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Some Integrals involving Legendre Functions
Authors: Collins W. D.
Published: January 2009

In this note generalisations of certain integrals involving Legendre functions including the Mehler-Dirichlet integral for Legendre functions of the first kind are given, these new results expressing associated Legendre functions of the first or second kinds as integrals involving corresponding functions of the same degree but different order. These integrals appear to be analogous to Sonine's integral in the theory of Bessel functions.
Enumeration of Indices of given Altitude and Potency
Authors: Minc H.
Published: January 2009

Indices of the free logarithmetic correspond to bifurcating root-trees (cf.(4)), to Evans' non-associative numbers (3) and to Etherington's partitive numbers (2). The free commutative logarithmetic is the homomorph of f determined by the congruence relation P + Q ∼ Q + P. Formulæ for aδ and pα, i.e. the numbers of indices of of a given potency* δ and the number of indices of a given altitude α respectively, were given by Etherington (1), who also gave corresponding formulæ for commutative indices of . Other enumeration formulæ are contained in (5).
A Problem in Partitions: Enumeration of Elements of a given Degree in the free commutative entropic cyclic Groupoid
Authors: Minc H.
Published: January 2009

A groupoid is a set closed with respect to a binary operation. It is commutative and entropic if xy = yx and xy.zw = xz.yw hold for all its elements. It is cyclic if it is generated by one element. Let x be the generator of the free commutative entropic cyclic groupoid . Then any element of can be written in the form xP where x1 = x and xQ+R = xQxR. Two indices P, Q are equal (called “concordant” in (3)) if and only if xP = xQ. The groupoid of these indices, the free additive commutative entropic logarithmetic (cf. (3)), is clearly isomorphic to .
Zeros and Periodicity of Functions of Infinite Matrices
Authors: Ibrahim Shafik Asaad
Published: January 2009

Certain functions of infinite matrices are known to exist.† This gives rise to the following questions:

1. Whether the power series of matrices

has a zero in the field ‡ of infinite matrices, and

2. If f(A) exists for a certain infinite matrix A, is there an infinite matrix B such that

In other words, is there a matrix period for f(A)?

In this paper theorems concerning zeros and periodicity of functions of semi block infinite matrices § (defined below) are established.
On extension of characters from normal subgroups
Authors: Karpilovsky G.
Published: January 2009

In what follows, character means irreducible complex character.

Let G be a finite group and let % be a character of a normal subgroup N. If χ extends to a character of G then χ is stabilised by G, but the converse is false. The aim of this paper is to prove the following theorem which gives a sufficient condition for χ to be extended to a character of G.
Polynomial algebras over the Steenrod algebra variations on a theorem of Adams and Wilkerson
Authors: Smith Larry , Switzer R. M.
Published: January 2009

The problem of deciding which graded polynomial algebras over the field of p elements can occur as the -cohomology of a space has played a central rôle in the development of algebraic topology beginning as early as 1950. In the case where the polynomial generators do not occur in dimensions divisible by p, Adams and Wilkerson [1] have given a complete solution by showing that the spaces constructed by Clark and Ewing [3] suffice to realize all such algebras as -cohomology rings. The main result of Adams and Wilkerson for odd primes can be stated as follows.
Low codimensional embeddings of Sp(n) and Su(n)
Authors: Walker G. , Wood R. M. W.
Published: January 2009

In [4] Elmer Rees proves that the symplectic group Sp(n) can be smoothly embedded in Euclidean space with codimension 3n, and the unitary group U(n) with codimension n. These are special cases of a result he obtains for a compact connected Lie group G. The general technique is first to embed G/T, where T is a maximal torus, as a maximal orbit of the adjoint representation of G, and then to extendto an embedding of G by using a maximal orbit of a faithful representation of G. In thisnote, we observe that in the cases G = Sp(n) or SU(n) an improved result is obtained byusing the “symplectic torus” S3 x … x S3 in place of T = S1 x … x S1. As in Rees's construction, the normal bundle of the embedding of G is trivial.
Partitioned groups and the additive structure of centralizer near-rings
Authors: Pettet Martin R.
Published: January 2009

If G is a finite group and A is a group of automorphisms of G, the “centralizer” nearring C(A, G) consists of the identity-preserving maps from G to itself which commute with the action of A. The main concern of this paper will be with the additive structur of C(A, G) in the case that this near-ring is semisimple.
The tensor product of distributive lattices: structural results
Authors: Bell Andrew M. , Brown Michael R. , Fraser Grant A.
Published: January 2009

The tensor product A ⊗ B of the distributive lattices A and B was first investigated by Fraser in [4] and [5]. In this paper, we present some results relevant to the structure and construction of this tensor product. In particular, we establish a sufficient condition for join-irreducibility in the tensor product and show that this condition characterizes join-irreducibility in the case that A and B satisfy the descending chain condition. We also show that if A and B satisfy the descending chain condition then so does A ⊗ B; this insures the compact generation of A ⊗ B by its join-irreducibles. We conclude with some examples and applications of our results to the tensor product of finite distributive lattices.
Indicial equivalents of multiparameter definiteness conditions in finite dimensions
Authors: Binding Paul
Published: January 2009

