bluuebunny/crossref_metadata_2025 huggingface dataset:
On maximal transitive subtournaments
Authors: Moon J. W.
Published: January 2009

A tournamentTn consists of a finite set of nodes1, 2, …, n such that each pair of distinct nodes i and j is joined by exactly one of the arcsij or ji. If the arc ij is in Tn we say that i beats j or j loses to i and write i→j. If each node of a subtournament A beats each node of a subtournament B we write A→B and let A + B denote the tournament determined by the nodes of A and B.
Characterizations of linear differential systems with a regular singular point
Authors: Harris W. A.
Published: January 2009

The linear differential system

where w is a vector with n components and A is an n by n matrix is said to have z = 0 as a regular singular point if there exists a fundamental matrix of the form

such that S is holomorphic at z = 0 and R is a constant matrix ((1), p. 111; (2), p. 73). For such systems A will have at most a pole at z = 0 and we may write

where p is an integer, Ã is holomorphic at z = 0, and Ã(0) ≠ 0. However, the converse is not true. When p ≦ − 1, A is holomorphic at z = 0, and every fundamental matrix is holomorphic at z = 0. If p ≧ 1, the non-negative integer p is called (after Poincaré) the rank of the singularity and there is a significant difference between the cases p = 0 and p ≧ 1. If p = 0 the linear differential system (1) is known to have z = 0 as a regular singular point ((1), p. 111) ; whereas, if p ≧ 1, z = 0 may or may not be a regular singular point.
Convergence criteria for bounded sequences
Authors: Borwein D.
Published: January 2009

Let {Kn} be a sequence of complex numbers, let

and let

Let D be the open unit disc {z: |z| <1}, let be its closure and let .

The primary object of this paper is to prove the two theorems stated below, the first of which generalises a result of Copson (1).
A configuration of lines in three dimensions
Authors: Hirschfeld J. W. P.
Published: January 2009

In 1849, Cayley and Salmon discovered that a general cubic surface in projective space of three dimensions over the complex numbers has twenty-seven lines on it. They remarked that all the properties of the twenty-seven lines would not become apparent until a better notation than they had given was found. This notation was discovered by Schläfli in 1858 in the double-six theorem (henceforth referred to as given five skew linesa1, …, a5with a single transversal b6such that no four of the ai lie in a regulus, the four ai excluding aj have a second transversal bj and the five lines b1, …, b5thus obtained have a transversal a6—the completing line of the double-six. The other fifteen lines of the cubic surface are then , where ai bj is the plane containing ai and bj.
A generalization of an early multivariate integral
Authors: Jack Henry
Published: January 2009

Earlier this year Professor W. L. Edge drew my attention to the paper (1). Since Black probably wrote his paper about 1890, the integral he considered must have been one of the earliest n-variate integrals to be evaluated. The present paper generalizes Black's result from a column vector as variate to a rectangular matrix—his integral is the case p = m = k = 1 of the integral J below.
Pseudo-umbilical surfaces with constant Gauss curvature
Authors: Chen Bang-Yen
Published: January 2009

Let M be a surface immersed in an m-dimensional space form Rm(c) of curvature c = 1, 0 or −1. Let h be the second fundamental form of this immersion; it is a certain symmetric bilinear mapping for x ∈ M, where Tx is the tangent space and the normal space of M at x. Let H be the mean curvature vector of M in Rm(c) and 〈, 〉 the scalar product on Rm(c). If there exists a function λ on M such that 〈h(X, Y), H〉 = λ〈X, Y〉 for all tangent vectors X, Y, then M is called a pseudo-umbilical surface of Rm(c). Let D denote the covariant differentiation of Rm(c) and η be a normal vector field. If we denote by D*η the normal component of Dη, then D* defines a connection in the normal bundle. A normal vector field η is said to be parallel in the normal bundle if Dη = 0. The length of mean curvature vector is called the mean curvature.
The maximum idempotent-separating congruence on a regular semigroup
Authors: Meakin John
Published: January 2009

It has been established by G. Lallement (3) that the set of idempotent-separating congruences on a regular semigroup S coincides with the set ∑() of congruences on S which are contained in Green's equivalence on S. In view of this and Lemma 10.3 of A. H. Clifford and G. B. Preston (1) it is obvious that the maximum idempotent-separating congruence on a regular semigroup S is given by
A characterisation of a certain class of convolution algebras
Authors: Baker J. W.
Published: January 2009

In (6) Taylor has introduced the notion of a convolution measure algebra. In the same paper he constructed a canonical embedding of an arbitrary, semisimple commutative convolution measure algebra A into the algebra M(S) of all bounded, regular Borel measures on a compact semigroup S. This embedding has the properties that A is σ(M(S), C(S)-dense in M(S), that if μ is in A and ν is absolutely continuous with respect to μ, then ν is in A and that the set A^ of non-zero complex homomorphisms of A can be identified with the set S^ of continuous semicharacters of S where h ∈ A^ is identified with χ ∈ S^ by the equation
Inequalities for certain cyclic sums II
Authors: Boarder J. C. , Daykin D. E.
Published: January 2009

Let k ≧ 2 be an integer and each of ν1, ν2, …, νk and δ1, δ2, …, δk be 0 or 1. Then given any positive integer M and non-negative reals a1, a2, …, aM we put

where

and
Contributions to genetic algebras II
Authors: Gonshor Harry
Published: January 2009

