors: Rao Aribindi Satyanarayan
Published: January 2009

Suppose X is a Banach space and J is the interval −∞<t<∞. For 1 ≦ p<∞, a function is said to be Stepanov-bounded or Sp-bounded on J if

(for the definitions of almost periodicity and Sp-almost periodicity, see Amerio-Prouse (1, pp. 3 and 77).
Solution of hypergeometric integral equations involving generalised functions
Authors: McBride Adam C.
Published: January 2009

In a previous paper (9), we introduced the spaces Fp, μ of testing-functions and the corresponding spaces of generalised functions. For 1≦p≦∞, Fp(=Fp, 0) is the linear space of all complex-valued measurable functions φ defined on (0, ∞) which are infinitely differentiable on (0, ∞) and for which

for each k = 0, 1, 2, …. In symbols,
Boolean algebras of projections
Authors: Spain P. G.
Published: January 2009

Bade, in (1), studied Boolean algebras of projections on Banach spaces and showed that a σ-complete Boolean algebra of projections on a Banach space enjoys properties formally similar to those of a Boolean algebra of projections on Hilbert space. (His exposition is reproduced in (7: XVII).) Edwards and Ionescu Tulcea showed that the weakly closed algebra generated by a σ-complete Boolean algebra of projections can be represented as a von Neumann algebra; and that the representation isomorphism can be chosen to be norm, weakly, and strongly bicontinuous on bounded sets (8): Bade's results were then seen to follow from their Hilbert space counterparts. I show here that it is natural to relax the condition of σ-completeness to weak relative compactness; indeed, a Boolean algebra of projections has σ-completion if and only if it is weakly relatively compact (Theorem 1). Then, following the derivation of the theorem of Edwards and Ionescu Tulcea from the Vidav characterisation of abstract C*-algebras (see (9)), I give a result (Theorem 2) which, with its corollary, includes (1: 2.7, 2.8, 2.9, 2.10, 3.2, 3.3, 4.5).
On groups which are residually -commutative
Authors: Arrell David G.
Published: January 2009

In this paper we study a property which we call residual -commutativity. This idea was kindled by a paper of Ayoub (1) and one of Stanley (6). Durbin has defined a similar property in (2) which has also been studied by Slotterbeck in (4). Durbin's property implies residual -commutativity but we have not been able to decide if they are equivalent. However, we have shown that they coincide in certain circumstances. The notation, unless otherwise stated, is that of Robinson (3).
A distributional generalised Stieltjes transformation
Authors: Pathak R. S.
Published: January 2009

The generalised Stieltjes transform of a function f(t) ∈ L(0, ∞) is defined by

The integral converges for complex s in the region Ω, where Ω is the s-plane cut from the origin along the negative real axis.
On a singular measure of D. M. Connolly and J. H. Williamson
Authors: Ludvik P. M.
Published: January 2009

In (1) a measure λ∈M(ℝ) is constructed and shown to satisfy the following:

(i) is absolutely continuous, where denotes the measure with for all Borel sets E,

(ii) λ*λ = λ2 is singular,

(iii) λk = λk−1*λ is absolutely continuous for K ≧6.
Approximations to the norm of the singular part of a measure
Authors: Pigno Louis
Published: January 2009

Let G be a non-discrete LCA group with dual group Γ. Denote by M(G) the usual convolution algebra of bounded Borel measures on G and Ma(G) those μ ∈ M(G) which are absolutely continuous with respect to mG—the Haar measure on G.
An extension of a Hardy-Littlewood-Pólya inequality
Authors: Erdélyi A.
Published: January 2009

The Hardy-Littlewood-Pólya inequality in question can be written in the form

Here and throughout, all functions are assumed to be locally integrable on ]0,∞[, 1≤p≤∞,p-1+(p′)-1=1 (with similar conventions for q,r,s), is the usual norm on Lp(0,∞), and if the right hand side is finite, then (1.1) is understood to mean that

defines a locally integrable function Kf for which (1.1) holds.
The lattice theoretic part of topological separation properties
Authors: Simmons H.
Published: January 2009

In this paper we show that for each n∈{2,3,4,5,} the topological separationproperty Tn can be decomposed

where C, N2,…,Nn are purely lattice theoretic properties with the expected im-plications holding between them.
Finitely Embedded Modules Over Group Rings
Authors: Smith P. F.
Published: January 2009

Let R be a ring and X a right R-module (all rings have identities and all modulesare unitary). The intersection of all non-zero submodules of X is denoted by μ(X). The module X is called monolithic if and only if μ(X)≠0 and in this case μ(X) is anessential simple submodule of X. (Recall that a submodule Y of X is essential if and only if Y ⊂ A ≠ 0 for every non-zero submodule A of X.) It is well known that a module X is monolithic if and only if there is a simple right R-module U such that X is a submodule of the injective hull E(U) of U. If x is a non-zero element of an arbitrary right. R-module X then by Zorn's Lemma there is a submodule Yx of X maximal with the property x ∉ Yx. It can easily be checked that X/Yx is monolithic and ⊂ Yx = 0, where the intersection is taken over all non-zero elements x of X.
The integral formula for the reduced algebraic multiplicity of meromorphic operator functions
Authors: Bart H. , Kaashoek M. A. , Lay D. C.
Published: January 2009