Let Tm, Vmn be Hermitean linear operators on complex Hilbert spaces Hm, m=1…k. A nonzero column vector satisfying

will be called an eigenvalue. This type of problem has been studied extensively by Atkinson [2] from the viewpoint of determinantal operators on the tensor product We shall connect his work with more recent investigations [5,7] of eigenvalue indices based on minimax principles for , which can be viewed as an operator on .
On the unique solvability of a class of modified boundary integral equations
Authors: Kleinman R. E. , Roach G. F.
Published: January 2009

In a recent paper the authors considered the transmission problem for the Helmholtz equation by using a reformulation of the problem in terms of a pair of coupled boundary integral equations with modified Green's functions as kernels. In this note we settle the question of the unique solvability of these modified boundary integral equations.
On the mutagenic radical property
Authors: Tzintzis G.
Published: January 2009

In their paper N. Divinsky and A. Sulinski [6] have introduced the notion of mutagenic radical property—that is, a radical property which is far removed from hereditariness—and constructed two such examples. The first is the lower radical property determined by a ring Swo (N. Divinsky [5]) and is an almost subidempotent radical property in the sense of F. Szász [9], and the second is a weakly supernilpotent radical property, that is the lower radical property determined by Swo and all nilpotent rings.
Young tableaux and longest monotone subsequences: An inequality and a conjecture
Authors: Gordon A. D.
Published: January 2009

Much work has been done on the following problem, which is sometimes referred to as Ulam's problem: what is the distribution of , the length (i.e. number of terms) of a longest monotone increasing [decreasing] subsequence of (not necessarily consecutive) terms in a random permutation of the first N integers? For example, it has been shown that converges almost surely to 2 [6,7,9]. In some cases, it is important to know the value of βN(j), the number of permutations for which αN=j
On pseudo-distributive near-rings
Authors: Mason Gordon
Published: January 2009

If G is a group and N a ring, the elements of the group ring NG can be thought of either as formal sums or as functions Φ:G→Nwith finite support. If N is a nearring, problems arise in trying to construct a group near-ring either way. In the first case, Meldrum [7] was abl to exploit properties of distributively generated near-rings (N, S) to build free (N,S)-products and hence a near-ring analogue of a group ring. For the latter case, Heatherly and Ligh [3] observed that the set of functions could be made into a near-ring under multiplication given by provided N satisfies

for all ai,bin∈N and k∈Z+. Such near-rings are called pseudo-distributive. In fact these are precisely the conditions under which the set Nk of k x k matrices over N is also a near-ring and then both NG and Nk are pseudo-distributive.
Almost diagonal systems in asymptotic integration
Authors: Gingold H.
Published: January 2009

Consider the ordinary linear matrix differential system

ψ(x) is a scalar mapping, X and A(x) are n by n matrices. Both belong to C1([a,∞)) for some integer l. The stability and asymptotic behaviour of its solutions have been subject to much investigation. See Bellman [2], Levinson [24], Hartman and Wintner [20], Devinatz [9], Fedoryuk [11], Harris and Lutz [16,17,18] and Cassell [30]. The special interest in eigenvalue problems and in the deficiency index problem stimulated a continued interest in asymptotic integration. See e.g. Naimark [36], Eastham and Grundniewicz [10] and [8,9]. Harris and Lutz [16,17,18] succeeded in explaining how to derive many known theorems in asymptotic integration by repeatedly using certain “(1 + Q)” linear transformations.
Initial boundary value problems for coupled nerve fibres
Authors: Grindrod P.
Published: January 2009

In this paper we analyse the electrical behaviour within systems of long and short coupled nerve axons by using a geometric approach to obtain a priori bounds on solutions. In [4[ we developed a general model for a bundle of n-uniform unmylinated nerve fibres. If FitzHugh-Nagumo dynamics, [3[ are used to describe the ionic membrane currents, then the model takes the form

Here W=(w1,…wn)T denotes the membrane action potentials for each fibre in the bundle and Z=(Z1,…Zn)T represents the recovery variables for each fibre, which control the return to the resting equilibrium after any transmission of signals.
The Poisson Correlation Function
Authors: Campbell J. T.
Published: January 2009

In this study, the use of factorial moments and factorial moment generating functions as applied (2) to the Poisson frequency function and Charlier's Type B function is further extended towards developing a theory of these distributions for the case of two or more correlated variables.
On the squares of Weber's parabolic cylinder functions and certain integrals connected with them
Authors: Mitra S. C.
Published: January 2009

The parabolic cylinder functions Dn(x) and D−(n+1) (± ix) are defined by

for all values of n and x.
The Operational Representations of Dn(x) and
Authors: Mitra S. C.
Published: January 2009

1. A given function f(x) is said to be represented operationally by another function φ(p), if

provided that the integral converges.
On some definite integrals involving Legendre functions
Authors: Shabde N. G.
Published: January 2009

A few definite integrals involving more than two Legendre functions in the integrand have been considered by Ferrers, Adams, Dougall, Nicholson and Bailey. We take for example the following integrals.
On Determinantal Symmetric Functions
Authors: Zia-ud-Din
Published: January 2009

§ 1. The theory of symmetric polynomials abounds in dual identities and symmetries of various kinds. It has been investigated from the determinantal standpoint largely by means of quotients of alternants, such as,

the denominator being the difference product of a, b, c, …, a simple alternant.
Theorems connecting different classes of Self-Reciprocal Functions
Authors: Mehrotra Brij Mohan
Published: January 2009

