bluuebunny/crossref_metadata_2025 huggingface dataset:
Presentations of faithful d.g. near-rings
Authors: Meldrum J. D. P.
Published: January 2009

Many basic definitions and results in the theory of near-rings can be found in G. Pilz (4). We follow these for the most part, except that we use left near-rings rather than right near-rings. We follow exactly an earlier paper, Meldrum (2), where there are detailed definitions and many results relating to faithful d.g. near-rings. Let R be a d.g. near-ring, distributively generated by the semigroup S, which need not be the semigroup of all distributive elements. Denote such a d.g. near-ring by (R, S). Then (R, +) = Gp < S; > where is a set of defining relations in S. Let (T, U) be a d.g. near-ring. Then a d.g. homomorphism θ from (R, S) to (T, U) is a near-ring homomorphism from R to T which satisfies Sθ ⊆ U. If (G, +) is a group, let T0(G) be the near-ring of all maps from G to itself with pointwise addition and map composition. Let End G be the semigroup of all endomorphisms of G. Then (E(G), End G) is a d.g. near-ring. A d.g. near-ring (R, S) is faithful if there exists a d.g. monomorphism θ:(R, S) → (E(G), End G) for some group G.
Centres for near-rings: applications to commutativity theorems
Authors: Bell Howard E.
Published: January 2009

Let R be an arbitrary near-ring and define the multiplicative centre Z(R) by

In previous papers (2,3,5) we have established additive or multiplicative commutativity for various near-rings R in which selected elements were restricted to lie in Z(R); the near-rings involved were usually distributively-generated (d-g) and were frequently assumed to have a multiplicative identity element as well.
The near-rings hosted by a class of groups
Authors: Laxton R. R. , Lockhart R.
Published: January 2009

Clay (3), Johnson (5) and Krimmel (6) have each considered the near-rings with identity on dihedral groups. Krimmel actually generalised the class of dihedral groups and investigated the class of finite non-abelian groups with a cyclic normal subgroup of prime index: we shall call this class . Krimmel considered the near-rings with identity that might be denned on members of and he determined the subclass of groups in which support near-rings of this kind. He also managed to calculate the number of non-isomorphic near-rings involved for certain cases. His methods were essentially combinatorial, and his results were expressed in terms of various integers which characterised the individual members of . Certain features of this work led us to investigate the structure of near-rings on members of from a more algebraic point of view and thereby to complete and extend Krimmel's programme. Part of the work in this paper formed the basis of the second author's thesis (7). We should like to thank Dr. J. Krimmel for permission to include some of his results, and Dr. J. Meldrum who detected an error in our original formulation of Theorem 7.1.
Near-rings of compatible functions
Authors: Pilz Günter
Published: January 2009

In this paper we study near-rings of functions on Ω-groups which are compatible with all congruence relations. Polynomial functions, for instance, are of this type. We employ the structure theory for near-rings to get results for the theory of compatible and polynomial functions (affine completeness, etc.). For notations and results concerning near-rings see e.g. (10). However, we review briefly some terminology from there. (N, +,.) is a near-ring if (N, +) is a group and . is associative and right distributive over +. For instance,M(A): = (AA, +, °) is a near-ring for any group (A, +) (° is composition). IfNis a near-ring thenN0: = {n∈N/n0 = 0}. A group (Γ, +) is anN-group (we writeNΓ) if a “product”nyis defined with (n+n‛)γ =nγ +n‛γ and (nn‛)γ =n(n‛γ). Ideals of near-rings andN-groups are kernels of (N-) homomorphisms. If Γ is a vector-space,Maff(Γ) is the near-ring of all affine transformations on Γ.Nis 2-primitive onNΓ ifNΓ is non-trivial, faithful and without properN-subgroups. The (2-) radical and (2-) semisimplicity are defined similarly to the ring case.
Near-rings with chain conditions on right annihilators
Authors: Oswald A.
Published: January 2009

Throughout this note, N will denote a (Left) near-ring with two-sided zero. Definitions of basic concepts can be found in (9).

We prove first that a right ideal I in a d.g. near-ring has a right identity if and only if x ∈ xI for each x ∈I. This enables us to study the structure of regular d.g. near-rings with chain conditions on right annihilators. Specifically we will prove that a regular d.g. near-ring with both the maximum and the minimum conditions on right annihilators is a finite direct sum of near-rings which are either rings of matrices over division rings or non-rings of the form MG(Γ) for a suitable type 2 N-module Γ. Finally we consider the case of maximum condition on N-subgroups.
On derivations of fields of algebraic functions
Authors: Bastida Julio R.
Published: January 2009

In this note, we shall prove that the transcendency degree of a finitely generated field extension is equal to a certain integer associated with a restriction mapping of spaces of derivations.
A triple in CAT
Authors: Wells Charles
Published: January 2009

A triple (or monad) in a category K is a triple = (T, μ, η) where, T: K → K is a functor and μ: TT →, T, η: 1k → T are natural transformations for which (1.1) and (1.2) commute:

In these diagrams the component of a natural transformation α at an object x is denoted xα. Thus for example (kη)T is the value of the functor T applied to the component of η at k, whereas (kT)η is the component of η at the object kT. I write functions and functors on the right and composition from left to right.
Functorial radicals and non-abelian torsion
Authors: Feigelstock Shalom , Klein Aaron
Published: January 2009

