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A Method of Evaluating Certain Determinants
Authors: Burchnall J. L.
Published: January 2009

1. Let [ast] (s, t=0, 1, … n) be a square matrix of order n+1 and determinant |ast| and suppose that by repeated “isolation” of the variables the corresponding bilinear form has been expressed as

where, for all r,

Then

Now (1) implies, and is implied by, the identities

Thus, from any known identity of the form (4), subject to the condition (2), we may at once infer, using (3). the value of the corresponding determinant |ars|.
Integral Functions with Gap Power Series
Authors: Erdös P. , Macintyee A. J.
Published: January 2009

1. Let

be an integral function, λn being a strictly increasing sequence of nonnegative integers. We shall use the notations

describing M (r) as the maximum modulus, m (r) as the minimum modulus and μ(r) as the maximum term of f (z).
The Elementary Divisors, Associated with 0, of a Singular M-matrix
Authors: Schneider Hans
Published: January 2009

1. Many investigations have been concerned with a squaro matrix P with non-negative coefficients (elements). It is remarkable that many interesting properties of P are determined by the set Σ of index pairs of positive (i.e. non-zero) coefficients of P, the actual values of these coefficients being irrelevant. Thus, for example, the number of characteristic roots equal in absolute value to the largest non-negative characteristic root p depends on Σ alone, if P is irreducible. If P is reducible, then Σ determines the standard forms of P (cf. § 3). The multiplicity of p depends on Σ, and on the set S of indices of those submatrices in the diagonal in a standard form of P which have p as a characteristic root. It has apparently not been considered before whether Σ and S also determine the elementary divisors associated with p. We shall show that, in general, the elementary divisors do not depend on these sets alone, but that necessary and sufficient conditions may be found in terms of Σ and S (a) for the elementary divisors associated with p to be simple, and (b) that there is only one elementary divisor associated with p.
On Compact Riemannian Manifolds with Zero Ricci Curvature
Authors: Willmore T. J.
Published: January 2009

In this note I prove the following result:—

Theorem. A compact, orientable, Riemannian manifold Mn, with positive definite metric and zero Ricci curvature, is flat if the first Betti number R1 exceeds n — 4.

In this statement of the theorem it is assumed that the dimensions of Mn are not less than four. If this is not the case, the result is still valid but appears as a purely local result and is true for a metric of arbitrary signature.
A Determinantal Expansion for a Class of Definite Integral
Authors: Shenton L. R.
Published: January 2009

We have shown in [1] that under certain conditions the definite integral may be approximated by a determinantal ratio. It is our object now to develop the theory when C (x) is a polynomial, showing the relation to the continued fraction form for . In particular we shall give various forms for the approximants, and an integral form for the numerator.
An application of the Kontorovich-Lebedev Transform
Authors: Lowndes J. S.
Published: January 2009

In this note a simple application of the Kontorovich-Lebedev transform is described. The problem to which the transform is applied is the one considered in a recent paper (1).
A General Basis Theorem
Authors: Robertson A. P. , Weston J. D.
Published: January 2009

The well-known “basis theorem” of elementary algebra states that in a finite-dimensional vector space, any two bases have the same number of elements; or, equivalently, that a vector space is n-dimensional if it has a basis consisting of n vectors (where the dimension of a vector space is defined to be the least upper bound of the numbers k for which there exist k linearty independent vectors, and a basis is defined to be a maximal set of linearly independent vectors). This theorem has an analogue in the theory of groups : if an Abelian group has a finite maximal set of independent non-cyclic elements, the number of elements in one such set being n, then no set of independent non-cyclic elements can have more than n members.
Matrices with Prescribed Characteristic Polynomials
Authors: Farahat H. K. , Ledermann W.
Published: January 2009