1. The question, “How, from a given function which is self-reciprocal for a transform of a particular order, can we construct other functions which are self-reciprocal for transforms of different orders?” was first raised by Hardy and Titchmarsh who gave some rules for constructing such functions. Following their method, I have shown, in a recent paper, that there are certain general theorems of the following type:—

If f (x) is its own Jμtransform, g (x) is its own Jv transform. In this note I add a few more such theorems, the interest lying mainly in the results themselves and not in a rigorous proof thereof; and hence only the formal procedure is given here.
On the application of the operational calculus to the expansion of a function in a series of Legendre's functions of the second kind
Authors: Banerjee D. P.
Published: January 2009

In the present note we shall obtain the expansion in a series of Legendre functions of the second kind of an integral function φ (ω) represented by Laplace's integral

where f (x) is an analytic function of x, regular in the circle

where an are constants and qn (ω) = in+1Qn (iω).
Note on a distance invariant and the calculation of Ruse's invariant
Authors: Walker A. G.
Published: January 2009

Several papers on the subject of spatial distance in General Relativity appeared a few years ago, and a simple extension of this idea to any pair of points in any Riemannian space was given by me in a thesis. A distance invariant was defined, and this was found to depend upon a certain two-point invariant which was first introduced by H. S. Ruse in a study of Laplace's Equation. This invariant, now written ρ and defined in (3), has lately re-appeared, and it may now be of interest to publish the results found earlier. These include a geometrical interpretation of ρ, a simple method of calculation, and an expansion as a power series in the geodesic arc. The dependence of ρ upon the geodesic arc is also considered.
A note on convergence factors
Authors: Fuchs W. H. J.
Published: January 2009

In this note denotes a divergent series of positive, decreasing terms for which are real numbers (convergence factors) such that is convergent. We put

H. Rademacher has shown that
On the methods of extending Dirac's equation of the electron to general relativity
Authors: Martin D
Published: January 2009

1. Introduction. The problem of extending Dirac's equation of the electron to general relativity has been attacked by many authors, by methods which fall roughly into either of two classes according as the formulation does or does not require the introduction of a local Galilean system of coordinates at each point of space-time. As examples of the former class we mention the methods of Fock (1929) and of Cartan (1938), and as representing the latter class the method described by Ruse (1937). Also, Whittaker (1937) discovered a vector whose vanishing is completely equivalent to the Dirac equations, but this method, unlike the others in the second category, does not apply the Riemannian technique to spinors but only to vectors and tensors derived from these. Now Cartan has denied the possibility of fitting a spinor into Riemannian Geometry if his point of view of spinors is adhered to, and this he argues accounts for the “choquant” properties with which they have been endowed by the geometricians in order to enable them to write down an expression of the usual form for the covariant derivative of a spinor. Consequently, doubt has been cast on the compatibility of the various methods, so in this paper an attempt is made to clarify the matter by working out explicitly the case of the general metric by some of the more important of these methods.
The Cesàro Summability of Fourier Integrals
Authors: Cossar J.
Published: January 2009

The functions f(t) and h(t) that occur in what follows are supposed to be integrable (L) in every finite interval in which they are defined; and the order of summability, which need not be an integer, is not negative.
The geometrical construction of Maschke's quartic surfaces
Authors: Edge W. L.
Published: January 2009

There is, in the second (Cambridge, 1911) edition of Burnside's Theory of Groups of Finite Order, an example on p. 371 which must have aroused the curiosity of many mathematicians; a quartic surface, invariant for a group of 24.5! collineations, appears without any indication of its provenance or any explanation of its remarkable property. The example teases, whether because Burnside, if he obtained the result from elsewhere, gives no reference, or because, if the result is original with him, it is difficult to conjecture the process by which he arrived at it. But the quartic form which, when equated to zero, gives the surface, appears, together with associated forms, in a paper by Maschke1, and it is fitting therefore to call both form and surface by his name.
Transposed algebras
Authors: Etherington I. M. H.
Published: January 2009

A linear algebra of order n, in general non-commutative and non-associative, may be regarded as being determined by the “cubic matrix ”consisting of its n3 constants of multiplication, and conversely. This requires that the n basis elements (units) of the algebra should be specified, and should be given in a definite order. Then the various “transpositions” of the cubic matrix induce corresponding “transpositions” of the algebra, for which a notation is given in §2.
A note on approximation by rational functions
Authors: Kober H.
Published: January 2009

In the present paper two problems on approximation by rational functions will be treated. The one concerns rational functions whose poles are of any order but lie at two preassigned points. The other problem relates to rational functions which have simple poles only.
Some Remarks on de Longchamps Chain
Authors: Brown L. M.
Published: January 2009

The chain of circle theorems now associated with the name of de Longchamps may be stated as follows.

(i) Any two lines L (A), L (B) meet in a point (AB).

(ii) Any three lines L(A), L (B), L(C) determine three points (AB) (BC) (CA) which lie on a circle S (ABC) with centre (ABC),

(iii) Any four lines L(A), L(B), L(C), L(D) determine four circles S(ABC) S(ABD) S(ACD) S(BCD) which meet in a point P (ABCD) and four centres (ABC) (ABD) (ACD) (BCD) which lie on a circle S(ABCD) with centre (ABCD).