Radicals appear in many algebraic contents. For modules over a ring, they give rise to pre-torsion and torsion theories, Goldman (5), Lambek (14). In the category of groups, Kurosh, Plotkin and others have introduced radicals (6), (13), (21), but unlike the radicals in module theory these radicals are not necessarily functorial, as for example the nil radical and the Hirsch-Plotkin radical (6). The functorial method in module theory has been extended to abelian categories, Dickson (2), to the category of nilpotent groups, Hilton (8), Warfield (25), and to the category of groups, Plotkin (22), and to general categories, Wiegandt (26), Holcombe and Walker (10).
Ergodic properties of Brownian Motion
Authors: Ugbebor Olabisi O.
Published: January 2009

Since Brownian motion is point recurrent in R1, recurrent in R2 and transient in Rn, n ≧ 3 (see (7)), it follows that the total time spent in a bounded open set in R1 or R2 is unbounded. With the following ergodic theorems for Brownian motion in R1 and R2 as motivation, we examine the rate of convergence in these theorems. Note that there is no ergodic property in Rn for n ≧ 3 since Brownian motion is not dense there.
Nonimmersions of complex Grassmann manifolds
Authors: Ilori Samuel A.
Published: January 2009

If an oriented manifold M immerses in codimension k, then the normal bundle has dimension k such that its Euler class χ є Hk(M; Z) and χ2 є H2k(M; Z). (Cf. (3)).

If M is the complex Grassmann manifold G2(Cn) of 2-planes in Cn (n = 4, 5,…, 15, 17), then dim M = 4n – 8 ≡ d and we shall show that although M immerses in R2d–1 by classical results (3), M does not immerse in Rd+d/2.

The same result was obtained for n = 4 and 5 by Connell (2) and for n = 6 and 7 by the author (6). The nonimmersion results of this paper are new for n = 8, 9, …, 15, 17 and they are an improvement over the result for the general G2(Cn) obtained in (5). In this paper, we use generators of the cohomology ring of G2(Cn) different from those used in (2) and (6) and this simplifies the calculations considerably.
On the attached prime ideals of certain Artinian local cohomology modules
Authors: Sharp R. Y.
Published: January 2009

The study of the cohomological dimensions of algebraic varieties has produced some interesting results and problems in local algebra: the general local problem is that posed by Hartshorne and Speiser in (4, p. 57). We consider a (commutative, Noetherian) local ring A (with identity), a proper ideal a of A, and ask the following question.
On certain cycles in graphs
Authors: Grant Douglas D.
Published: January 2009

We show that every simple graph of order 2rand minimum degree ≧4r/3 has the property that for any partition of its vertex set into 2-subsets, there is a cycle which contains exactly one vertex from each 2-subset. We show that the bound 4r/3 cannot be lowered tor, but conjecture that it can be lowered tor+ 1.
Higher point derivations on commutative Banach algebras, III
Authors: Dales H. G. , McClure J. P.
Published: January 2009

Let A be a commutative Banach algebra with identity 1 over the complex field C, and let d0 be a character on A. We recall that a (higher) point derivation of order q on A at d0 is a sequence d1, …, dq of linear functionals on A such that the identities

hold for each choice of f and g in A and k in {1, …, q}. A point derivation of infinite order is an infinite sequence {dk} of linear functionals such that (1.1) holds for all k. A point derivation is continuous if each dk is continuous, totally discontinuous if dk is discontinuous for each k≧1, and degenerate if d1 = 0.
Characters on C*-algebras, the joint normal spectrum, and a pseudo-differential C*-algebra
Authors: Power S. C.
Published: January 2009

Let {Ai; i ∈ Ω} be a family of C*-algebras acting on a Hubert space H and let A be the C*-algebra that they generate. We shall assume throughout that C*-algebras always contain the identity operator. Let M(A) denote the space of characters, that is, multiplicative linear functionals, acting on A, with the weak star topology. We obtain here a natural characterisation of M(A) as a subset of the product space determined by {M(Ai); i ∈ Ω}. In the case of singly generated C*-algebras this characterisation is related to the joint normal spectrum (5) of a family of operators.
On cyclic subgroups of finite groups
Authors: Dempwolff U. , Wong S. K.
Published: January 2009

In [3] Laffey has shown that if Z is a cyclic subgroup of a finite subgroup G, then either a nontrivial subgroup of Z is normal in the Fitting subgroup F(G) or there exists a g in G such that Zg∩Z = 1. In this note we offer a simple proof of the following generalisation of that result:

Theorem. Let G be a finite group and X and Y cyclic subgroups of G. Then there exists a g in G such that Xg∩Y⊴F(G).
Semisimple classes of alternative rings
Authors: Anderson T. , Wiegandt R.
Published: January 2009

Recently A. D. Sands [7] solved a problem posed in [6], and characterised the semisimple classes of associative rings as classes being regular, coinductive and closed under extensions. It is the purpose of this note to prove the same assertion for alternative rings. This result is perhaps not surprising, nevertheless its proof cannot be considered an easy one, and it requires a technique of dealing with ideals of ideals. In addition, semisimple classes of hereditary radicals and those of supernilpotent radicals will be characterised as easy consequences of our theorem.
On positively complemented subspaces of c0
Authors: Tsekrekos Panayotis C.
Published: January 2009