It is well known that every monic polynomial of degree n with coefficients in a field Φ is the characteristic polynomial of some n × n matrix A with elements in in Φ . However, it is clear that this result is an extremely weak one, and that it should be possible to impose considerable restrictions upon the matrix A. In this note we prove two results in this direction. In section 2, we show that it is possible to prescribe all but one of the diagonal elements of A. This result was first proved by Mirsky (2) when the ground field Φ is the field of complex numbers. In section 3, we see that we can require A to have any prescribed non-derogatory n–l × n–1 matrix in the top left-hand corner.
Longitudinal Elastic Waves in Cylinders and Tubes—Including Thermoelastic Effects
Authors: Lockett F. J.
Published: January 2009

Longitudinal elastic waves in solid circular cylinders were first investigated by Pochhammer (1) and Chree (2) but they did not take into account the effects that thermal properties have on the propagation of these waves. In this paper we shall consider waves in solid and hollow cylinders as well as in the infinite medium with a cylindrical cavity, and in each case we shall take account of the thermoelastic effects.
A Theorem on Continued Fractions
Authors: Watson G. N.
Published: January 2009

One of the outstanding theorems in the theory of continued fractions is the result described by O. Perron, Die Lehre von den Kettenbrüchen (1912, 1929, 1954–7) as the transformation of Bauer and Muir (for brevity I shall call it the BM theorem); this theorem and limiting cases of it give rise to numerous extremely interesting consequences.
A Theorem on Algebras of Measures on Topological Groups
Authors: Williamson J. H.
Published: January 2009

Let G be a locally compact Abelian group, and the set of bounded complex (regular countably-additive Borel) measures on G. It is well known that becomes a Banach space if the norm is defined by

the supremum being over all finite sets of disjoint Borel subsets of G.
Inequalities for certain Classes of Convex Functions
Authors: Mirsky L.
Published: January 2009

Making use of properties of doubly-stochastic matrices, I recently gave a simple proof (4) of a theorem of Ky Fan (Theorem 2b below) on symmetric gauge functions. I now propose to show that the same idea can be employed to derive a whole series of results on convex functions ; in particular, certain well-known inequalities of Hardy-Littlewood-Pólya and of Pólya will emerge as specìal cases.
Propagation of Thermal Stresses in an Infinite Medium
Authors: Lockett F. J. , Sneddon I. N.
Published: January 2009

In the full linear theory of thermoelasticity there is a coupling between the thermal and the purely mechanical effects so that not only does a nonuniform distribution of temperature in the solid produce a state of stress but dynamical body forces or applied surface tractions produce variations in temperature throughout the body. In a recent paper (Eason and Sneddon, (2)) an account was given of the calculation of the dynamic stresses produced in elastic bodies, both infinite and semi-infinite, by uneven heating. In this paper we shall consider the propagation of thermal stresses in an infinite medium when, in addition to heat sources, there are present body forces which vary with the time.
On semisimple classes of associative and alternative rings
Authors: Puczyłowski E. R.
Published: January 2009

In [6] Sands proved that the semisimple classes of associative rings are exactly the coinductive and closed under ideals and extensions classes. This characterization was transferred to the alternative case by Van Leeuwen, Roos and Wiegandt in [3]. Answering a question of [9], Sands [7] has recently proved that in the associative case the condition of being closed under ideals can be replaced by the regularity of the class. The same result for alternative rings has been proved by Anderson and Wiegandt in [2]. Thus the following result holds.
Generalized near-fields
Authors: Murty C. V. L. N.
Published: January 2009

By analogy with the concept of “inverse semi-group” in semi-group theory, in this paper we introduce the concept of “generalized near-field” in near-rings. A near-ring N is called a generalized near-field (GNF) if for each a ε N there exists a unique b ε N such that a = aba and b = bab, that is (N, ·) is an inverse semi-group. Surprisingly, this concept in rings coincides with that of “strong regularity”. But this is not true in the case of near-rings. Every GNF is strongly regular, but in general the converse is not true.
The centralizer of the general linear group
Authors: Maxson C. J. , Oswald A.
Published: January 2009