Throughout this paper A denotes an operator function, holomorphic on a deletedneighborhood of a complex number λo, with values in the space ℒ(X,Y) of boundedlinear operators between two complex Banach spaces X and Y. In his survey article(7), I. C. Gohberg has defined for such an arbitrary operator function A the algebraic multiplicity RM(A;λo) and the reduced algebraic multiplicity RM(A;λo) of A at λo. In earlier papers (e.g., (8, 16)) these notions have been defined and studied for morerestricted classes of operator functions. In (8) Gohberg and Sigal treated the case when A is finite-meromorphic at λo, A(λ) is bijective for λ in some deleted neighbor-hood of λo and the constant term A0 in the Laurent expansion of A at λo is aFredholm operator. They proved that in this case
Rigid Rings
Authors: Maxson C. J.
Published: January 2009

If R is a ring with identity and E is the semigroup of identity preserving ring endomorphisms of R then the fixed ring of E is denned by for each φɛE}. R is said to be rigid if RE = R.Equivalently, R is rigid if and only if
Jordan Subalgebras of Banach algebras
Authors: Bonsall F. F.
Published: January 2009

We recall that a JC-algebra (Størmer (3)) is a norm closed Jordan algebra of self-adjoint operators on a Hilbert space. Recently, Alfsen, Shultz, and Størmer (1) have introduced a class of abstract normed Jordan algebras called JB-algebras, and have proved that every special JB-algebra is isometrically isomorphic to a JC-algebra. We show that this result brings to a satisfactory conclusion the discussion in (2) of certain wedges W in Banach algebras and their related Jordan algebras W–W, and leads to two characterisations of the bicontinuously isomorphic images of JC-algebras.
The structure mappings on a regular semigroup
Authors: Meakin John
Published: January 2009

In (5) the author showed how to construct all inverse semigroups from their trace and semilattice of idempotents: the construction is by means of a family of mappings between ℛ-classes of the semigroup which we refer to as the structure mappings of the semigroup. In (7) (see also (8) and (9)) K. S. S. Nambooripad has adopted a similar approach to the structure of regular semigroups: he shows how to construct regular semigroups from their trace and biordered set of idempotents by means of a family of mappings between ℛ-classes and between ℒ-classes of the semigroup which we again refer to as the structure mappings of the semigroup. In the present paper we aim to provide a simpler set of axioms characterising the structure mappings on a regular semigroup than the axioms (R1)-(R7) of Nambooripad (9). Two major differences occur between Nambooripad's approach (9) and the approach adopted here: first, we consider the set of idempotents of our semigroups to be equipped with a partial regular band structure (in the sense of Clifford (3)) rather than a biorder structure, and second, we shall enlarge the set of structure mappings used by Nambooripad.
Adequate Semigroups
Authors: Fountain John
Published: January 2009

A monoid in which every principal right ideal is projective is called a right PP monoid. Special classes of such monoids have been investigated in (2), (3), (4) and (8). There is a well-known internal characterisation of right PP monoids using the relation ℒ* which is defined as follows. On a semigroup S, (a,b) ∈ℒ* if and only if the elements a,b of S are related by Green's relation ℒ* in some oversemigroup of S. Then a monoid S is a right PP monoid if and only if each ℒ*-class of S contains an idempotent. The existence of an identity element is not relevant for the internal characterisation and in this paper we study some classes of semigroups whose idempotents commute and in which each ℒ*-class contains an idempotent. We call such a semigroup a right adequate semigroup since it contains a sufficient supply of suitable idempotents. Dually we may define the relation ℛ* on a semigroup and the notion of a left adequate semigroup. A semigroup which is both left and right adequate will be called an adequate semigroup.
A De Rham Theorem for Generalised Manifolds
Authors: Verona Maria Elena
Published: January 2009

In (2) Bruhat has developed a theory of differentiable functions and distributions on a locally compact group in order to apply it to the study of the irreducible representations of the p-adic groups. Later, Whyburn (8) defined differentiable forms on a locally compact group and proved an analog of the de Rham theorem concerningthe relationship between the Čech cohomology and the De Rham cohomology. In (4) Ihave introduced the notions of “generalised manifold” (roughly speaking a projective limit of smooth manifolds) and of “differentiable forms” on it, extending some of the results due to Bruhat and Whyburn.
Hermitian Operators on Banach Jordan Algebras
Authors: Youngson M. A.
Published: January 2009

In this note, we examine some of the properties of Hermitian operators on complex unital Banach Jordan algebras, that is, those operators with real numerical range. Recall that a unital Banach Jordan algebra J, is a (real or complex) Jordan algebra with product a ˚ b, having a unit 1, and a norm ∥·∥, such that J, with norm ∥·∥, is a Banach space, ∥1∥ = 1, and, for all a and b in j,
Hypercentral injectors in infinite soluble groups
Authors: Tomkinson M. J.
Published: January 2009