This paper is a continuation of (4). The main aim of this paper is the introduction of the concept of sex-linked duplication. In addition, we shall give several equivalent definitions for the concept of a genetic algebra and make several remarks on overlapping of generations.
Unconditional convergence and bases
Authors: Tweddle I.
Published: January 2009

The recent papers (6), (7) of J. T. Marti have revived interest in the concept of extended bases, introduced in (1) by M. G. Arsove and R. E. Edwards. In the present note, two results are established which involve this idea. The first of these, which is given in a more general setting, restricts the behaviour of the coefficients for an extended basis in a certain type of locally convex space. The second result extends the well-known weak basis theorem (1, Theorem 11).
Inequalities concerning the inverses of positive definite matrices
Authors: Muir W. W.
Published: January 2009

Much has been written on inequalities concerning positive definite matrices, but a new insight may be gained by examining inequalities from the standpoint of the inverse matrix. The standard inequality of Hölder can then be used in a more fruitful manner. This leads to some new results and a rediscovery of some known results.
An application of a theorem of Singer
Authors: Connolly D. M. , Williamson J. H.
Published: January 2009

The authors have recently treated (2) the problem of finding subsets E of the real line , of type Fσ, such that E–E contains an interval and the k-fold vector sum (k)E is of measure zero. Positive results can be obtained, for all k, on the basis of a recent theorem of J. A. Haight (3), following earlier partial results (1), (4) for k ≦ 7; and indeed in these cases the problem has a solution with E a perfect set. An analogous problem, apparently in most respects subtler than the first, is the following. Do there exist finite regular Borel measures μ on such that is absolutely continuous (where is the adjoint of μ) and the kth convolution power μk is singular? Both problems are of interest in the general context of elucidating the properties of the measure algebra or, more generally, M(G) for locally compact abelian G. The second problem may be regarded as an attempt to provide (at least one aspect
of) a multiplicity theory for the first.
Topological convexity spaces
Authors: Bryant Victor
Published: January 2009

We shall start by recalling the definition and some basic properties of a convexity space; a topological convexity space (tcs) will then be a convexity space together with an admissible topology, and will be a generalisation of a topological vector space (tvs). After showing that the usual tvs results connecting the linear and topological properties extend to this new setting we then prove a form of the Krein-Milman theorem in a tcs.
On rings with quasi-injective cyclic modules
Authors: Ahsan J.
Published: January 2009

A ring R is called a qc-ring if each cyclic R-module is quasi-injective. For various properties of these rings we refer to Ahsan (1) and Koehler (15). In this paper we shall obtain some additional results related to qc-rings. The scheme of the paper is as follows. Section 2 contains various preliminary definitions and results. In Section 3, we shall prove that every commutative hypercyclic ring is a qc-ring. In this section, we shall also show that a qc-ring which satisfies the ascending chain condition on its annihilators has nilpotent Jacobson-radical. Finally, in Section 4, we shall study rings all of whose proper factor rings are qc. Such rings will be called “ restricted qc ”.
Measurable selections in normed spaces
Authors: Leese S. J
Published: January 2009

It is sometimes desirable to know in what circumstances a measurable set valued function admits a measurable selector; this problem occurs regularly in the theory of optimal control (see for example (3) and (7)). In this paper we demonstrate the existence of measurable selectors in two particular cases where the choice of selector has a simple geometrical interpretation, namely that of being a “ nearest-point ” selector, as is explained in detail below. This work derives in part from that of C. Castaing, particularly from Théorème 3.4 of (2), of which this is an extension.
Outer automorphisms of ordered permutation groups
Authors: Holland W. Charles
Published: January 2009

A well-known fact is that every automorphism of the symmetric group on a set must be inner (whether the set is finite or infinite) unless the set has exactly six elements (4, § 13). A long-standing conjecture concerns the analogue of this fact for the group A(S) of all order-preserving permutations of a totally ordered set S. The group A(S) is lattice ordered (l-group) by defining, for f, g ∈ A (S), f ≦ g whenever xy ≦ xg for all x ∈ S. From the standpoint of l-groups, A(S) is of considerable interest because of the analogue of Cayley's theorem proved in (2), namely every l-group may be embedded in some A(S). Unlike the non-ordered symmetric groups, which must be highly transitive, A(S) is severely restricted if assumed transitive. Of special interest are those cases when A(S) is doubly transitive (relative to the order), since the building blocks (primitive) of all transitive A(S) are either doubly transitive or uniquely transitive. It is easily seen that each inner automorphism of A(S) must preserve the lattice ordering (that is, must be an l-automorphism). It is tempting to conjecture that every l-automorphism of A(S) must be inner. However, an easy counterexample is at hand. If S consists of two copies of the ordered set of integers, one entirely above the other, then A(S) is the direct product of two copies of the l-group of integers, which has an outer l-automorphism exchanging the two factors. The conjecture referred to above is that if A(S) is transitive on S, then every l-automorphism of A(S) is inner.
Near-rings of quotients of endomorphism near-rings
Authors: Holcombe Michael
Published: January 2009