It has been proved, see [1], that a closed infinite dimensional subspace of c0 is isomorphic to c0 if and only if it is the range of a bounded linear projection. In [6] we proved half of the order-theoretic analogue of this result. In fact we showed that an infinite dimensional subspace of c0 which is the range of a positive projection is order-isomorphic to c0. We left open the question whether the converse holds also true. In this paper we answer this question negatively by providing an example in Section 4. In Section 3 we give necessary and sufficient conditions in order that an ordered-subspace of c0 be the range of a positive projection.
Zero sets—consequences for primitive near-rings
Authors: Scott S. D.
Published: January 2009

Throughout this paper all near-rings will be left distributive. We shall denote the zero-symmetric part of a near-ring N by N0. The fact that the near-rings under consideration may not be zero-symmetric has important consequences for what follows, particularly the results of the last section.
A remark on the radical of a group algebra
Authors: Brockhaus Peter
Published: January 2009

This paper contains a new proof of a theorem of D. A. R. Wallace [5] in case of a p-solvable group. An alternative proof has been given by K. Motose [4].

Let G be a finite group, F a field with prime characteristic p, P a Sylow p-subgroup of G. JFG will denote the Jacobson radical of FG.
The subnormal subgroup structure of the infinite general linear group
Authors: Arrell David G.
Published: January 2009

Let R be a ring with identity, let Ω be an infinite set and let M be the free R-module R(Ω). In [1] we investigated the problem of locating and classifying the normal subgroups of GL(Ω, R), the group of units of the endomorphism ring EndRM, where R was an arbitrary ring with identity. (This extended the work of [3] and [8] where it was necessary for R to satisfy certain finiteness conditions.) When R is a division ring, the complete classification of the normal subgroups of GL(Ω, R) is given in [9] and the corresponding results for a Hilbert space are given in [6] and [7]. The object of this paper is to extend the methods of [1] to yield a classification of the subnormal subgroups of GL(Ω, R) along the lines of that given by Wilson in [10] in the finite dimensional case.
Rigid Artinian rings
Authors: McLean K. R.
Published: January 2009

In [4], Maxson studied the properties of a ring R whose only ring endomorphisms φ: R → R are the trivial ones, namely the identity map, idR, and the map 0R given by φ(R) = 0. We shall say that any such ring is rigid, slightly extending the definition used in [4] by dropping the restriction that R2 ≠ 0. Maxson's most detailed results concerned the structure of rigid artinian rings, and our main aim is to complete this part of his investigation by establishing the following

Theorem. Let R(≠0) be a left-artinian ring. Then R is rigid if and only if

(i) , the ring of integers modulo a prime power pk,

(ii) R ≅ N2, the null ring on a cyclic group of order 2, or

(iii) R is a rigid field of characteristic zero.
Finite automorphism groups of laminated near-rings
Authors: Magill K. D. , Misra P. R. , Tewari U. B.
Published: January 2009

In [3] we initiated our study of the automorphism groups of a certain class of near-rings. Specifically, let P be any complex polynomial and let P denote the near-ring of all continuous selfmaps of the complex plane where addition of functions is pointwise and the product fg of two functions f and g in P is defined by fg=f∘P∘g. The near-ring P is referred to as a laminated near-ring with laminating element P. In [3], we characterised those polynomials P(z)=anzn + an−1zn−1 +…+a0 for which Aut P is a finite group. We are able to show that Aut P is finite if and only if Deg P≧3 and ai ≠ 0 for some i ≠ 0, n. In addition, we were able to completely determine those infinite groups which occur as automorphism groups of the near-rings P. There are exactly three of them. One is GL(2) the full linear group of all real 2×2 nonsingular matrices and the other two are subgroups of GL(2). In this paper, we begin our study of the finite automorphism groups of the near-rings P. We get a result which, in contrast to the situation for the infinite automorphism groups, shows that infinitely many finite groups occur as automorphism groups of the near-rings under consideration. In addition to this and other results, we completely determine Aut P when the coefficients of P are real and Deg P = 3 or 4.
On Brown–McCoy radical classes in categories
Authors: Veldsman S.
Published: January 2009

What does a simple ring with unity, a topological T0-space and a graph that has at most one loop but may possess edges, have in common? In this note we show that they all are Brown–McCoy semisimple. Suliński has generalised the well-known Brown–McCoy radical class of associative rings (cf. [1]) to a category which satisfies certain conditions. In [3] he defines a simple object, a modular class of objects and the Brown–McCoy radical class as the upper radical class determined by a modular class in a category which, among others, has a zero object and kernels. To include categories like that of topological spaces and graphs, we use the concepts of a trivial object and a fibre. We then follow Suliński and define a simple object, a modular class of objects and then the Brown–McCoy radical class as the upper radical class determined by a modular class.
Upper faithful d.g. near-rings
Authors: Meldrum J. D. P.
Published: January 2009

This paper is concerned with d.g. near-rings and their relationship to faithful d.g. near-rings. For general definitions and results, we refer to Pilz [5]. We use left near-rings where he uses right near-rings, but otherwise there is little difference. This work follows earlier work [3], [4] and Mahmood [2]. Before outlining the contents of the paper we present a précis of the definitions.
Metric and algebraic perturbations of function algebras
Authors: Jarosz Krzysztof
Published: January 2009