The Carter subgroups of a finite soluble group may be characterised either as theself-normalising nilpotent subgroups or as the nilpotent projectors. Subgroups with properties analogous to both of these have been considered by Newell (2, 3) in the class of -groups. The results obtained are necessarily less satisfactory than in the finite case, the subgroups either being almost self-normalising (i.e. having finite index in their normaliser) or having an almost-covering property. Also the subgroups are not necessarily conjugate but lie in finitely many conjugacy classes.
Dual integral equations with a trigonometric Kernel
Authors: Singh Brij M. , Dhaliwal Ranjit S.
Published: January 2009

In this paper, we solve the following dual integral equations

where δ is a real positive constant and f(x) is a continuous and integrable function of x in [0, a]. The dual integral equations (1) and (2) arise in a crack problem of elasticity.
A representation for the reproducing kernel for elliptic systems
Authors: Ahner John F. , Hamilton Eugene P.
Published: January 2009

The kernel function method of Bergman and Schiffer (see (1) and (2)) has recently been used by Colton and Gilbert (see (3) and (4)) in connection with approximation theory and the numerical treatment of elliptic differential equations. In (5) Gilbert and Weinacht have successfully extended the kernel function method to elliptic systems of differential equations. Essential to their work is the concept of a matrix kernel satisfying the reproducing property. This reproducing kernel is defined initially as the difference of the Neumann matrix and the Dirichlet matrix. Thus actually to obtain the kernel matrix from this definition one has to solve both a Neumann problem and a Dirichlet problem. In view of this restriction, Gilbert and Weinacht derive an ingenious representation for the reproducing kernel in terms of purely geometric quantities which are obtained directly from the fundamental matrix for the differential system.
Involution near-rings
Authors: Scott S. D.
Published: January 2009

Throughout this paper all near-rings considered will be zero-symmetric and left distributive. All groups will be written additively, but this does not imply commutativity. The near-ring of all zero-fixing maps of a group V into itself will be denoted by Mo(V). If N is a near-ring withan identity and α ≠ 1 is an element of N such that α2 = 1, then α will be called an involution of N. Let V be a group. An involution a of Mo(V) will be called an involution on V.
On Baire-hyperplane spaces
Authors: Valdivia M.
Published: January 2009

In this article we prove that in every infinite dimensional separable Fréchet space there is a dense barrelled subspace which is not the inductive limit of Bairehyperplane spaces.
Positive functionals and oscillation criteria for second order differential systems
Authors: Etgen Garret J. , Lewis Roger T.
Published: January 2009

Let ℋ be a Hilbert space, let ℬ = (ℋ, ℋ) be the B*-algebra of bounded linear operators from ℋ to ℋ with the uniform operator topology, and let ℒ be the subset of ℬ consisting of the self-adjoint operators. This article is concerned with the second order self-adjoint differential equation
A comparison of mean concentrations in a diffusion problem
Authors: Brown A.
Published: January 2009

In some biological problems, a parasite-host system is immersed in a toxic solution in order to kill off the parasite while leaving the host as little affected as possible. A problem of this type was considered by Clements and Edelstein (2), who treated both the host and the parasite as cylindrical in shape. In a separate paper Clements (1) considered the corresponding problem where the parasite is spherical and the host cylindrical. In both cases, the concentration of the toxic solution at the boundary is taken as having a constant value, c, and the penetration of the poison into the host and parasite is treated as a linear diffusion problem with an appropriate diffusion coefficient. It is assumed also that the host and the parasite are free of the toxic substance initially. The process is terminated when the average concentration in the parasite reaches a lethal level, τ, and the problem is to see how M, the average concentration in the host, is affected by the choice of c (for a given value of τ).
On a generalisation of a result of Ramanujan connected with the exponential series
Authors: Paris R. B.
Published: January 2009

One of the many interesting problems discussed by Ramanujan is an approximation related to the exponential series for en, when n assumes large positive integer values. If the number θn is defined by

Ramanujan (9) showed that when n is large, θn possesses the asymptotic expansion

The first demonstrations that θn lies between ½ and and that θn decreases monotoni-cally to the value as n increases, were given by Szegö (12) and Watson (13). Analogous results were shown to exist for the function e−n, for positive integer values of n, by Copson (4). If φn is defined by

then πn lies between 1 and ½ and tends monotonically to the value ½ as n increases, with the asymptotic expansion

A generalisation of these results was considered by Buckholtz (2) who defined, in a slightly different notation, for complex z and positive integer n, the function φn(z) by
Subdirect products of rings and distributive lattices
Authors: Bandelt Hans-J. , Petrich Mario
Published: January 2009