Let be a category with finite products and a final object and let X be any group object in . The set of -morphisms, (X, X) is, in a natural way, a near-ring which we call the endomorphism near-ring of X in Such nearrings have previously been studied in the case where is the category of pointed sets and mappings, (6). Generally speaking, if Γ is an additive group and S is a semigroup of endomorphisms of Γ then a near-ring can be generated naturally by taking all zero preserving mappings of Γ into itself which commute with S (see 1). This type of near-ring is again an endomorphism near-ring, only the category is the category of S-acts and S-morphisms (see (4) for definition of S-act, etc.).
Extensions relative to a Serre class
Authors: Cormack S.
Published: January 2009

Consider a class C of projective R-modules, where R is a commutative ring with identity, which satisfies the conditions of (2), namely that C is closed under the operations of direct sum and isomorphism and C contains the zero module. Following (2) a module M is said to have C-cotype n (respectively C-type n) if it has a projective resolution … → Pn → P0 → M → 0 with Pi ∈ C for i>n (respectively Pi ∈ C for i≦n). Let S be the class of modules of C-cotype −1, equivalently of C-type infinity. It is assumed throughout that S is a Serre Class. We define an abelian category of modules with the property that C-cotype is homological dimension in while in the case C = 0, S is just the category of R-modules. It follows that all categorical results on homological dimension also hold for cotype.
Topological aspects of suitable theories
Authors: Simmons H.
Published: January 2009

Roughly speaking a suitable theory is a theory T together with its formal provability predicate Prv (.). A pseudo-topological space is a boolean algebra B which carries a derivative operation d and its associated closure operation c. Thus we can pretend that B is a topological space. We show that the Lindenbaum algebra B(T) of a suitable theory becomes, in a natural way, a pseudotopological space, and hence we can translate properties of T into topological language, as properties of B(T). We do this translation for several properties of T, including (1) satisfying Gödel's first theorem, (2) satisfying Löb's theorem and (3) asserting one's own inconsistency. These correspond to the topological properties (1) having an isolated point, (2) being scattered, (3) being discrete.
A result on the convolution of distributions
Authors: Fisher B.
Published: January 2009

In a recent paper (1), Jones extended the definition of the convolution of two distributions to cover certain pairs of distributions which could not be convolved in the sense of the previous definition. The convolution ω1 * ω2 of two distributions ω1 and ω2 was defined as the limit of the sequence ω1n * ω2n, provided the limit ω exists in the sense that

for all fine functions φ in the terminology of Jones (2) where

and τ is an infinitely differentiate function satisfying the following conditions:
The tensor product of distributive lattices
Authors: Fraser Grant A.
Published: January 2009

The well-known algebraic concept of tensor product exists for any variety of algebras.The tensor product of groups and of rings have been studied extensively. For other varieties, such as the variety of semigroups, the tensor product has been investigated more recently (5). In this paper we investigate the tensor product of distributive lattices.
On the density of the invertible group in C*-algebras
Authors: Robertson Guyan
Published: January 2009

In what follows the term C*-algebra will mean a complex C*-algebra with identity. We denote the identity element by 1. We shall also use the notation and terminology of Dixmier (3) without comment.
Badly approximable functions and interpolation by Blaschke products
Authors: Rubel L. A. , Shields A. L.
Published: January 2009

A continuous function φ on the unit circle is called badly approximable if ‖ φ − p ‖∞ ≧ ‖ φ |∞ for all polynomials p, where ‖ |∞ is the essential supremum norm. In (4), Poreda asked whether every continuous φ may be
written φ = φW+φB, where φW is the uniform limit of polynomials (i.e. φW belongs to the disc algebra A) and φB is badly approximable. We call such a function φ decomposable. In (4), he characterised the badly approximable functions as those of constant non-zero modulus and negative winding number around the origin, i.e. ind (φ)<0. (See (3) for two new proofs of this result.) We show that the answer to Poreda's question is no in general, but give a necessary and sufficient condition for a given φ to have such a decomposition. Then we apply this criterion to solve an interpolation problem.
Exponentiation in Banach star algebras
Authors: Dales H. G.
Published: January 2009

In their recently published book Bonsall and Duncan ask the following question ((1), p. 65): if A is a Banach star algebra with an identity, is it true that
On power-bounded prespectral operators
Authors: Dowson H. R.
Published: January 2009

Foguel (8) and Fixman (7) independently proved that an invertible spectral operator, which is power-bounded, is of scalar type. Their proofs rely heavily on a result of Dunford on spectral operators whose resolvents satisfy a growth condition. (See Lemma 3.16 of (6, p. 609).) Observe that the resolvent of an invertible power-bounded operator T satisfies an inequality of the form
An Integral Transform Solution of the Differential Equation for the Transverse Motion of an Elastic Beam
Authors: Fulton J.
Published: January 2009

It is well known* that certain types of partial differential equation may be solved using integral transforms with suitable kernels. In general, these equations may be solved by the classical method of separating variables, but the use of an integral transform yields the solution in a more direct way in the sense that the boundary values are contained in the solution.