Let A and B be function algebras. We generalise the Nagasawa theorem by proving that the Banach–Mazur distance between the underlying Banach spaces of A and B, is close to one if and only if they are almost isomorphic, that is if and only if there is a linear map T from A onto B such that ∥T−1(Tf · Tg)−fg∥≦ε∥f∥∥g∥.
On semigroup algebras and semisimple semilattice sums of rings
Authors: Teply Mark L.
Published: January 2009

Let P be a semilattice. In (5), a ring T is called a supplementary semilattice sum of subringsTα (α∈P) if the following conditions hold: TαTβ⊆Tαβ for all α,β∈P, and for each α∈P. Thus, as an abelian group, T is a direct sum of the additive subgroups Tα (α∈P), and the multiplicative structure of T is strongly influenced by the semilattice P. Properties of these rings have been studied extensively in (2), (3), (5), and (6).
Note on Partial Fractions and Determinants
Authors: Turnbull H. W.
Published: January 2009

In looking for a compact way of writing down the partial fraction formula in general, with repeated factors, I noticed how the expansion of a determinant by its top or bottom row suggested a method. The following gives a formula perfectly easy to write down in any given case where the factors of the denominator of the fraction are known. Incidentally it gives, as a determinant, the integral of a rational fraction f(x)/Q(x) where f(x) and Q(x) are polynomials, Q(x) having higher order.
On Determinants of Symmetric Functions
Authors: Aitken A. C.
Published: January 2009

The result of dividing the alternant |aαbβcγ…| by the simplest alternant |a0b1c2…| (the difference-product of a, b, c, …) is known to be a symmetric function expressible in two distinct ways, (1) as a determinant having for elements the elementary symmetric functions C, of a, b, c, …, (2) as a determinant having for elements the complete homogeneous symmetric functions Hr. For example

The formation of the (historically earlier) H-determinant is evident. The suffixes in the first row are the indices of the alternant; those of the other rows decrease by unit steps. This result is due to Jacobi.
On Differentiating a Matrix
Authors: Turnbull H. W.
Published: January 2009

The theorem is well known. So also is the theorem that if concerning a determinant Λ and its reciprocal expressed by means of cofactors Aij of aij. Not quite so well known is the Cayley Hamilton theorem that a matrix X =[xij] satisfies its own characteristic equation

Unlike as these three results are, they nevertheless can be looked upon as particular phases of a general theorem concerning a matrix differential operator acting upon a function of a matrix X or its transposed.
On Hardy's Theory of m-Functions
Authors: Copson E. T.
Published: January 2009

§1. The Cardinal Function of Interpolation Theory is the function

which takes the values an at the points x = n. Ferrar has recently proved

Theorem1. If are convergent, C(x)is an m-function3for

This means that C(x) is a solution of the intergral equation

Ferrar's proof deals with functions of a real variable and involves some rather difficult double limit considerations.
Spherical Simplexes in n-dimensions
Authors: SEN R. N.
Published: January 2009

All points in an n-space equidistant from a fixed point (the centre) constitute what may be called a spherical continuum of the nth order,—the continuum being of n — 1 dimensions ((n — l)-dimensional spread) and of the 2nd degree. Any region of this spherical continuum bounded by n (n — l)-dimensional linear continua or, primes (spaces of n — 1 dimensions), passing through the centre shall be called a spherical simplex of the nth order. This spherical simplex is bounded by n faces, spherical simplexes of the (n — l)th order, each of which in turn is bounded by n — 1 spherical simplexes of the (n — 2)th order, and so on till we reach spherical triangles, arcs and lastly points, the vertices. The total number of spherical simplexes of different orders connected with one of the nth order is 2n — 2. The n spherical continua of the (n — l)th order which contain the faces of the spherical simplex of the nth order determine a set of 2n spherical simplexes of the same order, 2n–1 pairs, the two spherical simplexes of a pair being symmetrically situated with respect to the centre and therefore congruent.
The Complete System of Linear Complexes and one Quaternary Quadric
Authors: Das Gupta P. N. , Turnbull H. W.
Published: January 2009

In the following pages we give a complete system of projective concomitants for any number of linear complexes and one quadric, in a quaternary field. The investigation follows the earlier1 paper on mixed quaternary forms, in which the corresponding system, omitting the quadric, was given. Such a system, for one quadric and one linear complex was given by Weitzenböck who used complex symbols. A detailed investigation of the case of two linear complexes and a quadric, together with their geometry, has been given elsewhere.
On the Number of Ways of Colouring a Map
Authors: Birkhoff George D.
Published: January 2009

It is well known that any map of n regions on a sphere may be coloured in five or fewer colours. The purpose of the present note is to prove the following

Theorem. If Pn(λ)denotes the number of ways of colouring any ma: of n regions on the sphere in λ (or fewer) colours, then

(1)

This inequality obviously holds for λ = 1, 2, 3 so that we may confine attention to the case λ > 4. Furthermore it holds for n = 3, 4 since the first region may be coloured in λ ways, the second in at least λ — 1 ways, the third in at least λ — 2 ways, and the fourth, if there be one, in at least λ — 3 ways.
On Rational Solutions of x3 + y3 +z3 = R.
Authors: Richmond H. W.
Published: January 2009