Let G be a group, written additively with identity 0, but not necessarily abelian and let S be a semigroup of endomorphisms of G. The set for all is a zero-symmetric near-ring with identity under the operations of function addition and composition, called the centralizer near-ring determined by the pair (S, G). Centralizer near-rings are general, for if N is any zero-symmetric near-ring with identity then there exists a group G and a semigroup S⊇ G such that For background material and definitions relative to near-rings in general we refer the reader to the book by Pilz [7]. For material on centralizer near-rings we refer the reader to [4] and [6].
Fricke's octavic curve
Authors: Edge W. L.
Published: January 2009

The paper is concerned with a curve F, the complete intersection of a quadric with a quartic surface, that admits a group of self-projectivities isomorphic to the symmetric group of degree 5. Every generator of the quadric is, as shown at the end of the paper, cut by F equianharmonically. F has 80 stalls, points where its osculating plane is stationary; they are of two kinds, 60 to be labelled ∑, the other 20 Ω. F also has inflections at 24 points which compose a figure encountered on earlier occasions. A search is made for tritangent planes of F of which, when reckoned according to proper multiplicity, there must be 2048. Among them are 60 all of whose three contacts are ∑ while a further 120 each involve a single ∑ among their contacts and 420 each involve a single Ω.
On radicals of finite near-rings
Authors: Kaarli K.
Published: January 2009

In this paper the study of radicals of finite near-rings is initiated. The main result (Theorem 4.3) gives a description of hereditary radicals having hereditary semisimple classes too. Also it is shown that there exist non-hereditary radicals having hereditary semisimple classes.
The strong maximum principle for the heat equation
Authors: Colton David
Published: January 2009

The strong maximum principle for harmonic functions is usually arrived at by appealing to the mean value theorem (c.f. [2], p. 53). It is also of course possible simply to appeal to the Hopf maximum principle [2], but using sledge hammers to kill flies is generally viewed as aesthetically unpleasing. In contrast to the case of harmonic functions, the only proof of the strong maximum principle for the heat equation that is known to me is to invoke Nirenberg's strong maximum principle for parabolic equations [2]. As in the case of harmonic functions, it seems desirable to provide a direct proof of this result without having to go through the subtle comparison arguments that are employed in the more general case. The purpose of this note is to provide a proof of the strong maximum principle for the heat equation based on a mean value theorem for solutions of the heat equation which we derive below. Such an approach provides a straightforward and simple proof of the strong maximum principle which avoids most of the detailed estimates of the proof of the maximum principle for more general parabolic equations. Unfortunately the proof of the maximum principle for the heat equation using the mean value theorem is not as short as the proof in the corresponding case of harmonic functions. It nevertheless seems worthwhile to show that such an alternate proof is possible, and it is to this purpose that we address this paper.
Counterexamples in intersections for C*-tensor products
Authors: Kye Seung-Hyeok
Published: January 2009

Let A and B be C*-algebras and A ⊗ B denote the minimal C*-tensor product of A and B. T. Huruya [1] gave examples of C*-tensor products A ⊗ B with C*-subalgebras A1⊗B2 and A2⊗B2 such that strictly contains , answering a question of S. Wassermann [3, Remark 23]. In this short note, we show that the same situation can occur even if A1 = A2
Stability of barrelledness and related concepts in topological vector spaces
Authors: Popoola J. O. , Tweddle I.
Published: January 2009

Let E be a separated locally convex barrelled space with continuous dual E′ and algebraic dual E* and let M be a subspace of E* with and dim Robertson, Tweddle and Yeomans have recently considered the question of barrelledness under the Mackey topology τ(E,E' + M) when E is given to be barrelled under its original topology τ(E,E') [5], [6], [7].
Sur l'image d'une fonction entiere de deux variables complexes
Authors: Couty D.
Published: January 2009