Rings and distributive lattices can both be considered as semirings with commutative regular addition. Within this framework we can consider subdirect products of rings and distributive lattices. We may also require that the semirings with these restrictions are regarded as algebras with two binary operations and the unary operation of additive inversion (within the additive subgroup of the semiring). We can also consider distributive lattices with the two binary operations and the identity mapping as the unary operation. This makes it possible to speak of the join of ring varieties and distributive lattices. We restrict the ring varieties in order that their join with distributive lattices consist only of subdirect products. In certain cases these subdirect products can be obtained via a general construction of semirings by means of rings and distributive lattices.
On the Chern classes of the regular representations of some finite groups
Authors: Mann Benjamin M. , Milgram R. James
Published: January 2009

In studying the cohomology of the symmetric groups and its applications in topology one is led to certain questions concerning the representation rings of special subgroups of . In this note we calculate the Chern classes of the regular representation of (Z/p)n where p is a fixed odd prime in terms of certain modular invariants first described by L. E. Dickson in 1911. In a later paper [9] we apply these results to study the odd primary torsion in the PL cobordism ring. Some indications of this application are given in Sections 10–12 where we apply the result above to obtain information about the cohomology of . After circulation of this note in preprint form we learned that H. Mui [10], has also proved Theorem 6.2.
The first hundred years (1883–1983)
Authors: Rankin R. A.
Published: January 2009

This account of our Society is based to some extent on my Presidential address, which was given on 19 October 1977 and was devoted to the first fifty years.

In the latter half of the nineteenth century there was an upsurge of interest in mathematics that resulted in the foundation of a number of mathematical societies in different countries. The London Mathematical Society (1865), the Moscow Mathematical Society (1867), the Société Mathématique de France (1873), the Edinburgh Mathematical Society (1883) and the New York (later American) Mathematical Society (1888) were all founded in this period. There had, of course, been earlier more local societies, such as the Spittalfields Mathematical Society, which flourished over a long period before becoming defunct, as well as one or two much older bodies, for example the Mathematische Gesellschaft in Hamburg (1690), which still survive.
Semi-embeddings of Banach spaces which are hereditarily c0
Authors: Drewnowski L.
Published: January 2009

Following Lotz, Peck and Porta [9], a continuous linear operator from one Banach space into another is called a semi-embedding if it is one-to-one and maps the closed unit ball of the domain onto a closed (hence complete) set. (Below we shall allow the codomain to be an F-space, i.e., a complete metrisable topological vector space.) One of the main results established in [9] is that if X is a compact scattered space, then every semi-embedding of C(X) into another Banach space is an isomorphism ([9], Main Theorem, (a)⇒(b)).
A note an homeomorphic measures on topological groups
Authors: Morris Sidney A. , Peck Vincent C.
Published: January 2009

The classical von Neumann–Oxtoby–Ulam Theorem states the following:

Given non-atomic Borel probability measures μ, λ on In such that

there exists a homeomorphism h of In onto itself fixing the boundary pointwise such that for any λ-measurable set S

It is known that the above theorem remains valid if In is replaced by any compact finite dimensional manifold [2], [4] or with I∞, the Hilbert cube, [8].
Eigenvalues of the Klein-Gordon equation
Authors: Najman Branko
Published: January 2009

Consider the Klein-Gordon equation

where q, Aj are real valued functions on Rn, m and e positive constants. Equation (1) describes the motion of a relativistic particle of mass m and charge e in an external field described by the electrostatic potential q and the electromagnetic potential A = (Aj); units are chosen so that the speed of light is one.
Finite P-groups with unique maximal classes
Authors: Macdonald I. D.
Published: January 2009

Is it possible for a finite p-group to have only one conjugacy class of maximal size? This question was opened to public consideration in a paper [2] of John Meldrum dealing with the breadth of the wreath product of finite p-groups. His Theorem 21 gives a formula for the breadth of A wr B in terms of various constants including the breadths of A and B, a formula which differs according to whether or not A has a unique largest class. Hence the question.
The duals of generic space curves and complete intersections
Authors: Bruce J. W.
Published: January 2009

In a previous paper we discussed the duals of generic hypersurfaces: both smooth hypersurfaces in ℝn and algebraic hypersurfaces in real or complex projective space ℙn. In this note we show how to extend the methods of [1] to cover the case of complete intersections in ℙn and preface this with a brief discussion on the contact of space curves in ℝn with planes. We shall use the notation of [1].
A sharpening of the Berkson-Glickfeld theorem
Authors: Magyar Zoltan
Published: January 2009

It is known that if in a Banach*-algebra with unit the following holds:

then it is a C*-algebra (see [3]).

We shall show that the above theorem can be sharpened in the following way: we replace the submultiplicativity of the norm by the weaker assumption

Observe that under this assumption even the existence of exp(ih) is not at all obvious, but it will be proved to be true below. Our main result is Theorem 2 which depends on Theorem 1. Our last remark is the equivalent-norm-version of the statement.
Subsimple, injective, retract
Authors: Feigelstock Shalom , Klein Aaron
Published: January 2009

Simple and subsimple objects were introduced in [6]. It was shown that if there are enough simple objects in a category , then there is no room for injectives in . This idea was exploited in [6] and [2] to show that several classes of groups, rings and classes belonging to other categories do not possess non-trivial injectives or retracts. In this note, the above results will be strengthened by introducing a weaker condition than subsimple of [6]. As a consequence, and by employing some embedding theorems, we show that some important classes do not possess non-trivial retracts.