The earliest proof that every rational number (R) can be expressed as a sum of cubes of three rational numbers (x, y, z), not necessarily positive, was published in 1825 by S. Ryley, a schoolmaster of Leeds: formulae were given for x, y, z in terms of a parameter, such that every value of the parameter led to a system of values of x, y, z satisfying the above relation, and every rational value of the parameter led to a system of rational values of x, y, z. The later solutions referred to by Dickson are found to give the same results as Ryley's formula, as does another method, quoted in a modified form by Landau from a paper by the present writer. Thus it might almost be believed that Ryley's century-old result embodies all that is known with regard to the resolution of a number into three cubes, and that his formula is unique. I propose to examine the rationale of his method and the causes of its success; it will then appear that an infinity of similar formulae exist, and that one of them is at least as simple as his. It is convenient to state Ryley's formula, and the modification made by Landau, in section 2; and to generalize the method in section 3.
The Absolute Summability (A) of Fourier Series
Authors: Prasad B. N.
Published: January 2009

In a recent paper Dr J. M. Whittaker has shown that the Fourier series

of a function f(θ) which has a Lebesgue integral in (— π, π), is absolutely summable (A) to sum l, if

exists, where

.
Note on Dual Symmetric Functions
Authors: Aitken A. C.
Published: January 2009

In an earlier paper, which this note is intended to supplement and in some respects improve, the writer gave a general theorem of duality relating to isobaric determinants with elements Cr and Hr, the elementary and the complete homogeneous symmetric functions of a set of variables. The result was shewn to include as special cases the dual forms of “bi-alternant” symmetric functions given by Jacobi and Naegelsbach, as well as two equivalent forms of isobaric determinant used by MacMahon as a generating function in an important problem of permutations.
The Milne Quadrics of the Trinodal Cubic Surface, and the Contact Conics of the Harmonic Envelope of the Plane Trinodal Quartic
Authors: Blakey J. , Milne W. P.
Published: January 2009

There are two planes, called “parabolic planes,” through eachline on the cubic surface cutting the surface in residual conics touching the line in question at two points called the “parabolic points” of that line.
On Professor Whittaker's solution of differential equations by definite integrals: Part II
Authors: Kermack W. O. , McCrea W. H.
Published: January 2009

In Part I it has been shown that, given a contact transformation, two equations

can be derived which lead to the compatible differential equations

It will be shown in the present communication that the necessary and sufficient condition that (1.3), (1.4) should be compatible is that

regarded as an equation in the non-commutative variables q, p which themselves satisfy the condition

We shall call functions satisfying this condition conjugate functions. From this point of view the method employed by Professor Whittaker in his original paper, involving the use of a contact transformation,, was really a particular method of generating conjugate functions. This powerful method may be supplemented and extended by the other methods developed in the following pages.
Matrix Differentiation of the Characteristic Function
Authors: Turnbull H. W.
Published: January 2009

The following work is a sequel to three previous communications, and more particularly to the first. The present object is to shew the effect of repeated operation with the matrix differential operator , when it acts upon a scalar matrix formed from an n rowed determinant |xij|, or sums of principal minors, the n2 elements xij being treated as independent variables. Thus when z is a scalar quantity ω z means the matrix [∂z/∂xij], whose ijth element is the derivative.
Professor GEORGE A. GIBSON, M.A., LL.D., F.R.S.E.
Authors: Bell R. J. T.
Published: January 2009

George Alexander Gibson was born at Greenlaw, Berwickshire, in 1858. His father was a man of rare capacity and character who wrote a history of Greenlaw and taught himself Latin so as to be able to obtain the material for his history from original documents.

After attending an elementary school at Greenlaw, Gibson enrolled in 1874 as a student at the University of Glasgow, where he gained the highest places in all the classes of the Arts curriculum. At this period of his life his health was not robust and it threatened to terminate his university studies. Consequently, in 1881, he took the degree of M.A. without honours. In the following year he was able to sit the honours examination in Mathematics and Natural Philosophy, and he so distinguished himself that he was awarded the Ewing Fellowship. When the degree of D.Sc. was afterwards insti tuted and the regulation required that a candidate for it should have an honours degree in Arts or Science, his ordinary degree precluded Gibson from becoming a candidate for the doctorate which his published work would doubtless have gained for him.
Sets of Semi-Commutative Matrices
Authors: Williamson J.
Published: January 2009

In a paper entitled “Sets of anticommuting matrices” Eddington proved that if El, E2, …., Eqform a set of q four-rowed square matrices satisfying the relations,

,

where E is the unit matrix, then the maximum value of q is five. Later Newman showed that this result is a particular case of the general theorem that ifE1, E2, …., Eqform a set of q t-rowed square matrices satisfying (1), where t = 2Pτ and τ is odd, then the maximum value of q is 2p + 1.
A Matrix Representation of Ascending and Descending Continued Fractions
Authors: Milne-Thomson L. M.
Published: January 2009

The present paper describes briefly a notation for representing continued fractions in many dimensions, which has the advantage providing a direct method of attack and of rendering intuitive, results which are usually proved by induction. The notation is the outcome of a generalisation which I previously made [1] in connection with the solution of certain difference equations. Only formal theorems are considered here. For a discussion of convergence reference may be made to the works [2, 3, 4, 5] cited at the end. The paper by Paley and XJrsell is particularly important since these authors discuss very fully the non-cyclic simple continued fraction
An Approximation connected with e-x
Authors: Copson E. T.
Published: January 2009