Soit Θ un automorphisme analytique de , Θ ayant un point fixe attractif zo (c'está-dire il existe un voisinage V de zo tel que pour tout élément z de V, on ait Alors d'aprés un résultat bien connu, il existe une fonction F entiére de dans , injective, telle que Ceci provient du fait qu'il existe une fonction F, définie au voisinage de z0, telle que F'(zo)=I, vérifiant Fequation fonctionnelle est un automorphisme analytique de vérifiant pour tout z élément de .
Associativity of the tensor product of semilattices
Authors: Fraser Grant A. , Albert John P.
Published: January 2009

The tensor product of semilattices has been studied in [2], [3] and [5]. A survey of this work is given in [4]. Although a number of problems were settled completely in these papers, the question of the associativity of the tensor product was only partially answered. In the present paper we give a complete solution to this problem.
The countable neighbourhood property and tensor products
Authors: Bonet José
Published: January 2009

This article is intended to enlarge the study of spaces satisfying the countable neighbourhood property and to clarify the incidence of this property in the stability of some locally convex properties of tensor products.

We shall use the standard notations of locally convex spaces as in [17] and [18]. The word space will always mean separated locally convex space. If (£, t) is a space, the set of all continuous seminorms on it will be denoted by cs(E). The linear hull and the absolutely convex hull of a subset C of a space will be written lin(C) and г(C) respectively.
The interpolation proof of Grothendieck's inequality
Authors: Jameson G.J.O
Published: January 2009

This note is an exposition of the simple and elegant approach to Grothendieck's inequality given in [2] and lsqb;4], with one further simplification. The process of factorizing through L2 ([2], p. 21) introduces a factor of into the final constant. We show that this step can be avoided.
Joint numerical ranges for unbounded normal operators
Authors: Danrun Huang
Published: January 2009

For bounded operators, the theory of the joint numerical range has been developed by various authors [1,2,3,4,5]. Especially, the properties of commuting normal n-tuples are discussed in detail. Our purpose here is to show that many results in the above references still hold in the case of unbounded normal operators (see Theorem 2.3, Corrollary 3.5, Theorem 4.1, Theorem 4.2). Besides, the operator algebras are closely related to the theory of joint spectrum and joint numerical ranges in the boundedcase (cf. [1,3]). How about unbounded operators? It seems that one must consider unbounded operator algebras. Some work has been done in this direction for the joint spectrum of unbounded normal operators [9]. In the last section of this paper, we provide some intimate relations between the joint numerical range and the unbounded operator algebras for unbounded normal operators.
Note on a Resolution of Linear Differential Systems
Authors: I. S.  , Sokolnikoff E. S.
Published: January 2009

In a recent paper J. M. Whittaker considered the problem of resolving a linear differential system into a product of two or more systems of lower order. This note is a contribution to this problem and furnishes the necessary and sufficient conditions for the resolution of the system into two equivalent systems of lower order of the type considered by Whittaker.
Integral equations for Lamé functions
Authors: Erdélyi A.
Published: January 2009

In the theory of ordinary linear differential equations with three regular singularities and in the theory of their special and limiting cases, integral representations of the solutions are known to be very important. It seems that there is no corresponding simple integral representation of the solutions of ordinary linear differential equations with four regular singularities (Heun's equation) or of particular (e.g. Lamé's equation) or limiting (e.g. Mathieu's equation) cases of such equations. It has been suggested (Whittaker 1915 c) that the theorems corresponding in these latter cases to integral representations of the hypergeometric functions involve integral equations of the second kind. Such integral equations have been discovered for Mathieu functions (Whittaker 1912, cf. also Whittaker and Watson 1927 pp. 407–409 and 426) as well as for Lame functions (Whittaker 1915 a and b, cf. also Whittaker and Watson 1927 pp. 564–567) and polynomial or “quasi-algebraic” solutions of Heun's equation (Lambe and Ward 1934). Ince (1921–22) investigated general integral equations connected with periodic solutions of linear differential equations.
A note on small vibrations
Authors: Schlapp Robert
Published: January 2009

This note exhibits some old results in what is possibly a new form.