All the categories are assumed to have a zero object.
Length functions on hypercentral groups
Authors: Wilkens David L.
Published: January 2009

In [6] the structure of any real valued length function on an abelian group G is determined. It is shown there, in Theorem 6.1., that such a length function is an extension of a non-Archimedean length function l1 on N by an Archimedean length function l2 on H=G/N. Any non-Archimedean length function is given by a chain of subgroups, as described in [5], and following from results of Nancy Harrison [2], the length l2 is essentially the absolute value function on a subgroup of R. In the situation above if N≠G then N is a subgroup of G whose elements have bounded lengths. In this paper we show that it is an easy consequence of techniques developed in [1] that this result can be extended to hypercentral groups, thus determining the structure of any length function in this case. We point out that the result does not extend to soluble groups. The infinite dihedral group D∞ is soluble. However if D∞ is regarded as a free product of two cyclic groups of order 2 and is given the length function associated with a free product, as described by Lyndon [3], then N is not a subgroup of D∞, and the lengths of its elements are unbounded.
On a relation between the topology and the intrinsic and extrinsic geometries of a compact submanifold
Authors: Leung Pui-Fai
Published: January 2009

Let Mn be an n-dimensional smooth compact Riemannian manifold. By a theorem of Nash, we can think of it as an isometrically immersed submanifold in some higher dimensional Euclidean space ℝn+m. Viewing in this way we can compare the intrinsic geometry of M to its extrinsic geometry. Classically, the Gauss equation

where K(X,Y) denotes the sectional curvature in M corresponding to the plane spanned by the two orthonormal vectors X, Y and B denotes the second fundamental form gives one of the most important relations between the intrinsic and extrinsic geometries of M. In this note we shall prove the following.
Extensions of characters from Hall π-subgroups of π-separable groups
Authors: Isaacs I. M.
Published: January 2009

The main result of this paper is the following

Theorem A. Let G be a π-separable finite group with Hall πsubgroup H. Suppose θεIrr(H). Then there exists a unique subgroup M, maximal with the property that it contains H and θ can be extended to a character of M.
The ideal structure of idempotent-generated transformation semigroups
Authors: Reynolds M. A. , Sullivan R. P.
Published: January 2009

Let X be a set and the semigroup (under composition) of all total transformations from X into itself. In ([6], Theorem 3) Howie characterised those elements of that can be written as a product of idempotents in different from the identity. We gather from review articles that his work was later extended by Evseev and Podran [3, 4] (and independently for finite X by Sullivan [15]) to the semigroup of all partial transformations of X into itself. Howie's result was generalized in a different direction by Kim [8], and it has also been considered in both a topological and a totally ordered setting (see [11] and [14] for brief summaries of this latter work). In addition, Magill [10] investigated the corresponding idea for endomorphisms of a Boolean ring, while J. A. Erdos [2] resolved the analogous problem for linear transformations of a finite–dimensional vector space.
A property of Bernstein-Schoenberg spline operators
Authors: Goodman T. N. T. , Sharma A.
Published: January 2009

Let Bnf; x) denote the Bernstein polynomial of degree n on [0,1] for a function f(x) defined on this interval. Among the many properties of Bernstein polynomials, we recall in particular that if f(x) is convex in [0,1] then (i) Bn(f;x) is convex in [0,1] and (ii) Bn(f;x)≧Bn+1(f;x), (n = l,2,…). Recently these properties have been the subject of study for Bernstein polynomials over triangles [1].
Semigroups of high rank II doubly noble semigroups
Authors: Giraldes Emilia
Published: January 2009

This paper is a sequel to [2]. By asemigroup of high rankwe mean a semigroup such that for s1≠s2, 〈S\{s1,s2}〉⊂S(properly). Semigroups of high rank such that 〈S\{s}〉⊂S(royal semigroups) were classified in [2], where it was also shown that for anoble semigroup(i.e. a semigroup of high rank such that there exists a superfluous elementzinSfor which 〈S\{z}〉S) there exists eithere exactly one superfluous element or exactly two superfluous elements [2, Theorem 3.7].
Local invariance of free topological groups
Authors: Khan M. S. , Morris Sidney A. , Nickolas Peter
Published: January 2009

In 1948, M. I. Graev [2] proved that the free topological group on a completely regular Hausdorff space is Hausdorff, by showing that the free group admits a certain locally invariant Hausdorff group topology. It is natural to ask if Graev's locally invariant topology is the free topological group topology. If X has the discrete topology, the answer is clearly in the affirmative. In 1973, Morris-Thompson [6] showed that if X is not totally disconnected then the answer is negative. Nickolas [7] showed that this is also the case if X has any (non-trivial) convergent sequence (for example, if X is any non-discrete metric space). Recently, Fay and Smith Thomas handled the case when X has a completely regular Hausdorff quotient space which has an infinite compact subspace (or more particularly a non-trivial convergent sequence).(Fay-Smith Thomas observe that their class of spaces includes some but not all those dealt with by Morris-Thompson.)
On the structure of the set of solutions of the Darboux problem for hyperbolic equations
Authors: de Blasi F. S. , Myjak J.
Published: January 2009