One of the many interesting problems discussed by Ramanujan is concerned with the effect of truncating at its maximum term nn/n! the exponential series for en, where n is a positive integer. When n is large, the sum of the first n terms is, roughly speaking, half the sum of the whole series.
Reduction of Singular Pencils of Matrices
Authors: Ledermann W.
Published: January 2009

Let ρA + σB =[ραμν + σbμν] be a pencil of type m × m′, i.e. with m rows and m′ columns, where A and B are matrices with constant elements which are not mere scalar multiples of each other; and ρ and σ are homogeneous parameters.
The problem of the in-and-circumscribed polygon for a plane quartic curve
Authors: Edge W. L.
Published: January 2009

Consider a plane curve C of order n and class X; it is to be supposed throughout that C has only ordinary Plücker singularities, i.e. nodes, cusps, inflections and bitangents. Through any point P1 of C there pass, apart from the tangent at P1 itself, X – 2 lines which touch C; let T12 be the point of contact of any one of these tangents and P2 any one of the n – 3 further intersections of P1T12 with C. Through P2 there pass, apart from the tangent at P2 itself and the line P2P1, X — 3 lines which touch C; let T23 be the point of contact of any one of these with C and P3 any one of its n — 3 further intersections with C.
Gauss' Theorem in a General Space-time
Authors: Ruse H. S.
Published: January 2009

In classical mechanics Gauss' Theorem for a gravitational field states that, if S is a closed surface and N the component of gravitational force along the outward normal, then

where β is the Newtonian constant of gravitation and M is the total mass inside S. This result has recently been extended to general relativity by E. T. Whittaker,1 who, however, considered only the case of a statical gravitational field, the line-element of which is given by2

where the coefficients U and αμν are independent of t. It is not immediately clear from his work whether the results are extensible to more general space-times.
Note on Relativistic Mechanics
Authors: Walker A. G.
Published: January 2009

The object of this note is to derive a form of Poisson's equation from general relativistic mechanics, without assuming the field to be either static or “weak”. The problem is essentially a “local” problem, all observations being made by one observer; this observer determines the apparent gravitational field in his vicinity by observing the motions of free (isolated) particles. Defining gravitational mass by means of Poisson's equation, we find the relation between the densities of gravitational and inertial mass relative to any observer. We also find what may be called the non-rotating frame of reference belonging to any observer.
The Veronesean of quadrics and associated loci
Authors: Head J. W.
Published: January 2009

In this paper we consider the correspondence between tangential quadrics of [3] and points of [9]. Godeaux has considered this geometrically, with the object of obtaining a representation for a twisted cubic of three dimensions. We have considered it from a standpoint more algebraic than that of Godeaux, with particular reference to the types of pencils of quadrics that correspond to special lines of [9], and to the interpretation in [9] of the fact that the condition for a net of quadrics to be part of the polar system of a cubic surface is poristic.
On a function which is self-reciprocal in the Hankel transform
Authors: Phillips E. G.
Published: January 2009

It is well-known that if v≧–½ and

then ø (x) is said to be self-reciprocal in the Hankel transform and may be described as Rv. If v = ±½(1) reduces to the Fourier sine or cosine transform. Functions of these two classes may be described as Rg and Re.

In a list of pairs of reciprocal functions, G. A. Campbell gives the example that

is Rc, but there does not seem to be any explicit reference to this function in the literature. It suggests that there is a corresponding function which is Rv.
A further note on multivariate selection
Authors: Aitken A. C.
Published: January 2009

In an earlier “Note on Selection from a Normal Multivariate Population” the author considered the transformation of an n-variate normal population induced by “selection” operating to alter the parameters of frequency in a subset of p variates, while preserving normal frequency in the subset. The results were new in notation only; in substance they were first found by K. Pearson.
On extensions of Pascal's theorem
Authors: Richmond H. W.
Published: January 2009

The object of this paper is firstly to extend the theorem of Pascal concerning six points of a conic to sets of 2 (n + 1) points of the rational normal curve of order n in space of n dimensions; secondly to explain why a wider extension to other sets of 2 (n + 1) points in [n] must be sought; and lastly to give briefly an extension to [3] and [4] which will be further generalised in a later paper. The striking feature of Pascal's theorem—that each of the sixty ways of arranging the points in a cycle, or as vertices of a closed polygon, leads to a different version of the theorem—is retained in the following extension to [n].
On bounded skew-symmetric forms
Authors: Wintner Aurel
Published: January 2009

It is well known that the real, skew-symmetric, non-singular, bilinear forms of n + n variables have no invariants. In fact, each of these forms may be transformed into one and the same form, for instance into the one which occurs in the usual representation of the complex group. The standard proofs of this theorem break down in case of infinite forms which are bounded in the sense of Hilbert, one of the impediments being the possibility of a continuous spectrum. The object of this note is to show that, while the usual proofs break down, the theorem itself is true in Hubert's case also.
On the concept of gravitational force and Gauss's theorem in general relativity
Authors: Synge J. L. , Ruse H. S.
Published: January 2009