The simple harmonic oscillator, whose kinetic and potential energies are given in terms of a single coordinate x by

where b is a positive constant, has the equation of motion

The solution, appropriate to the initial conditions at t = 0 is conveniently written
On the relations between the numbers of standard tableaux
Authors: Rutherford D. E.
Published: January 2009

§ 1. Corresponding to any partition

which we denote by [a1, . …, ak] or briefly by [a], of the integer n, we can construct a shape which has a1 spaces in the first row, a2 in the second row, . …, ak in the kth and last row. Thus the shape corresponding to the partition [5, 3, 3, 2] of 13 has the form:
On a type of canonical transformation
Authors: Davidson P. M.
Published: January 2009

1. A well known theorem in classical dynamics tells us that if Q, P, and are written as functions of q, p and t by means of the equations

where W (q, Q, t) is an arbitrary function, and if two functions F (Q, P, t) and f (q, p, t) are related by

then q and p are conjugate with respect to f(q, p, t) if Q and P are conjugate with respect to F (Q, P, t).
Conics on a Maschke Surface
Authors: Edge W. L.
Published: January 2009

The six quaternary quartic forms

were first obtained by Maschke; it has recently been explained that the quartic surfaces obtained by equating these forms to zero are important constituents of Klein's famous configuration derived from six linear complexes that are mutually in involution. The quartic surface Φi = 0 will be denoted, for each of the six suffixes i, by Mi.
Two infinite integrals
Authors: Erdélyi A.
Published: January 2009

1·1. The two functions F (λ, θ) and G (λ, θ), defined by the infinite integrals (1) and (2) respectively, below, occur in Kottler's theoretical discussion of the diffraction of a monochromatic plane wave by a perfectly black half plane. Some properties of these functions have been investigated by several recent writers.
Some theorems on absolute Cesàro summability
Authors: Hyslop J. M.
Published: January 2009

1. It is convenient to begin with a brief statement of the notation which will be used throughout this paper.

Let k be any positive number and let

where is the coefficient of xn in the formal expansion of (1 – x )–k–1, and let

Then the series Σαn is said to be summable(C, k) if is convergent, that is, if tends to a limit, and absolutely summable (C, k), or summable |C, k|, if is absolutely convergent.
Matrices with finite period
Authors: Taussky Olga , Todd John
Published: January 2009

The study of the primitive solutions of the equation

where A = (aij) is an n × n matrix whose elements are rational integers, was begun a long time ago. In most cases this equation occurred incidentally in another theory; for instance Jordan encountered it in connection with linear differential equations having algebraic solutions, Minkowski in connection with quadratic forms and Turnbull in geometry. An important fact about these matrices is that any unimodular matrix can be represented as the product of matrices with finite period.
On some generalisations of Laguerre polynomials
Authors: Kober H.
Published: January 2009

Let , (n = 0, 1, ….) be the Laguerre, Hn(x) the Hermite polynomial. Let , be the space of all functions f(x) the pth powers of which are integrable over (a, b), with the norm
Note on the hodograph of non-holonomic dynamical systems
Authors: Spain B.
Published: January 2009

Consider a non-holonomic dynamical system specified by the N coordinates qi, and kinetic energy defined by

where amn are functions of qi and the dot denotes differentiation with respect to the time t.
A proof of Aronhold's theorem upon quartic curves
Authors: Richmond H. W.
Published: January 2009

It is to be expected that a finite number of plane curves of order four should have seven given lines as bitangents, because the number of conditions imposed is equal to the number of effective free constants in the equation of such a curve, viz., 14. Aronhold made the interesting discovery that one curve could be determined in which no three of the given lines have their six points of contact on a conic. The method, due to Geiser, of obtaining the bitangents as projections of the lines of a cubic surface leads to a simple proof of the existence of this quartic.
Expansions in terms of Parabolic Cylinder Functions
Authors: Cherry T. M.
Published: January 2009