Consider the Darboux problem

where φ,ψ:I→Rd (I=[0,1]) are given absolutely continuous functions with φ(0)=ψ(0), and the mapping f : Q × Rd→Rd (Q = I × I) satisfies the following hypotheses:

(A1) f(.,.,z) is measurable for every z ∈ Rd;

(A2) f(x, y,.) is continuous for a.a. (almost all) (x, y) ∈ Q;

(A3) there exists an integrable function α:Q →[0, + ∞) such that |f(x, y, z)|≦α(x, y) for every (x, y, z)∈ Q × Rd.
Decompositions of vector measures in Riesz spaces and Banach lattices
Authors: Schmidt Klaus D.
Published: January 2009

The present paper is mainly concerned with decomposition theorems of the Jordan, Yosida-Hewitt, and Lebesgue type for vector measures of bounded variation in a Banach lattice having property (P). The central result is the Jordan decomposition theorem due to which these vector measures may alternately be regarded as order bounded vector measures in an order complete Riesz space or as vector measures of bounded variation in a Banach space. For both classes of vector measures, properties like countable additivity, purely finite additivity, absolute continuity, and singularity can be defined in a natural way and lead to decomposition theorems of the Yosida-Hewitt and Lebesgue type. In the Banach lattice case, these lattice theoretical and topological decomposition theorems can be compared and combined.
The computation of the Hodgkin-Snaith operation in K*(Z × BU, Z/p)
Authors: Hegenbarth Friedrich
Published: January 2009

This paper builds on the work of V. P. Snaith [5] (in particular Section 6) and gives a more explicit determination of the result.

Recently J. E. McLure has given a satisfactory account of Dyer-Lashof operations in X-theory. For any E∞-space Y and for any r≧2 there is a map

with specific properties (see [3], Theorem 1). Earlier, L. Hodgkin [2] and V. P. Snaith [5] constructed an operation

where Ind(Y) = {xp\x∈Kα(Y,Z/p)} is the indeterminancy of .
On functions with derivative of bounded variation: An analogue of Banach's indicatrix theorem
Authors: Stadje Wolfgang
Published: January 2009

A simple, but nice theorem of Banach states that the variation of a continuous function F:[a, b]→ ℝ is given by where t(y) is defined as the number of x ∈ [a, b[ for which F(x)= y (see, e.g., [1], VIII.5, Th. 3). In this paper we essentially derive a similar representation for the variation of F′ which also yields a criterion for a function to be an integral of a function of bounded variation. The proof given here is quite elementary, though long and somewhat intriciate.
A weighted hyperplane mean associated with harmonic majorization in half-spaces
Authors: Armitage D. H.
Published: January 2009

The purpose of this paper is to introduce a new kind of weighted hyperplane mean for subharmonic functions and to use this mean in giving results on the harmonic majorization of subharmonic functions of restricted growth in half-spaces.
Locally nilpotent skew linear groups
Authors: Wehrfritz B. A. F.
Published: January 2009

Throughout this paper D denotes a division ring with centre F and n a positive integer. A subgroup G of GL(n,D) is absolutely irreducible if the F-subalgebra F[G] enerated by G is the full matrix ring Dn ×n. It is completely reducible (resp. irreducible) if row n-space Dn over D is completely reducible (resp. irreducible), as D–G bimodule in the obvious way. Absolutely irreducible skew linear groups have a more restricted structure than irreducible skew linear groups, see for example [7],[8], [8] and [10]. Here we make a start on elucidating the structure of locally nilpotent suchgroups.
An aspect of the invariant of degree 4 of the binary quintic
Authors: Edge W. L.
Published: January 2009

A binary form of odd degree,

has a quadratic covariant Г, (ab2maxbx in Aronhold's notation, and the discriminant Δ of Г is an invariant of ƒ For m = 2Δ was obtained by Cayley in 1856 [3, p. 274]; it was curiosity as to how Δ could be interpreted geometrically that triggered the writing of this note. An interpretation, in projective space [2m + 1], that does not seem to be on record, of Γ and Δ is found below. If m = 1 one has merely the Hessian and discriminant of a binary cubic whose interpretations in the geometry of the twisted cubic are widely known [5, pp. 241–2].
On normed Lie algebras with sufficiently many subalgebras of codimension I
Authors: Kissin E. V.
Published: January 2009