(iv) Any five lines L(A), L(B), L(C), L(D), L(E) determine five points like P (ABCD), which lie on a circle SM (ABCDE) (Miquel's Theorem), five circles like S (ABCD), which meet in a point P (ABCDE), and their five centres like (ABCD), which lie on a circle S (ABCDE) with centre (ABCDE).

(v) Any six lines L (A), … . , L (F) determine six circles like SM (ABCDE) which meet in a point PM (ABCDEF), six circles like S (ABCDE), which meet in a point P (ABCDEF) and six centres like (ABCDE) which lie on a circle S (ABCDEF) with centre (ABCDEF).
A. G. D. Watson's principal directions for a Riemannian V4
Authors: Ruse H. S.
Published: January 2009

A. G. D. Watson (1939-41), remarking that there are no Ricci principal directions ata world-point of space-time at which the Einstein equations are satisfied, shows how to define at any world-point a set of principal directions intrinsically related to the Riemann tensor Rijkl itself. These directions are unique except when the space-time has any kind of rotational symmetry about the world-point.
On the limits of oscillation of a function and its Cesàro means
Authors: Rajagopal C. T.
Published: January 2009

There is a group of Tauberian theorems of which the simplest isone due to K. Ananda Rau [Theorem 2 of the paper numbered 1 inthe list of references at the end of the note]. More complicatedtheorems of the same group are discussed in a paper by S.Minakshisundaram and myself to be published by the LondonMathematical Society [4].
An infinite integral formula
Authors: Lambe C. G.
Published: January 2009

§ 1. The object of this note is to discuss the formula

the integral being supposed convergent for certain ranges of values of x and z. The contour is such that the poles of Γ(– s)lie to its right and the other poles of the integrand to its left. It will be seen that all the Pincherle-Mellin-Barnes integrals are particular cases of this formula.
An extension of de Longchamps' chain of theorems
Authors: Richmond H. W.
Published: January 2009

1. Points, n in number, A, B, C, D, E, … ., are taken at random in a plane, and through each is drawn a line in a random direction. The only condition imposed is that no two of these lines may be parallel.

(i). Two points A, B, define a circle S (AB) which passes through A, B and the intersection of the random lines through A and B. Its centre is denoted by (AB). Each pair of the points gives such a circle and centre.
A self-reciprocal function
Authors: Mohan Brij
Published: January 2009

1. The object of this note is to discover a new function which is its own reciprocal in the Hankel Transform of order zero.

I will make use of the following theorem of Hardy and Titchmarsh1:–

A necessary and sufficient condition that a Junction f (x) should be its own Jv transform is that it should be of the form

where 0 < c < 1, and
A closed dimensionless linear set
Authors: Best E.
Published: January 2009

The problem which we discuss in this paper can be easily settled for a closed plane set; after a brief introduction giving the definitions and theorems used later we indicate how this may be done; we then proceed to the main problem.
Compléments à un théorème de T. S. Broderick concernant les événements dépendants
Authors: Fréchet Par Maurice
Published: January 2009

Introduction. Nous nous proposons ici de pousser un peu plus loin les conséquences que Broderick a tirées d'un théorème dont il nous a fait connaître l'existence au dernier Colloque de St Andrews.

L'un des buts de Broderick était de prouver que certaines formules de King restent valables quand les événements considéréd ne sont plus indépendants, démonstration dont il n'avait pas trouvé trace dans la littérature, sauf relativement à des cas particuliers.
On Lambe's infinite integral formula
Authors: Erdélyi A.
Published: January 2009

In a recent paper (these Proceedings (2), 6 (1939), 75–8), C. G. Lambe established, and gave some applications of, the formula

in which Ds is the symbol for the derivative of fractional order s. Lambe's proof of (1) is not quite rigorous and it does not bring out the conditions which have to be imposed upon f(x) in order to make (1) true. Furthermore this proof does not give any evidence as to the definition of fractional derivative which is to be used in connection with (1).
Some applications of the generalised Poisson-Jensen formula
Authors: Scott S. A.
Published: January 2009

The object of this paper is to show how some formulae in Analytic Number Theory, in particular, the formula for N(T), the number of zeros of the Zeta-function between t = 0 and t = T, are easy deductions from the Generalised Poisson-Jensen formula. A similar method, using Green's function instead of the general function g(s) of § 2, has been published by F. and R. Nevanlinna (Math. Zeitschrift, 20 (1924), and 23 (1925), but the result contained in (vi) below appears to be new, although the writer has not been able, as yet, to make any effective use of it. It is clear that other applications could be made, but it seems sufficient to give here an indication of the method. The notation throughout is the usual one, and the references are to the Cambridge Tract by E. C. Titchmarsh on “The Zeta-function of Riemann”. Finally, I am indebted to the referee for the reference to the papers of F. and R. Nevanlinna.
A property of Cesàro-Perron integrals
Authors: Bosanquet L. S.
Published: January 2009

It is well known that if f(t) is (a) integrable in the Lebesgue sense, or more generally (b) integrable in the Perron sense, over every interval (α, β) interior to (a, b), and if

exists, then f(t) is integrable in the Perron sense over (a, b) to the value (1·1).
On two linear vector spaces associated with a vector in an Ln
Authors: Wong Yung-Chow
Published: January 2009