When the plane wave equation is expressed in terms of parabolic co-ordinates x, y, the variables are separable, and the elementary solutions have the form

where x, y, μ are real. In this context, therefore, the functions Dν (z) which are directly significant are those where amp z = ± π/4 and ν + ½ is purely imaginary, rather than those where z is real and ν is a positive integer. The expansion of an arbitrary function in terms of the latter sort of D-function (substantially, in terms of Hermite polynomials) is well known. This paper is concerned with the expansion in terms of the former sort of D-function.
A Generalization of Minimal Varieties
Authors: Freeman J. G.
Published: January 2009

I consider an n-dimensional generalized metric space Sn with coordinates xi(h, i, j, k… run from 1 to n throughout), with each point of which is associated a contravariant vector-density with components ui and weight p, called the element of support. The unit vector in the direction of the element of support has components denoted by li.
Two Proofs of the 6Ψ6 Summation Theorem
Authors: Slater L. J. , Lakin A.
Published: January 2009

The 6Ψ6 summation theorem was first proved by Bailey1, who deduced it indirectly from a transformation of a well-poised 8Φ7 series into two 4Φ3 series. No direct proof of the theorem has been published, and, since it has interesting applications in the proofs of various identities which occur in combinatory analysis, for example the A series of Rogers2 and some elegant identities due to Ramanujan3, we give two new proofs of the theorem in this paper.
Enumeration of Indices of Given Altitude and Degree
Authors: Etherington I. M. H.
Published: January 2009

This note is a sequel to the article by Mine (2) on the same problem.

I described in (1) a notation for indices of powers in non-associative algebra, defined the degree † and altitude of a power or index, and observed that powers can be represented by bifurcating root-trees. For example, the power xx.x is denoted x2 + 1, with index 2 + 1, and is represented by the tree ; the degree (the number of factors, or free knots in the tree) is 3, and the altitude (the height of the tree) is 2. Multiplication being non-commutative or commutative, one maintains or ignores the distinction between left and right in the tree.
Asymptotic Formulæ for the Number of Partitions of a Multi-Partite Number
Authors: Robertson M. M.
Published: January 2009

A multi-partite number of orders j is a j dimensional vector, the components of which are non-negative rational integers. A partition of (n1, n2, …, nj) is a solution of the vector equation

in multi-partite numbers other than (0, 0, …, 0). Two partitions, which differ only in the order of the multi-partite numbers on the left-hand side of (1), are regarded as identical. We denote by P1(n1, …, nj) the number of different partitions of (n1 …, nj) and by p2(n1, …, nj) the number of those partitions in which no part has a zero component. Also, we write P3(n1, …, nj) for the number of partitions of (n1; …, nj) into different parts and p4(n1, …, nj) for the number of partitions into different parts none of which has a zero component.
The Propagation of Thermal Stresses in a Semi-infinite Medium
Authors: Lockett F. J.
Published: January 2009

When an elastic solid is subjected to a dynamical system of surface or body forces, not all of the work done by these forces is employed in deforming the material. The remainder is converted into heat energy producing a distribution of temperature throughout the body. Similarly the application of a surface temperature distribution, or the introduction of heat sources within the body, produces elastic as well as thermal effects. Thus we see that in the dynamical case there is a link between these two types of condition—thermal and elastic.
On the Eigenvector belonging to the Maximal Root of a Non-negative Matrix
Authors: Ostrowski A. M.
Published: January 2009

By a theorem of Perron, a non-negative irreducible (n × n) matrix A = (aμν) has a positive fundamental root σ, the “ maximal root of A ”, such that the moduli of all other eigenvalues of A do not exceed σ. If we put

σ lies between R and r. Since σ is not changed if A is transformed by a positive diagonal matrix D(p1,…pn, σ lies also between the expressions