It is the purpose of this note to apply this technique to obtain the solution of the differential equation associated with the transverse motion of an elastic beam for a wide class of boundary conditions.
A Banach Space which is not Equivalent to an Adjoint Space
Authors: Weston J. D.
Published: January 2009

A Banach space which is not reflexive may or may not be equivalent (in Banach's sense) to an adjoint space. For example, it is an elementary fact that the space (l), though not reflexive, is equivalent to (co)*, where (co) is the space of all sequences that converge to zero, normea in the usual way. On the other hand, (co) itself is not equivalent to any adjoint space : this can be proved by means of the Krein-Milman theorem, but here we obtain the result by an elementary argument which is scarcely more complicated than the standard proof that (co) is not reflexive.
An Algebraic Problem in Control Theory
Authors: Wouk Arthur
Published: January 2009

In the series of papers in the early forties summarised in (1), (2), A. I. Lur'e showed how to utilise Liapunov's second or direct method in the investigation of the stability of linear automatic control systems with a single nonlinear actuator. His approach consists of

1. the transformation of the original system of differential equations via the so-called Lur'e transformation into canonical coordinates in which the construction of the Liapunov function is direct, and

2. the conversion of the differential problem into a purely algebraic problem.

We will be concerned here with the questions of the existence and construction of the Lur's transformation.
Note on Quasigroups and Trees
Authors: Etherington I. M. H.
Published: January 2009

If letters a, b, c, … are used to denote points of a nondegenerate plane cubic curve, other than the singular point if any, and if the product ab is defined as the third point of the curve collinear with a and b, we obtain an algebraic system having nonassociative multiplication (ab . c ≠ a . bc in general). It is in fact a totally symmetric entropic quasigroup (these terms are defined below). This idea, which was put forward at a meeting of the Edinburgh Mathematical Society a few years ago, will be exploited in a forthcoming paper. Such quasigroups have many properties which can be interpreted geometrically. Or, conversely, known properties of cubic curves suggest theorems about totally symmetric entropic quasigroups, which if established will involve a gain in generality, since not every such quasigroup can be “placed” on a cubic.
On the Solution of some Axisymmetric Boundary Value Problems by means of Integral Equations
Authors: Collins W. D.
Published: January 2009

This paper concludes a series of papers (1) on a group of axisymmetric boundary value problems in potential and diffraction theory by considering some potential problems for a circular annulus. The Dirichlet problem for an annulus has recently been considered by Gubenko and Mossakovskiǐ (2), who, by a somewhat complicated method, show it to be governed by either one of two Fredholm integral equations of the second kind. The purpose of the present paper is to show how the method developed in previous papers, by which certain integral representations of the potentials in problems for circular disks arid spherical caps are used to reduce such problems to the solutions of either single Abel integral equations or Abel and Fredholm equations, can be applied to both the Dirichlet and Neumann problems for the annulus to give reasonably straightforward derivations of the governing Fredholm equations.
On k-Conjugacy in a Group
Authors: Yff Peter
Published: January 2009

All elements mentioned herein are in a group G. A well-known definition states that x and y are conjugate if there exists an element a such that y = α–1xα. Conjugacy is an equivalence relation in G. In the present paper this will be called 1-conjugacy.
Asymptotic Solutions of Integral Equations with Convolution Kernels
Authors: Hutson V.
Published: January 2009

The equations considered are Fredholm integral equations of the second kind with regular kernels, whose argument depends only on the difference of the variables. Approximate solutions are sought for a given finite range of the eigenvalues, and for large values of the range of integration. Certain special conditions are imposed on the general form of the Fourier transforms of the kernel. Then it is shown that approximate solutions may be obtained in terms of the solutions of the corresponding (singular) Wiener-Hopf equations. Approximations to the eigenvalues are also found. It is shown that the eigenfunctions are unique, and that except possibly near the end points of the range, the solutions are of trigonometric type with the zeros of successive solutions interlacing.
The Maximum Idempotent-Separating Congruence on an Inverse Semigroup
Authors: Howie J. M.
Published: January 2009

A congruence ρ on a semigroup will be called idempotent-separating if each ρ-class contains at most one idempotent. It is shown below that there exists a maximum such congruence µ on every inverse semigroup S. Two characterisations of µ are found, and it is shown (a) that S/µ⋍E, the semilattice of idempotents of S, if and only if E is contained in the centre of S; (b) that µ is the identical congruence on S if and only if E is self-centralising, in a sense explained below.
B# Modular Annihilator Algebras
Authors: Duncan J.
Published: January 2009

The structure of various classes of annihilator algebras has been known for some time. Bonsall and Goldie (1) considered semi-simple Banach algebras with the properties

(i)r(L)≡{x:xε,yx=0(yεL)} ≠(0) for each proper closed left ideal L of ,

(ii)l(K)≡{x:xε,xy=0(yεK)}≠(0) for each proper closed left ideal K of ,
On Certain Integral Equations of Convolution Type with Bessel-Function Kernels
Authors: Srivastav R. P.
Published: January 2009

In this paper we first obtain elementary solutions of the integral equations

and

Using these solutions we then define operators of fractional integration. These operators may be regarded as a generalisation of the operators of fractional integration introduced by Sneddon (1) as a modification of Erdé1yi–Kober operators. In fact Erdélyi–Kober–Sneddon operators may be obtained by multiplying both sides of the equations by α-1½β and considering the limiting case α→0. We employ these operators to find a generalisation of the Mellin transform.
Derivation of Solutions of the Klein-Gordon Equation from Solutions of the Wave Equation
Authors: Chambers LL. G.
Published: January 2009

The Klein–Gordon equation

Ω being a constant of dimensions [time]-1 and c being a constant velocity, appears in nuclear physics (1) and, when the Laplacian operator is twodimensional, in the theory of long gravity waves on a rotating earth (2). If Ω is zero it reduces to the wave equation
On Triple Series Equations Involving Series of Jacobi Polynomials
Authors: Srivastava K. N.
Published: January 2009