Let υλ be a vector (i.e. a vector field) in an affinely connected space Ln, ∇k the symbol of covariant differentiation, and r the rank of the matrix ‖∇k υλ‖ then there exist two sets of n − r independent vectors and which satisfy respectively the equations
W. von Tschirnhaus and algebraic equations with roots in arithmetical progression
Authors: Loria Gino
Published: January 2009

In an interesting appendix to a letter written by John Collins to James Gregory on August 3, 1675, but not published until a few months ago, appear some formulae, given without proof, for expressing the roots of an equation of any degree from the 2nd to the 9th in terms of the coefficients, under the assumption that these roots are in arithmetical progression. The formulae were discovered by the well known contemporary of Leibniz, Baron W. von Tschirnhaus. It is evident that in the case of an equation of degree n this particular assumption imposes n − 2 conditions on the coefficients; so that two of these coefficients can be chosen ad libitum. Tschirnhaus did not go to the trouble of obtaining these relations explicitly, in fact he makes no mention of them, but he gives expressions, in the cases indicated above, for the roots as functions of the first two coefficients of the equation in question, and these coefficients, as we have observed, are arbitrary. It is not known by what approach he arrived at his formulae; it seems likely to us, however, that he expressed the desired roots in terms of two arbitrary unknowns, that he evaluated the sum of these, and the sum of their products two at a time, and that, finally, he equated the results to the first two coefficients of the equation. In this way two equations are obtained, sufficient to determine the two auxiliary unknowns; and the problem can be considered as solved. Without seeming to imply that this procedure was the same as that adopted by the eminent German mathematician, we shall show that by its means one can not only derive his results, but also solve the question in the case of an algebraic equation of any degree.
An algebraic derivation of the distribution of rectangular coordinates
Authors: Hsu P. L.
Published: January 2009

Let xir (i = 1,…., q; r = 1, …., m; q ≦ m) be random variables which have an elementary probability law p(x11, …., xqm) Let
Extension of a Formula by Cayley to Symmetric Determinants
Authors: Gårding Lars
Published: January 2009

It has been proved by CAYLEY that if x11, x12, x21 … are independent variables, x = det (xik), ξ = det (ξik), (i, k = 1, … n) where ξik =∂/∂xik then by formal derivation ξxα = α(α + 1)…(α + n − 1)xα−1. This is a special case of the formula

where m=1,…,n and with i = i1,..im; k= k1,…km and xi,… is the algebracial complement of i = i1,..im; k = k1,…km, in .
A geometrical note on the Binary Cubic Form
Authors: Scott T.
Published: January 2009

An extensive investigation into the geometry associated with the binary cubic form as representing a triad of points on a twisted cubic curve has been carried out by Sturm. The object of the present note is to derive two further geometrical properties, one associated with the twisted cubic, and the other with the rational plane cubic curve.
A Theorem on General Regular Transformations of Series
Authors: Evans Arwel
Published: January 2009

Many papers have been written on transformations of sequences which can be written as

where is a function of a finite or infinite number of variables for fixed m. If the number of variables is finite it becomes large with m.

Almost all these papers are on linear transformations of the sr, e.g. Cesàro and Abel means, but there are obvious transformations of non-linear form which are regular, and it is a theorem on these which is considered here.
Some Theorems in Group Velocity
Authors: Davidson P. M.
Published: January 2009

We consider a medium in which the equation satisfied by a disturbance ø(x,t) is capable of solutions of the form sin(vx—wt), with v and ω real but not necessarily of the same sign. If the phase velocity U = ω/v is not of constant magnitude the medium is said to be dispersive, and the group velocity V may conveniently be defined as dω/v, an expression easily re-written in other familiar forms. From this definition the two physical interpretations of V may easily be seen. In one interpretation we consider a superposition of two harmonic waves with slightly different v (and ω); the velocity of advance of the group form, being the velocity of advance of a point at which the two waves have a constant phase difference, is dω/v.
A Generalised Hypergeometric Function
Authors: Fabian William
Published: January 2009

The hypergeometric function1F(a, b; c; z) is analytic in the domain |arg(−z)| < π, and, when |z| < 1, may be represented by the series

When |z| = 1 in the domain |arg(−z)| <π, this series converges2 to F(a; b; c; z) if R(a+b−c) < 0 (integral values of a, b and c are excluded in the present paper).
A Note on Convergence Factors
Authors: Walsh C. E.
Published: January 2009

Suppose that un ≥ un+1 > 0, for n = 1, 2, … and that e1, e2, … are factors which make convergent.