Whittaker and Ruse have developed forms of Gauss's theorem in general relativity, their theorems connecting integrals of normal force taken over a closed 2-space V2 with integrals involving the distribution of matter taken over an open 3-space bounded by V2. The definition of force employed by them involves the introduction of a normal congruence (with unit tangent vector λi), the “force” relative to the congruence being the negative of the first curvature vector of the congruence (– δλi/δs). This appears at first sight a natural enough definition, because – δλi/δs at an event P represents the acceleration relative to the congruence of a free particle travelling along a geodesic tangent to the congruence at P. In order to give physical meaning to this definition of force it is necessary to specify the congruence λi physically, and it would seem most natural to choose the congruence of world-lines of flow of the medium. Supposing certain conditions satisfied by this congruence (cf. Ruse, loc. cit.), the theory of Ruse is applicable, and from this follows a form of Gauss's theorem.
Some sufficient conditions for the absolute Cesàro summability of series
Authors: Hyslop J. M.
Published: January 2009

denote the n-th Cesàro mean of order k for the series aΣan, that is,

where

and let
The configuration determined by five generators of a quadric threefold
Authors: Brown L. M.
Published: January 2009

Through four generally placed lines in space of four dimensions there passes a doubly infinite system of quadric primals, but through five lines there pass in general no quadrics. It therefore follows that there must exist some special relationship between five lines in order that they may be generators of a quadric. This problem has been discussed by Richmond,1 who gives a condition which is in a restricted sense an extension of Pascal's Theorem. The five lines being taken in order certain points may be obtained which lie in a space. In Section I we state Richmond's criterion and show that it is sufficient as well as necessary. Section II is concerned with the twelve spaces which arise if all the different possible orders of the lines are considered. They cut by pairs in six planes whose configuration is developed. In Section III other lines connected with the configuration are introduced. It is shown that by taking crossers of the lines of our original figure in a certain manner five further generators are obtained, and that the same entire configuration of generators arises whether we begin with the five original or the five final lines. Furthermore, though the twelve spaces analogous to Pascal lines obtained from the final five are new, yet the six planes, their intersections by pairs, and the configuration dependent from them, are the same as those constructed from the original five.
Functions with Dominant Polar Singularities on the Circle of Convergence
Authors: Wilson R.
Published: January 2009

1. Introduction. In an earlier paper1 the function

was considered, having on its circle of convergence, taken to be |z|=1 only isolated essential points of finite exponential order, situated at the points eiav (v = 1, 2, .., k). It was there proved,2 in the cases k = 2, k = 3, that the upper density of small coefficients is positive only if the points eiav are situated at some of the vertices of a regular polygon inscribed in the circle of convergence and if the singularities are virtually identical or linearly related. In this case the sequence of small coefficients possesses a density. The coefficients cn can be interpolated in the form

where the Gv(z) are integral functions of order less than 1, so that the result stated may be regarded as expressing an arithmetical property of a certain type of integral function at positive integer points.
The Linear j-Differential Equation
Authors: Ingram W. H.
Published: January 2009

The basic reciprocity of j-differential and LM-integral

for bounded functions f(x) with simple discontinuities but continuous on the left at each point and for g(x) in the somewhat restricted class B of functions of bounded variation and also left-continuous, was established in (2) and (3); the dot here indicates the lower product of and (jg, g+ (x+)dx), with , and the integral indicated is the RJDS-integral, equivalent to (LM) .
A Generalisation of Dirichlet's Multiple Integral
Authors: Jack Henry
Published: January 2009

A previous note (2) showed how the integral of f(1x1+2x2++nxn) over the interior of a simplex could be reduced to a contour integral. The same idea is applied here in Theorems 1 and 2 to give a generalisation of Dirichlet's multiple integral ((1), pp. 169172). These results are then used in Theorem 3 to reduce an integral over all real n-dimensional space to a contour integral. In Theorem 4 an integral over the group of all 3 3 orthogonal matrices of determinant 1 is reduced to a contour integral. This result can be extended formally to the case of 4 4 matrices; beyond this it seems difficult to go.
A Generalisation of a Result of Abel with an Application to Tree Enumerations
Authors: Robertson M. M.
Published: January 2009

We prove the following theorem, which was established by Abel (1) for the case u = 1.

Theorem. If u, k are positive integers and x, 1, , u, are real numbers, then

where the sum is taken over all distinct ordered solutions (s1, , su) in non-negative integers of the equation .
Quasigroups and cubic curves
Authors: Etherington I. M. H.
Published: January 2009

A note on this subject was read to the Edinburgh Mathematical Society in June 1951. Subsequently I had the benefit of conversations and correspondence with J. G. Brennan and I. R. Porteous, and substantial contributions from them are incorporated here, more than are acknowledged in detail below.
On some steady state thermoelastic stress distributions in a slab
Authors: Shail R.
Published: January 2009

The calculation of the steady state thermal stresses in an isotropic elastic half space or slab with traction free faces has been the subject of several investigations. Steinberg and McDowell (1), using an extension of the Bousinesq-Papkowitch method of isothermal elasticity, first derived the now well-known result that in such a body which contains no heat sources there exists a plane state of stress parallel to the boundary planes. Sneddon and Lockett (2) approached this class of problems by direct solution of the equations of thermoelasticity using a double Fourier integral transform method, the results being transformed to Hankel type integrals in the case of axial symmetry. A further approach due to Nowinski (3) exploits the fact that in steady state thermoelasticity each component of the displacement vector is a biharmonic function which can be expressed as a combination of harmonics. However, possibly the most economical method of solution of this type of problem is that of Williams (4) who expressed the displacement vector in terms of two scalar potential functions, one of which is directly related to the temperature field. The same principle has also been used by Fox (5) in treating thermoelastic distributions in a slab containing a spherical cavity.
Local uniqueness in boundary problems
Authors: Martin M. H.
Published: January 2009