Recently Collins (2) has studied triple series equations involving series of Legendre polynomials. These equations arise in the study of mixed boundary value problems and can be regarded as extensions of the dual series equations considered by Collins in (1).
The Idempotent-separating Congruences on a Regular 0-bisimple Semigroup
Authors: Munn W. D.
Published: January 2009

A congruence ρ on a semigroup is said to be idempotent-separating if each ρ-class contains at most one idempotent. For any idempotent e of a semigroup S the set eSe is a subsemigroup of S with identity e and group of units He, the maximal subgroup of S containing e. The purpose of the present note is to show that if S is a regular O-bisimple semigroup and e is a non-zero idempotent of 5 then there is a one-to-one correspondence between the idempotentseparating congruences on 5 and the subgroups N of He with the property that aN ⊆ Na for all right units a of eSe and Nb ⊆ bN for all left units b of eSe. Some special cases of this result are discussed and, in the final section, an application is made to the principal factors of the full transformation semigroup on a set X.
Jordan algebras arising in population genetics
Authors: Holgate P.
Published: January 2009

The non-associative algebras arising in genetics (1), are rather isolated from other branches of non-associative algebra (6). However, in a paper (5), in which he studied these algebras in terms of their transformation algebras, Schafer proved that the gametic and zygotic algebras for a single diploid locus are Jordan algebras.
Some remarks on countably barrelled and countably quasibarrelled spaces
Authors: Iyahen Sunday O.
Published: January 2009

Barrelled and quasibarrelled spaces form important classes of locally convex spaces. In (2), Husain considered a number of less restrictive notions, including infinitely barrelled spaces (these are the same as barrelled spaces), countably barrelled spaces and countably quasibarrelled spaces. A separated locally convex space E with dual E' is called countably barrelled (countably quasibarrelled) if every weakly bounded (strongly bounded) subset of E' which is the countable union of equicontinuous subsets of E' is itself equicontinuous. It is trivially true that every barrelled (quasibarrelled) space is countably barrelled (countably quasibarrelled) and a countably barrelled space is countably quasibarrelled. In this note we give examples which show that (i) a countably barrelled space need not be barrelled (or even quasibarrelled) and (ii) a countably quasibarrelled space need not be countably barrelled. A third example (iii)shows that the property of being countably barrelled (countably quasibarrelled) does not pass to closed linear subspaces.
The Dirichlet problem for a thin rectangle
Authors: Knowles J. K.
Published: January 2009

We consider the Dirichlet problem for Laplace's equation in a rectangle with a view toward determining the asymptotic behaviour of the solution for the case in which the width of the rectangle is small in comparison with its length. Although the construction of an explicit representation of the solution is an elementary matter, the resulting formula is inconvenient for present purposes, and we accordingly proceed along different lines.
The distribution of irreducible polynomials in several indeterminates over a finite field
Authors: Cohen Stephen D.
Published: January 2009

We consider non-zero polynomials f(x1, …, xk) in k variables x1, …, xk with coefficients in the finite field GF[q] (q = pn for some prime p and positive integer n). We assume that the polynomials have been normalised by selecting one polynomial from each equivalence class with respect to multiplication by non-zero elements of GF[q]. By the degree of a polynomial f(x1, …, xk) will be understood the ordered set (m1, …, mk), where mi is the degree of f(x1 ,…, xk) in x1(i = 1, 2, …, K). The degree (m,…, mk) of a polynomial will be called totally positive if mi>0, i = 1, 2, …, k.
Homomorphism theories for universal algebras
Authors: Hoehnke Hans-Jürgen
Published: January 2009

It is well-known that a homomorphism ø(A→B) between groups A and B induces a homomorphism ø*(ZA→ZB) between the corresponding group rings ZA and ZB over the ring of integers Z. The identical congruence O on B and the unit element eB of B can be characterised by the equations x–y = 0 and x–eB = 0 (x,y ∈ B) respectively. Similarly the congruence Γø corresponding to ø and the corresponding normal subgroup of A are

and {x∈A1 = A,(x–eA)ø = 0} respectively.
A note on logarithmic summability (L):Corrigendum
Authors: Sitaraman Y.
Published: January 2009

I am indebted to Dr. B. Kuttner for having kindly drawn my attention to certain corrections necessary in my paper with the above title, appearing in these Proceedings, Vol. 15 (Series II), Part 1, June 1966, pages 47–55.

The major corrections are as follows. On pages 48-9, Lemma 1 should appear without the factor log (1 – z)−1 of the denominator in the integrand at the end of the lemma; the formula in the proof of Lemma 1 is incorrect, its correct form being
-spaces and the closed-graph theorem
Authors: Iyahen S. O.
Published: January 2009

The problem considered in this paper is that of finding conditions on a range space such that the closed-graph theorem holds for linear mappings from a class of linear topological spaces. The concept of a -space, which is a result of this investigation, is meaningful for commutative topological groups but we limit our consideration in this paper to linear topological spaces. On restricting ourselves to locally convex linear topological spaces, we see that the notion of a -space is an extension of the powerful idea of a B-complete space.
Lower bounds for the radical of the group algebra of a finite p-soluble group
Authors: Wallace D. A. R.
Published: January 2009