Let where all the a's are positive, diverges, and Define En by the equation

For the case where an = 1 it has been shewn by Fuchs1 and Karamata1 that, under various conditions, En = 0(1) The object of this note is to extend some of their results.
Some Finite Identities Connected with Poisson's Summation Formula
Authors: Guinand A. P.
Published: January 2009

It is known that there is a connexion between the functional equation of the theta-function

and the reciprocity formula for Gauss sums. The latter formula is usually written as

where p and q are positive integers, mutually prime, and not both odd.
The Definition of Lie Derivative
Authors: Willmore T. J.
Published: January 2009

The differential operation known as Lie derivation was introduced by W. Slebodzinski in 1931, and since then it has been used by numerous investigators in applications in pure and applied mathematics and also in physics. A recent monograph by Kentaro Yano (2) devoted to the theory and application of Lie derivatives gives some idea of the wide range of its uses. However, in this monograph, as indeed in other treatments of the subject, the Lie derivative of a tensor field is defined by means of a formula involving partial derivatives of the given tensor field. It is then proved that the Lie derivative is a differential invariant, i.e. it is independent of a transformation from one allowable coordinate system to another. Sometimes some geometrical motivation is given in explanation of the formula, but this is seldom very satisfying.
An Example of a Plane Shock of Variable Strength
Authors: Smith P.
Published: January 2009

A particular solution of the equations of one-dimensional anisentropic flow of a polytropic gas is linked by a shock to gas at rest in which the density is non-uniform. The approach is inverse in that the density distribution is derived from the position of the shock and the prescribed flow behind it. The velocity and strength of the shock each vary with time. The result is an example of the propagation of a shock through an inhomogeneous gas.
Some Further Triple Integral Equation Solutions
Authors: Cooke J. C.
Published: January 2009

A solution of a triad of integral equations involving Bessel functions is given. This, like earlier ones, is in the form of a pair of Fredholm integral equations, which may be solved by iteration in certain cases. In spite of a slightly more general formulation of the problem, the kernels of these equations are simpler than those given in earlier solutions. Certain extensions are considered and a formal solution given. Application is made to the problem of incompressible inviscid flow normal to an annular disc, and to the flow due to the slow rotation of such a disc in a viscous fluid.
Integral Equation Formulation of some Three Part Boundary Value Problems
Authors: Williams W. E.
Published: January 2009

One of the simplest three part boundary value problems is the electrostatic problem for the circular annulus and, at present, there seems to be no method available for obtaining the solution in a closed form. It has recently been shown by the author (1) and Cooke (2) that this problem can be reduced to the solution of a Fredholm integral equation of the second kind. The equation obtained in (1, 2) is fairly simple and is suitable for obtaining a numerical solution but, unfortunately, it cannot be solved iteratively to give a simple form of solution valid for small values of the ratio (inner radius/outer radius).
The Difference between Consecutive Prime Numbers V
Authors: Rankin R. A.
Published: January 2009

Let pn denote the nth prime and let ε be any positive number. In 1938 (3) Ishowed that, for an infinity of values of n,

where, for k≧1, logk+1x = log (logk x) and log1x = log x. In a recent paper (4) Schönhage has shown that the constant ⅓ may be replaced by the larger number ½eγ, where γ is Euler's constant; this is achieved by means of a more efficient selection of the prime moduli used. Schönhage uses an estimate of mine for the number B1 of positive integers n≦u that consist entirely of prime factors p≦y, where

Herer x is large and α and δ are positive constants to be chosen suitably.
On the Number of Crossings in a Complete Graph
Authors: Harary Frank , Hill Anthony
Published: January 2009

The purpose of this article is to describe two problems which involve drawing graphs in the plane. We will discuss both complete graphs and complete bicoloured graphs. The complete graphKn with n points or vertices has a line or edge joining every pair of distinct points, as shown in fig. 1 for n = 2, 3, 4, 5, 6.
Heat Conduction in Hollow Cylinders with Radiation
Authors: Marchi E. , Zgrablich G.
Published: January 2009

A new finite integral transformation (an extension of those given by Sneddon (1)), whose kernel is given by cylindrical functions, is used to solve the problem of finding the temperature at any point of a hollow cylinder of any height, with boundary conditions of radiation type on the outside and inside surfaces, with independent radiation constants. It is to be noticed that all possible problems on boundary conditions in hollow cylinders can be solved by particularising the method described here.
On Measure of Sum Sets III
Authors: Macbeath A. M.
Published: January 2009

In my paper on the continuous (α + β) – theorem (these Proceedings 12 (1961), 209–211) the proof of Lemma 6 assumes that if A and B are closed sets in the relative topology as subsets of the space P of positive reals then C = A + B is closed.
On well-bounded operators of type (B)
Authors: Spain P. G.
Published: January 2009

The notion of a well-bounded operator was introduced by Smart (9). The properties of well-bounded operators were further investigated by Ringrose (6, 7), Sills (8) and Berkson and Dowson (2). Berkson and Dowson have developed a more complete theory for the type (A) and type (B) well-bounded operators than is possible for the general well-bounded operator. Their work relies heavily on Sills' treatment of the Banach algebra structure of the second dual of the Banach algebra of absolutely continuous functions on a compact interval.
Some axially symmetric potential problems
Authors: Leppington F. G. , Levine H.
Published: January 2009

Some axially symmetric boundary value problems of potential theory are formulated as integral equations of the first kind. In each case the kernel admits an expansion, for small values of a parameter of the problem, that leads to an approximate integral equation whose solution provides a direct asymptotic estimate for the physical quantity of primary interest. A manipulation of the original and modified integral equations provides an efficient formula for calculating higher order terms in the asymptotic expansion.
Some non-embedding theorems for the Grassmann manifolds G2,n and G3,n
Authors: Oproiu V.
Published: January 2009