Let H be a finite or infinite dimensional Lie algebra. Barnes [2] and Towers [5] considered the case when H is a finite-dimensional Lie algebra over an arbitrary field, and all maximal subalgebras of H have codimension 1. Barnes, using the cohomology theory of Lie algebras, investigated solvable algebras, and Towers extended Barnes's results to include all Lie algebras. In [4] complex finite-dimensional Lie algebras were considered for the case when all the maximal subalgebras of H are not necessarily of codimension 1 but when

where S(H) is the set of all Lie subalgebras in H of codimension 1. Amayo [1]investigated the finite-dimensional Lie algebras with core-free subalgebras of codimension 1 and also obtained some interesting results about the structure of infinite dimensional Lie algebras with subalgebras of codimension 1.
On infinite groups of Fibonacci type
Authors: Campbell Colin M. , Thomas Richard M.
Published: January 2009

Let Fn = 〈 a1,a2,…,an〉 denote the free group of rank n, and let θ denote the automorphism of Fn which permutes the generators cyclically, in other words:
Stability of linear delay equations under a small noise
Authors: Mohammed S. E. A.
Published: January 2009

The stochastic stability of linear systems driven by white noise has been treated by several authors e.g. Has'minskii [7], Kushner [11], Kleimann and Arnold [9], Pinsky, [14], Friedman and Pinsky [4], Itô and Nisio [8] , Mohammed [13]. Following R. Kubo [10, 1966], the Brownian movement of a molecule in a “heat bath” is modelled in [4, pp. 223–226] by an asymptotically stable linear f.d.e. which is forced by white noise.
Approximation by partial isometries
Authors: Yuan Wu Pei
Published: January 2009

Let B(H) be the algebra of bounded linear operators on a complex separable Hilbert space H. The problem of operator approximation is to determine how closely each operator T ∈B(H) can be approximated in the norm by operators in a subset L of B(H). This problem is initiated by P. R. Halmo [3] when heconsidered approximating operators by the positive ones. Since then, this problem has been attacked with various classes L: the class of normal operators whose spectrum is included in a fixed nonempty closed subset of the complex plane [4], the classes of unitary operators [6] and invertible operators [1]. The purpose of this paper is to study the approximation by partial isometries.
Co-absolutely co-pure modules
Authors: Hiremath V. A.
Published: January 2009

B. Maddox [15] defined absolutely pure modules and derived some interesting properties of these modules. C. Megibben [17] continued the study of these modules and found more interesting properties. We introduce in this paper co-absolutely co-pure modules as dual to absolutely pure modules. We first prove that over a commutative classical ring these modules are precisely the flat modules. As a biproduct we get a projective characterization of flat modules over a commutative co-noetherian ring. Secondly, over a quasi-Frobenius ring R, co-absolutely co-pure right R-modules turn out to be projective modules. Finally we get a characterization of almost Dedekind domains in terms of co-absolutely co-pure modules.
Maximal subalgebras of Heyting algebras
Authors: Adams M. E.
Published: January 2009

AHeyting algebra is an algebra H;∨,∧ →, 0,1) of type (2,2,2,0,0) for which H;∨,∧,0,1) is a bounded distributive lattice and → is the binary operation of relative pseudocomplementation (i.e., for a,b,c∈H,ac ∧≦birr c≦a→b). Associated with every subalgebra of a Heyting algebra is a separating set. Those corresponding to maximal subalgebras are characterized in Proposition 8 and, subsequently, are used in an investigation of Heyting algebras.
On the number of eigenvalues in the spectral gap of a Dirac system
Authors: Hinton D. B. , Mingarelli A. B. , Read T. T. , Shaw J. K.
Published: January 2009

We consider the one-dimensional operator,

on 0<x<∞ with. The coefficientsp,V1andV2are assumed to be real, locally Lebesgue integrable functions;c1andc2are positive numbers. The operatorLacts in the Hilbert spaceHof all equivalence classes of complex vector-value functionssuch that.Lhas domainD(L)consisting of ally∈Hsuch thatyis locally absolutely continuous andLy∈H; thus in the language of differential operatorsLis a maximal operator. Associated withLis the minimal operatorL0defined as the closure ofwhereis the restriction ofLto the functions with compact support in (0,∞).
On a construction in bordism theory
Authors: Ray Nigel
Published: January 2009

In [2], R. Arthan and S. Bullett pose the problem of representing generators of the complex bordism ringMU*by manifolds which aretotally normally split; i.e. whose stable normal bundles are split into a sum of complex line bundles. This has recently been solved by Ochanine and Schwartz [8] who use a mixture ofJ-theory and surgery theory to establish several results, including the following.
On the other pαqβ theorem of Burnside
Authors: Bialostocki Arie
Published: January 2009

The “other” pαqβ theorem of Burnside states the following:

Theorem A.l. Let G be a group of order pαqβ, where p and q are distinct primes. If pα>qβ, then Op(G)≠1 unless

(a) p is a Mersenne prime and q = 2;

(b) p = 2 and q is a Fermat prime; or

(c) p = 2 and q = 7.
Length functions and pregroups
Authors: Chiswell I. M.
Published: January 2009