Over a field of characteristic p>0 the group algebra of a finite group has a unique maximal nilpotent ideal, the Jacobson radical of the algebra. The powers of the radical form a decreasing and ultimately vanishing series of ideals and it would be of interest to determine the least vanishing power. Apart from the work of Jennings (3) on p-groups little is known in general (cf. (5)) about this particular power of the radical (cf. Remarks of Brauer in (4), p. 144. Problem 15). In this paper we give non-trivial lower bounds for the index of the least vanishing power of the radical when the group is p-soluble. Of the lower bounds we give we show that that lower bound, which is dependent solely on the order of the group, is the best possible such bound and that this bound is invalid if the assumption of p-solubility is omitted.
On Kadison's condition for extreme points of the unit ball in a B*-algebra
Authors: Yood Bertram
Published: January 2009

Let B be a complex Banach algebra with an identity 1 and an involution x→x*. Kadison (1) has shown that, if B is a B*-algebra, [the set of extreme points of its unit ball coincides with the set of elements x of B for which
On a Boolean algebra of projections constructed by Dieudonné
Authors: Dowson H. R.
Published: January 2009

Dieudonné (4) has constructed an example of a Banach space X and a complete Boolean algebra B̃ of projections on X such that B̃ has uniform multiplicity two, but for no choice of x1, x2 in X and non-zero E in B̃ is EX the direct sum of the cyclic subspaces clm {Ex1:E∈B̃} and clm {Ex2:E∈B̃}. Tzafriri observed that it could be deduced from Corollary 4 (9, p. 221) that the commutant B̃′ of B̃ is equal to A(B̃), the algebra of operators generated by B̃ in the uniform operator topology. A study of (3) suggested the direct proof of the second property given in this note. From this there follows a simple proof that B̃ has the first property.
On partitions of N points
Authors: Watson Donald
Published: January 2009

In a paper (1) by Harding there is a tacit invitation to seek the connection between the following two problems:

(i) Find the number, ηk(N), of regions into which a k-dimensional space is partitioned by a set of N (k- l)-dimensional hyperplanes.

(ii) Find the number, vk(N), of distinct partitions of a given set of N points in a k-dimensional space E that can be induced by (k- 1)-dimensional hyperplanes.
Further Arithmetical Functions in Finite Fields
Authors: Cohen Stephen D.
Published: January 2009

In this paper, the author continues his investigation, initiated in (4) and (5), into the nature of certain “arithmetical” functions associated with the factorisation of normalised non-zero polynomials in the ring GF[q, X1, …, Xk], where k ≧ 1, GF(q) is the finite field of order q and X1, …, Xk are indeterminates. By normalised polynomials we mean that exactly one polynomial has been selected from equivalence classes with respect to multiplication by non-zero elements of GF(q). With this normalisation GF[q, X1, …, Xk] becomes a unique factorisation domain. The constant polynomial will be denoted by 1. By the degree of a polynomial A in GF[q, X1, …, Xk], we shall mean the ordered set (m1, …, mk), where mi is the degree of A in Xi, 1 ≦ i ≦.k. We shall assume that A(≠ 1), a typical polynomial in GF[q, X1, … Xk], has prime factorisation

where P1, …, Pr are distinct irreducible polynomials (i.e. primes).
Genetic Algebras Associated with Polyploidy
Authors: Holgate P.
Published: January 2009

The relationship between certain non-associative algebras and the deterministic theory of population genetics was first investigated by Etherington(3)-(8), who defined the concepts of baric, train and special train algebras. Gonshor(10)dealt with, among other topics, algebras corresponding to autopolyploidy, on the assumption that chromosome segregation operated. In this paper [ discuss algebras corresponding to more general systems of inheritance among polyploids, which have been discussed without using algebras by Haldane(11), Geiringer(9), Moran(13)and Seyffert (16). These algebras are special cases of what I have defined as segregation algebras, and mixtures of them. All the algebras corresponding to a fixed ploidy have a relationship which I have called special isotopy. An example shows that algebras arise in other genetic systems which are not isotopic to segregation algebras.
Two Enumerative Proofs of an Identity of Jacobi
Authors: Sudler C.
Published: January 2009

In (7), Wright gives an enumerative proof of an identity algebraically equivalent to that of Jacobi, namely

Here, and in the sequel, products run from 1 to oo and sums from - oo to oo unless otherwise indicated. We give here a simplified version of his argument by working directly with (1), the substitution leading to equation (3) of his paper being omitted. We then supply an alternative proof of (1) by means of a generalisation of the Durfee square concept utilising the rectangle of dimensions v by v + r for fixed r and maximal v contained in the Ferrers graph of a partition.
Note on Numerical Integration
Authors: Wolff C. E.
Published: January 2009

All the commonly used rules for the approximate quadrature of areas, such as those of Cotes, Simpson, Tchebychef and Gauss, are based on the assumption that y can be expressed as a rational integral function of x with finite coefficients. A tacit assumption is thus made that is not infinite within the range considered, and it is therefore hardly a matter for surprise that the degree of accuracy obtainable by the use of these rules in the case of a curve which touches the end ordinates is very poor.
Note on the Elementary Divisors of Some Related Determinants
Authors: Aitken A. C.
Published: January 2009