In recent years, the problem of embedding the projective spaces in Euclidean spaces was studied very much, by different methods. Usually, the negative results on the embedding problem are proved by using suitable homotopy invariants. The best known example of such homotopy invariants is given by the Stiefel–Whitney classes.
Nuclearity and Banach spaces
Authors: Valdivia Manuel
Published: January 2009

Let E be a nuclear space provided with a topology different from the weak topology. Let {Ai: i ∈ I} be a fundamental system of equicontinuous subsets of the topological dual E' of E. If {Fi: i ∈ I} is a family of infinite dimensional Banach spaces with separable predual, there is a fundamental system {Bi: i ∈ I} of weakly closed absolutely convex equicontinuous subsets of E'such that is norm-isomorphic to Fi, for each i ∈ I. Other results related with the one above are also given.
Finitep-soluble groups with irreducible modular representations of given degrees
Authors: Humphreys J. F.
Published: January 2009

LetGbe a finite group,pbe a prime andKbe a field of characteristicp. Let

be a decomposition of the group ring ofGoverKas a sum of indecomposable two-sided ideals. An irreducibleK(G)-module is said to be in the blockBiif it occurs as a composition factor ofBi. The block containing the trivialK(G)-module is called the principal block ofG.
On a problem about cyclic subgroups of finite groups
Authors: Barriga Oscar E.
Published: January 2009

Let G be a finite group and let S be a subgroup of G with core

We say that (G, S) has property (*) if there exists x ∈ G such that S ∩ x−1
Geometric inequalities involving the stretch of a set
Authors: Lillington J. N.
Published: January 2009

A number of papers have been written concerning the properties of triangles which circumscribe convex sets, see for example (1), (2), (3).

In this note we shall characterise two set functionals, one of which has already been introduced in (6). This will enable us to produce some new results in the spirit of those obtained in the papers mentioned above. These two functionals are interesting in that although they are defined in a rather abstruse manner, they turn out to have an intuitive geometric meaning, namely that they are the minimal widths of certain circumscribing triangles.
On the order of magnitude of Jacobsthal's function
Authors: Vaughan R. C.
Published: January 2009

Let n be an integer with n > 1. Jacobsthal (3) defines g(n) to be the least integer so that amongst any g(n) consecutive integers a + 1, a + 2, … a + g(n) there is at least one coprime with n. In other words, if

then

It is probably true that

where ω(n) denotes the number of different prime divisors of n, and Erdos (1) has pointed out that by the small sieve it is possible to show that there is a constant C such that
The graded Cartan matrix and global dimension of 0-relations algebras
Authors: Burgess W. D.
Published: January 2009

The Cartan matrix C of a left artinian ring A, with indecomposable projectives P1,…,Pn and corresponding simples Si=Pi/JPi, is an n×n integral matrix with entries Cij, the number of copies of the simple sj which appear as composition factors of Pi. A relationship between the invertibility of this matrix (as an integral matrix) and the finiteness of the global dimension has long been known: gl dim A < ∞⇒det C = ± 1 (Eilenberg [3]). More recently Zacharia [9] has shown that gl dim A ≦ 2⇒det C = 1, and in fact no rings of finite global dimension are known with det C = −1. The converse, det C = l⇒gl dim A < ∞, is false, as easy examples show ([[1) or [3]). However if A is left serial, gl dim A < ∞iff det C = l [1]. If A = ⊕n ≧ 0 An is ℤ-graded and the radical J = ⊕n ≧ 0 An, Wilson [8] calls such rings positively graded. Here there is a graded Cartan matrix with entries from ℤ[X] and gl dim A < ∞⇒det = 1 and, hence, det C = l [8, Prop. 2.2].
Complex inversion formula for the distributional Stieltjes transform
Authors: Pilipović Stevan
Published: January 2009

There are several approaches to the Stieltjes transform of generalized functions ([1, 10, 5, 6, 3, 2]). In this paper we use the definition of the distributional Stieltjes transform of index ρ (ρ ∈ ℝ\(−ℕ0); ℕ0 = ℕ∪{0}), Sρ-transform, given by Lavoine and Misra [3]. The Sρ-transform is defined for a subspace of the Schwartz space (ℝ) while in [10, 5, 6, 2] the Stieltjes transform is defined for the elements of appropriate spaces of generalized functions. In these spaces differentiation is not defined which means that the Stieltjes transform of some important distributions, for example δ(k)(x − a), a≧0, k ∈ ℕ, is meaningless in the sense of [10, 5, 6, 2]. It is easy to see that the distributions δ(k)(x − a), a≧0, k ∈ ℕ, have the Sρ-transform for ρ>−k, ρ∈ℝ\(−ℕ0). These facts favour the approach to the Stieltjes transform given in [3].
On the oscillation of solutions of certain linear differential equations in the complex domain
Authors: Bank Steven B. , Langley J. K.
Published: January 2009

Our starting point is the differential equation

where A(z) is a transcendental entire function of finite order, and we are concerned specifically with the frequency of zeros of a non-trivial solution f(z) of (1.1). Of course it is well known that such a solution f(z) is an entire function of infinite order, and using standard notation from [7],

for all , b∈C\{0}, at least outside a set of r of finite measure.
Domination of the supremum of a bounded harmonic function by its supremum over a countable subset
Authors: Bonsall F. F.
Published: January 2009