The idea of a pregroup was introduced by Stallings and provides an axiomatic setting for a well-known argument, due to van der Waerden, used to prove normal form theorems. Details are provided in [7], Section 3.
A sufficient condition for the second derived factor group to be finite
Authors: Howse J. R.
Published: January 2009

This paper concerns an application of an algorithm for the second derived factor group as described by Howse and Johnson in [3]. This algorithm has as its basis theFox derivative (see [1]), a mapping from the free group F to the group-ring ℤF, definedas follows: let X be a set of generators of a group G, and let w = y1…yk with each yi∈X±1.
The nonabelian tensor square of a finite split metacyclic group
Authors: Johnson D. L.
Published: January 2009

Given any group G, its tensor square G ⊗ G is defined by the followingpresentation(see [[3]):

where g,g′,h,h′ range independently over G, and gh=ghg−1. In what follows, gg′ ⊗ gh is often written in the abbreviated form g(g′ ⊗ h).
Some maximal normal subgroups of the modular group
Authors: Jones Gareth A.
Published: January 2009

For each finite group G, let G denote the set of all normal subgroups of the modular group Γ = PSL2(ℤ) with quotient group isomorphic to G; since Γ is finitely generated, the number NG = |G| of such subgroups is finite. We shall be mainly concerned with the case where G is the linear fractional group PSL2(q) over the Galois field GF(q), in which case we shall write (q) and N(q) for G and NG; for q>3, PSL2(q) is simple, so the elements of (q) will be maximal normal subgroups of Γ.
Construction of open sets of free K-tuples of matrices
Authors: Mandel A. , Gonçalves J. Z.
Published: January 2009

A k-tuple of members of a group is free if it freely generates a free group. In an earlier paper [1] the authors have shown:
On a Generalisation of the Laplace Transformation
Authors: Erdélyi A.
Published: January 2009

On reading a recent paper by R. S. Varma (Varma 1949) I recalled that in May 1942 I investigated an integral transformation which is very similar to Varma's. Varma has

and points out that this reduces to a Laplace integral for k = ¼, m = ± ¼. Instead of (1), one could consider the integral

which was introduced by C. S. Meijer (Meijer 1940 b); this integral reduces to a Laplace integral whenever k = m + ½. Now, apart from comparatively unimportant factors, the nucleus of (2) is a fractional derivative or integral, as the case may be, of e−st, and on carrying out a fractional integration by parts, it appears that (2) is essentially the Laplace transform of a fractional integral or derivative of f. Thus, the whole theory of the transformation (2), including inversion formulae, representation theorems, etc., can be deduced from the well-known theory of the Laplace transformation. It is not quite clear that a similar reduction is possible for (1), although it is certainly possible when k = 0.
Congruences on a Distributive Lattice
Authors: Thueston H. A.
Published: January 2009

Among the many papers on the subject of lattices I have not seen any simple discussion of the congruences on a distributive lattice. It is the purpose of this note to give such a discussion for lattices with a certain finiteness. Any distributive lattice is isomorphic with a ring of sets (G. Birkhoff, Lattice Theory, revised edition, 1948, p. 140, corollary to Theorem 6); I take the case where the sets are finite. All finite distributive lattices are covered by this case.
A Determinantal Expansion for a Class of Definite Integral
Authors: Shenton L. R.
Published: January 2009

In a previous paper (Shenton, 1953) we have given an expansion for integrals of the form This expansion may be expressed as a determinantal quotient or Schweinsian series. In the present paper we state more general terms under which the expansion holds and consider the case when the limits of integration are infinite and the weight function of the form In particular we giye expansions for the Psi function, and where C (x) is a positive polynomial.
Some Identities Associated with a Discriminant
Authors: Watson G. N.
Published: January 2009

This paper has to deal with some topics suggested by a theorem which is notorious in the theory of quadratic forms, namely that the roots of the characteristic equation of any real quadratic form are all real. For a proof of this theorem, the reader may consult Fcrrar (1), p. 146.
Note on the Problem of the Electrified Disc
Authors: Ashour A. A.
Published: January 2009

1. Prof. E. T. Copson has discussed the well-known problem of a circular disc kept at a constant potential Vo in an external field of potential Φ by reducing it to the solution of two integral equations. Tho solution is however fairly simple if we use oblate spheroidal co-ordinates. This is due to the fact that in this system of coordinates the disc can be represented in terms of one co-ordinate only. This method is applied to the above problem and Copson's results are obtained. The solution when Vo is not constant, but any surface function of the disc, is also obtained.
On Integral Transforms
Authors: Conolly B. W.
Published: January 2009

Titchmarsh and others [4] havo studied integral transform pairs of tho type

showing how such pairs may be generated as a consequence of the relation which holds between the Mellin transforms of the kernels.
A Matrix Problem Concerning Projections
Authors: Schneider Hans
Published: January 2009

The following problem is a slight generalisation of one posed partly solved by H. Nagler.1 We shall use A* for the conjugate transpose of a matrix A. A projection is an idempotent matrix, its latent roots consist of units and zeros.