The matrices considered in the following note are non-singular, and are related to a given matrix A having elements taken from the field of positive and negative integers and zero. The invariant properties of such a matrix A, under multiplications by matrices of determinant equal to unity, can be formulated, as is well known, in terms of the “elementary divisors” of the determinant |A|. Thus if A is of the nth order, and p is a prime occurring in |A| to the power hn, in the H.C.F. of the first minors of |A| to the power hn−1, in the H.C.F. of the second minors to the power hn−2, and so on, h0 by convention being zero, then the first differences of the h's,

are invariant under the transformations considered, and it is known that r ≥ er−1. The numbers

where the product includes all prime factors of |A|, are called the elementary divisors of |A| and
A Generalization of Circulants
Authors: Bell E. T.
Published: January 2009

In connection with Bertrand's algebraical exercise of 1850, Muir remarks that it is not unlikely that the divisibility of a3 + b3 + c3 − 3abc by a + b + c had been previously noted, although. there is no record of the fact. Bertrand's exercise is to the effect that the circulant of the third order repeats under multiplication or, what is the same, admits of composition; the formulae of composition are stated in the exercise precisely as they would follow from Spottiswoode's theorem on the linear factors of a circulant. The whole of this is implied as an immediate special case, and indeed as one that any reader would construct at once, in an identity due to Lagrange, reproduced by Legendre.
An Application of Abel's Lemma to Double Series
Authors: Walsh C. E.
Published: January 2009

Let bmn be a positive function of m and n which decreases steadily with n, so that bmn ≥ bm, n+1 for all values of m and n. Assume also that |am1 + am+2 + … + bmn| < K for all values of m and n, K being finite. Denote bySmn the sum of the first n terms in the first m rows of the double series
On the Stieltjes Integral
Authors: Whittaker J. M.
Published: January 2009

The conception of the integral of one function with respect to another was introduced by Stieltjes in his classical memoir on continued fractions. He denned the integral as

and gave the formula for integration by parts.
Further Notes on the Stieltjes Integral
Authors: Hyslop J.
Published: January 2009

The definition here used of the Stieltjes Integral is the same as that of a previous note, viz.:—

Let f (x), φ (x) be two real functions defined in (a, b) a finite interval on the axis of the real variable x. Let Δ1, Δ2, …, Δn, be a finite set of sub-intervals which together make up (a, b). Δrφ denotes the increment of φ (x) in Δr. Let ξr be any point of Δr, and form the sum
The Composition of Linear Differential Systems
Authors: Whittaker J. M.
Published: January 2009

A common method of solving a linear differential equation consists in expressing the differential operator as a product of factors. The possibility of doing so has been studied extensively by Vessiot, following the work of Picard and Drach, on the lines of the Galois theory of algebraic equations. The analogous process of resolving a, linear differential system, consisting of an equation together with boundary conditions, into two or more systems of lower order does not seem to have been investigated. Such a resolution is not always possible, even in cases where the differential equation can be factorised. Thus the system

is equivalent to the systems

but the system

cannot be so resolved.
The Polar Properties of the Plane Trinodal Quartic Curve
Authors: Rutter Dryden
Published: January 2009

Clebsch was the first to investigate the properties of the plane quartic curve. The object of the present paper is to study the properties of the plane trinodal quartic curve.
Solution with Axial Symmetry of Einstein's Equations of Teleparallelism
Authors: Parsons J. D.
Published: January 2009

In a recent paper Dr G. C. McVittie discussed the solution with axial symmetry of Einstein's new field-equations in his Unified Field Theory of Gravitation and Electricity. Owing to an error in his calculation of the field equations, Dr McVittie did not obtain the general solution, which we discuss in the present paper.
The Tritangent Circles of a Circular Quartic Curve
Authors: Richmond H. W.
Published: January 2009

§1. In a recent paper with this title Prof. W. P. Milne has discussed the properties of the conics which pass through two fixed points of a plane quartic curve and touch the curve at three other points. In dealing with a numerous family of curves such as this it is very desirable to have a scheme of marks or labels to distinguish the different members of the family; Hesse's notation for the double tangents of a C4 illustrates this. By using another line of approach to the subject, by projecting the curve of intersection of a quadric and a cubic surface from a point at which (under exceptional circumstances) the surfaces touch, I find that a fairly simple notation for the 64 conics, in harmony with that for the bitangents, can be obtained. This paper, let it be said, from start to finish is no more than an adaptation of results known for the sextic space-curve referred to; it will be sufficient therefore to state results with short explanations.
Factorial Moments and Frequencies of Charlier's Type B
Authors: Campbell J. T.
Published: January 2009

When a given frequency distribution is to be graduated, it is customary to express the constants of the fitted curve in terms of the moments of the frequency distribution. The rth moment of the distribution, in which the relative frequency of a measure x is φ(x) or, in the case of a continuous variable, the differential of frequency is φ(x)dx, is defined in the respective cases by

the summation or integration being taken over the whole range of possible values of x. In the present paper we make use of another kind of moment, the factorial moment, which has already been considered by several writers, and which is specially suited to the case when the frequencies of the distribution are given for discrete, equidistant values of the variable. The (r + 1)th factorial moment, for the case where x, measured from some arbitrary origin, can increase by increments h, 2h, 3h,.…, will be defined to be

where the summation extends over all possible values of x; it will be denoted by m(r+1). By a suitable choice of scale the increments of x may be taken as equal to unity in any given case.
On Errors in Determinants
Authors: Etherington I. M. H.
Published: January 2009