The study of periodic, irrotational waves of finite amplitude in an incompressible fluid of infinite depth was reduced by Levi-Civita (1) to the determination of a function

regular analytic in the interior of the unit circle ρ = 1 and which satisfies the condition
The solution of dual cosine series by the use of orthogonality relations
Authors: Noble B. , Whiteman J. R.
Published: January 2009

In this paper expressions are derived for the unknown coefficients in the dual cosine series:

where α is a given constant, and f1(x) and f2(x) are prescribed functions.
A generalization of summability-(Z, p) of Silverman and Szász
Authors: Sherry David L.
Published: January 2009

A sequence x = {si} has been defined to be summable-(Z, p) to s if

Where p is a positive integer and
Semimodularity and bisimple ω-semigroups
Authors: Scheiblich H. E.
Published: January 2009

Let S be a completely 0-simple semigroup and let Λ(S) be the lattice of congruences on S. G. Lallement (2) has described necessary and sufficient conditions on S for Λ(S) to be modular, and has shown that Λ(S) is always semimodular . This result may be stated: If S is 0-bisimple and contains a primitive idempotent, then Λ(S) is semimodular.
A note on orthogonal polynomials
Authors: Kershaw D.
Published: January 2009

During an investigation into the existence of Gauss-type quadrature formulae for the numerical solution of Fredholm integral equations with weakly singular kernels an intermediate result was found which is of independent interest.
Primitive idempotent measures on compact semigroups
Authors: Duncan J.
Published: January 2009

Let S be a compact semigroup (with jointly continuous multiplication) and let P(S) denote the probability measures on S, i.e. the positive regular Borel measures on S with total mass one. Then P(S) is a compact semigroup with convolution multiplication and the weak* topology. Let II(P(S)) denote the set of primitive (or minimal) idempotents in P(S). Collins (2) and Pym (5) respectively have given complete descriptions of II(P(S)) when S is a group and when K(S), the kernel of S, is not a group. Choy (1) has given some characterizations of II(P(S)) for the general case. In this paper we present some detailed and intrinsic characterizations of II((P(S)) for various classes of compact semigroups that are not covered by the results of Collins and Pym.
In what spaces is every closed normal cone regular?
Authors: McArthur C. W.
Published: January 2009

It is known (13, p. 92) that each closed normal cone in a weakly sequentially complete locally convex space is regular and fully regular. Part of the main theorem of this paper shows that a certain amount of weak sequential completeness is necessary in order that each closed normal cone be regular. Specifically, it is shown that each closed normal cone in a Fréchet space is regular if and only if each closed subspace with an unconditional basis is weakly sequentially complete. If E is a strongly separable conjugate of a Banach space it is shown that each closed normal cone in E is fully regular. If E is a Banach space with an unconditional basis it is shown that each closed normal cone in E is fully regular if and only if E is the conjugate of a Banach space.
The radical of the group algebra of a sub-group, of a polycyclic group and of a restricted SN-group
Authors: Wallace D. A. R.
Published: January 2009

Let G be a group and let K be an algebraically closed field of characteristic p>0. The twisted group algebra Kt(G) of G over K is defined as follows: let G have elements a, b, c, … and let Kt(G) be a vector space over K with basis elements , …; a multiplication is defined on this basis of Kt(G) and extended by linearity to Kt(G) by letting

where α(x, y) is a non-zero element of K, subject to the condition that

which is both necessary and sufficient for associativity. If, for all x, y ∈ G, α{x, y) is the identity of K then Kt(G) is the usual group algebra K(G) of G over K. We denote the Jacobson radical of Kt(G) by JKt(G). We are interested in the relationship between JKt(G) and JKt(H) where H is a normal subgroup of G. In § 2 we show, among other results, that if certain centralising conditions are satisfied and if JK(H) is locally nilpotent then JK(H)K(G) is also locally nilpotent and thus contained in JK(G). It is observed that in the absence of some centralising conditions these conclusions are false. We show, in particular, that if H and G/C(H) are locally finite, C(H) being the centraliser of H, and if G/H has no non-trivial elements of order p, then JK(G) coincides with the locally nilpotent ideal JK(H)K(G). The latter, and probably more significant, part of this paper is concerned with particular types of groups. We introduce the notion of a restricted SN-group and show that if G is such a group and if
G has no non-trivial elements of order p then JKt(G) = {0}. It is also shown that if G is polycyclic then JKt(G) is nilpotent.
The states of a Banach algebra generate the dual
Authors: Sinclair Allan M.
Published: January 2009

In this paper we prove that the states of a unital Banach algebra generate the dual Banach space as a linear space (Theorem 2). This is a result of R. T. Moore (4, Theorem 1(a)) who uses a decomposition of measures in his proof. In the proof given here the measure theory is replaced by a Hahn-Banach separation argument. We shall let A denote a unital Banach algebra over the complex field, and D(1) denote {f ∈ A′: ‖f‖ = f(1) = 1} where A′ is the dual of A. The motivation of Moore's results is the theorem that in a C*-algebra every continuous linear functional is a linear combination of four states (the states are the elements of D(1)) (see (2, 2.6.4, 2.1.9, 1.1.10)).