By a theorem of Frobenius, to σ as an eigenvalue of A belongs a positive eigenvector ξ, = (x1 …, xn), satisfying
The Angular Momentum of an Electromagnetic Field
Authors: Chambers LL. G.
Published: January 2009

It is known that the behaviour of an electromagnetic field is consistent with it possessing a linear momentum density (1, p. 103)

(Girogi Units are used and the system of notation is the usual one).
Asymptotic Representation of Certain Real Integrals with Parameter-dependent Limits of Integration
Authors: Ross E. W.
Published: January 2009

In this paper we consider a function f(x) defined by

All quantities are taken to be real, it is assumed that R is a function of the variable x, b is a constant, N and G are functions of the variable t and all the functions are such that the integral (1) exists when x is large enough. We wish to find an asymptotic representation of f(x) as x → + ∞, assuming that we are given certain information about the limiting behaviours of the functions R, N and G.
Combinantal Forms of a Quadric Net Applied to Irreducible Concomitant Systems
Authors: Scott T.
Published: January 2009

A complete system of combinantal forms (generalised and ordinary) of a pencil of quadrics fλ ≡ λ1f1 + λ2f2 can be chosen such that the coefficients of the various power products of λ1, λ2 in the former give a complete irreducible system of concomitants of the two quadrics f1, f2, and conversely. This result was proved by Todd (1), who used it in conjunction with Schur function analysis (2) (3) to derive the complete irreducible system of concomitants of two quaternary quadratics (4).
The Complete Irreducible System of Invariants of Three Quadrics
Authors: Scott T.
Published: January 2009

The present account is an application of the principles of combinantal forms and Schur function analysis given in a previous paper (A), the references therein being henceforward denoted by A1 to A9, and the complete irreducible system of invariants of three quaternary quadrics will now be obtained from the complete system (not necessarily itself irreducible) derived by Turnbull (A5, p. 483). This latter system comprises 47 invariants, viz. 15, 1, 6, 6, 1, 15 and 3 members of total degrees 4, 6, 8, 10, 12, 14 and 18 respectively in the coefficients of the quadrics. It will be proved that all of these are irreducible except for the one of degree 12 and the three of degree 18, the former being of especial interest as it is a real combinant and moreover, involves unusual features in the proof of its reduction and also in the derivation of the form expressing it in terms of irreducible invariants.
On an Integral in the Distribution of Eigenvalues
Authors: McLeod J. B.
Published: January 2009

In the case of a wave function with spherical symmetry, the wave equation can be separated using spherical polar coordinates, and the equation for the radial component becomes

where λ is a constant parameter, proportional to the energy of the particle under consideration, q(r) is proportional to the potential energy, and l is a positive integer or zero.
The Propagation of a Constant Strength Shock through a Simple Wave
Authors: Weir D. G.
Published: January 2009

When a shock wave moving through an isentropic ideal gas catches up with, and passes into a simple expansion wave, the shock decays. Because of this the gas will not be isentropic in the region behind the shock. The problem of determining the motion of the gas in this region is as yet unsolved. In this paper we introduce a simple compression wave behind the shock which catches up with it at the instant of its entry into the leading expansion wave. This second wave is chosen so as to counteract the decaying effect of the first, and keep the shock strength constant throughout the motion. We assume the first wave to be point-centred, and caused by the withdrawal of a piston at a finite velocity from a gas at rest in a shock tube. After a finite time the piston is halted causing the shock. The problem is then to determine the subsequent motion of the piston to produce a compression wave with the desired property.
Evaluation of Certain Integrals Involving Bessel Functions
Authors: Deutsch E.
Published: January 2009

In the solution of boundary value problems in mathematical physics by means of integral transforms we often find that the solution of a particular problem can be expressed in terms of integrals of the type

where r and z are positive and m and n are integers satisfying the convergence condition m+n>−2.
The Laplace Transform of the Modified Bessel Function K(t±mx) where m = 1, 2, 3, …, n
Authors: Ragab F. M.
Published: January 2009