§1. Introduction. Care is needed in dealing with determinants whose elements are subject to experimental error, particularly when a determinant itself is small compared with its first minors. For, as these examples show, a relatively tiny error in one element may be responsible for a large error in the determinant
On the Absolute Summability (A) of Infinite Series
Authors: Fekete M
Published: January 2009

§1. A series

has been defined by J. M. Whittaker to be absolutely summable (A), if

is convergent in (0 ≤ x < 1) and f (x) is of bounded variation in (0, 1), i.e.

for all subdivisions 0 = x0 < x1 < x2 < . … < xm < 1.
A Note on the Capelli Operators associated with a Symmetric Matrix
Authors: Wallace Andrew H.
Published: January 2009

In the Proceedings of the Edinburgh Mathematical Society, 1948, there appear two papers by Lars Gårding and Turnbull respectively (Gårding [1], Turnbull [2]) which formulate the theory of Cayley and Capelli operators associated with symmetric matrices. Turnbull derives the modification, appropriate to symmetric matrices, of Capelli's Theorem, which states that (taking a third order operator for the sake of ease in writing)

where the symbol (xyz)ijk stands for the determinant

with a similar meaning for the determinantal differential operator, while the symbols are polarisations (Capelli [1]; cf. Turnbull [1], p. 116). Gårding's theorem deals with the effect of such modified Capelli operators on powers of the determinant of the symmetric matrix in question. The subject of this note is an alternative derivation of the modified Capelli theorem and of Gårding's theorem.
A Type of Alternant
Authors: Ponting F. W.
Published: January 2009

We define

where ap ≠ aq when p ≠ q. If N = Σλi, then the partition (λ1, λ2, …, λn) of N with λ1 ≥ λ2 ≥ … ≥ λn is denoted by (λ) and we set

All partitions will be in descending order and the usual notation for repeated parts will be used.
Rings with Central Idempotent or Nilpotent Elements
Authors: Drazin M. P.
Published: January 2009

It is easy to see (cf. Theorem 1 below) that the centrality of all the nilpotent elements of a given associative ring implies the centrality of every idempotent element; and (Theorem 7) these two properties are in fact equivalent in any regular ring. We establish in this note various conditions, some necessary and some sufficient, for the centrality of nilpotent or idempotent elements in the wider class of π-regular rings (in Theorems 1, 2, 3 and 4 the rings in question are not even required to be π-regular).
On a Set of Conform-Invariant Equations of the Gravitational Field
Authors: Buchdahl H. A.
Published: January 2009

Eddington has considered equations of the gravitational field in empty space which are of the fourth differential order, viz. the sets of equations which express the vanishing of the Hamiltonian derivatives of certain fundamental invariants. The author has shown that a wide class of such equations are satisfied by any solution of the equations

where Gμν and gμν are the components of the Ricci tensor and the metrical tensor respectively, whilst λ is an arbitrary constant. For a V4 this applies in particular when the invariant referred to above is chosen from the set

where Bμνσρ is the covariant curvature tensor. K3 has been included since, according to a result due to Lanczos3, its Hamiltonian derivative is a linear combination of and , i.e. of the Hamiltonian derivatives of K1 and K2. In fact
A Determinantal Expansion for a Class of Definite Integral
Authors: Shenton L. R.
Published: January 2009

We develop here the recurrence relations for the generalised C.F.'s introduced in Part 3 (Shenton 1956). In the main the discussion will be limited to second order C.F.'s, but results for higher orders will be given when these are not complicated.
Sir Edmund Whittaker, F.R.S.
Authors: Martin D.
Published: January 2009

Edmund Taylor Whittaker was born at Birkdale, Lancashire, on 24th October 1873 and belonged to a family which had been known for several generations in the district where the Rabble forms the boundary between Lancashire and Yorkshire. He entered Manchester Grammar School in 1885 and spent two years in the Classical side before turning over to the Mathematical side. In December 1891 he won a scholarship to Trinity College, Cambridge, and, after an undergraduate career of unusual distinction, was bracketed with J. H. Grace as Second Wrangler in the Tripos of 1895 ; Bromwich was Senior Wrangler. In 1896 Whittaker was elected a Fellow of Trinity College and in 1897 was first Smith's prizeman, his essay being entitled “On the reduction of the theory of multiform functions to uniform functions.”
Parseval Relations for Kontorovich-Lebedev Transforms
Authors: Lowndes J. S.
Published: January 2009

The Kontorovich-Lebedev Transform of a function f(r), 0<r<∞, may be written in a general form as fg(μ) where,
On Integral Transform Pairs Arising from Second Order Differential Equations
Authors: Mercer A. McD.
Published: January 2009

We present in this paper a straight-forward method of deducing integral transform pairs directly from second order differential equations. In (1) this subject has been very thoroughly treated and in view of this we believe that the method described here has its greatest justification in its simplicity.
The j-Differential and its Integral
Authors: Ingram W. H.
Published: January 2009

The reciprocity, , has been shown to hold when g(x) is in B* [Proc. Edin. Math. Soc., 12 (1960), 85]. The purpose of the present note is to show that it holds more generally and in particular when g(x) is in a subclass B″, of functions of bounded variation B, such that B* ⊂ B″ ⊂ B′ ⊂ B; it is assumed that f(x) is any function in D1 and that f(x) and g(x) are both continuous on the left and possibly with simultaneous points of discontinuity.