For what sequences {an} of points of the open unit disc D does there exist a constant k such that

for all bounded harmonic functions f on D?
Maximal subsemigroups of Lie groups that are total
Authors: Lawson Jimmie D.
Published: January 2009

The major problem with which this paper is concerned is determining criteria that allow one to decide whether the subsemigroup generated by a subset B of a group G is all of G. Motivations for considering this problem arise from at least two sources.
Ideals in pseudo-rings
Authors: Jump K.
Published: January 2009

The concept of a pseudo-ring was introduced by Patterson (1). Briefly, a pseudo-ring is an algebraic system consisting of an additive abelian group A, a distinguished subgroup A*, and a multiplication operation A* × A→A under which A* is a ring and A a left A*-module. For convenience, we denote the pseudo-ring by = (A*, A). For the definitions of the various types of ideal, we refer the reader to (1).
A remark on continuous bilinear mappings
Authors: Baker J. W. , Pym J. S.
Published: January 2009

The main theorem of this paper is a little involved (though the proof is straightforward using a well-known idea) but the immediate corollaries are interesting. For example, take a complex normed vector space A which is also a normed algebra with identity under each of two multiplications * and ∘. Then these multiplications coincide if and only if there exists α such that ‖a ∘ b ‖ ≦ α ‖ a * b ‖ for a, b in A. This is a condition for the two Arens multiplications on the second dual of a Banach algebra to be identical. By taking * to be the multiplication of a Banach algebra and ∘ to be its opposite, we obtain the condition for commutativity given in (3). Other applications are concerned with conditions under which a bilinear mapping between two algebras is a homomorphism, when an element lies in the centre of an algebra, and a one-dimensional subspace of an algebra is a right ideal. An example shows that the theorem is false for algebras over the real field, but Theorem 2 gives the parallel result in this case.
Operator norms determined by their numerical ranges
Authors: McGregor Colin M.
Published: January 2009

This paper owes its origin to the following question posed by A. M. Sinclair, “If a linear algebra with identity has two equivalent unital algebra norms, |.|1 and |.|2, whose corresponding numerical radii, v1 and v2, are equal on the whole algebra, are |.|1 and |.|2 related? Are they, for example, necessarily equal?” We do not give a complete answer to this question but are able to give sufficient conditions on algebras of operators for v1 = v2 to imply |.|1 = |.|2 That this implication does not hold for an arbitrary algebra with identity is demonstrated by means of a counter-example. The result for operator algebras is used to deduce some essentially non numerical range results for equivalent operator norms.
Varieties and d.g. near-rings
Authors: Meldrum J. D. P.
Published: January 2009

In this note we show that the variety of near-rings generated by d.g. near-rings is the class of all near-rings R satisfying

This extends a result of J. J. Malone (3) on the embedding of near-rings.
On a conjecture of Guinand for the plane partition function
Authors: Andrews George E.
Published: January 2009

In (1; p. 38), A. P. Guinand discusses the plane partition function q(n). He observes that q(3), q(6), q(9), q(15), q(18), q(21), and q(24) are respectively 6, 48, 282, 1479, 6879, 29601, 118794, and 451194. As all these are multiples of 3 he suggests the conjecture that q(3n) ≡ 0 (mod 3) for all positive integers n.
The Jacobian Curve of a Net of Quadrics
Authors: Edge W. L.
Published: January 2009

Much has been written, from the algebraical as well as from the geometrical standpoint, on the subject of pencils of quadrics: algebraically the problem consists of the canonical reduction of a pencil of quadratic forms, and the classical paper on the subject is by Weierstrass. But among the different kinds of pencils of quadratic forms there is the “singular pencil,” in which the discriminant of every form belonging to the pencil is zero; interpreted geometrically this means that every quadric belonging to the pencil is a cone. This case was expressly excluded from consideration by Weierstrass, and the canonical reduction was only accomplished later by Kronecker. But, although Weierstrass and Kronecker together solved completely the problem of the canonical reduction of a pencil of quadratic forms, a much clearer insight into the nature of the problem was gained when Segre gave the geometrical solution. He published two papers, one dealing with the non-singular pencils and the other with the singular pencil.
On Straneo's Unified Theory of Gravitation and Electricity
Authors: Gormley P. G.
Published: January 2009

In a recent series of papers, Straneo has introduced into Riemannian space-time a teleparallelism defined by an asymmetric connection, and in this manner attempted to develop a unified theory of gravitation and electricity. The essential idea underlying his work is, therefore, similar to that of the Einstein-Mayer theory of 1929–31.
A Generalisation of a Theorem of E. Toeplitz
Authors: Anderson J.
Published: January 2009

In the Mathematische Annalen, 11 (1877), 440–444, E. Toeplitz has proved the following theorem: If three quadrics are polar quadrics of three points with respect to a cubic surface, then their (2, 2, 2) invariant vanishes; the invariant being of the second degree in the coefficients of each of the three quadrics.
Note on an “Alternant” with Factorial Elements
Authors: Zia-uddin
Published: January 2009

The alternant with powers of variables for its elements

has often been investigated and its properties are well known. In the present note an “alternant” with factorial elements,

will be discussed. Here a, b, c, …, z, α, β, γ, …, ω are integers such that

and

with the convention that a(0) = 1, O(0) = 1, and that, where negative indices occur (in the reductions of §2), the factorial is to be regarded as zero.