In the present paper we determine the Laplace transforms of the modified Bessel function of the second kind Kn(t±mx), where m is any positive integer. The Laplace transforms are given in (2) and (4) below, p being the transform parameter and having positive real part.
The solution of Problems Involving the Melting and Freezing of Finite Slabs by a Method due to Portnov
Authors: Jackson F.
Published: January 2009

In a recent paper (1) Portnov used a form of Poisson integral to find the exact solution for the temperature distribution in a freezing semi-infinite slab occupying the region x > 0, and having an arbitrary time dependent temperature applied at the face x = 0. Previously, Boley (2) had used a method based on Duhamel's theorem to find solutions for problems involving melting, in both finite and semi-finite regions, caused by time dependent heat fluxes. Steady-state solutions have been investigated by Landau (3), Masters (4) and others (5).
A Class of Homomorphism Theories for Groupoids
Authors: Losey Gerald
Published: January 2009

In a well-behaved homomorphism theory for a class of algebraic systems certain “closed objects” relative to a given G ∈ are distinguished which act as kernels of homomorphisms. For example, if is the class of groups then the closed objects relative to a given group G are the normal subgroups of G; if is the class of semigroups with zero element then one can devise a homomorphism theory in which the closed objects relative to a given S ∈ are the ideals of S[cf. Rees (3)]; in the class of groupoids one may define the closed objects relative to a given groupoid G to be the congruence relations on G, that is, subsets π⊆G×G which are equivalence relations having the property that (x1y1, x2y2) ∈ π whenever (x1, x2), (y1, y2) ∈ π. Given such a closed object N relative to G there exists a “factor” system G/N and a (canonical) homomorphism η: G→G/N characterised by the property: If σ G→H is a homomorphism with kernel N then there is a unique homomorphism : G/N→H such that . η = σ and the kernel of is trivial in the sense that the kernel of is the unique smallest closed object relative to G/N.
An Extension of the Almost Isolated Singularity of Finite Exponential Order
Authors: Wilson R.
Published: January 2009

Let f(z) be represented on its circle of convergence |z| = 1 by the Taylor series

and suppose that its sole singularity on |z| = 1 is an almost isolated singularity at z = 1. In the neighbourhood of such a singularity f(z) is regular on a sufficiently small disk, centre z = 1, with the outward drawn radius along the positive real axis excised. If also in this neighbourhood |f(z)| e−(1/δ)ρ remains bounded for some finite ρ, where δ is the distance from the excised radius, then the singularity is said to be of finite exponential order.
A Pair of Dual Integral Equations Involving Bessel Functions of the First and the Second Kind
Authors: Srivastav R. P.
Published: January 2009

In this paper we give a method for the solution of the dual integral equations

where Jv and Yv are Bessel functions of the first and second kind, −½≦α≦½, f1(ρ) and f2(ρ) are known functions and ψ(ξ) is to be determined. Such equations arise in the discussion of boundary value problems for half-spaces containing a cylindrical cavity. For example, let us take the problem of finding a potential function φ(ρ, θ, z) which satisfies Laplace's equation for

subject to the usual regularity conditions and the following boundary conditions:
Group algebras of some generalised soluble groups
Authors: Knott R. P.
Published: January 2009

In (8) Stonehewer referred to the following open question due to Amitsur: If G is a torsion-free group and F any field, is the group algebra, FG, of G over F semi-simple? Stonehewer showed the answer was in the affirmative if G is a soluble group. In this paper we show the answer is again in the affirmative if G belongs to a class of generalised soluble groups
On (J, pn)-summability of Fourier series
Authors: Khan F. M.
Published: January 2009

Let pn>0 be such that pn diverges, and the radius of convergence of the power series

is 1. Given any series σan with partial sums sn, we shall use the notation

and
Varieties of topological groups generated by Lie groups
Authors: Chen Su-Shing , Morris Sidney A.
Published: January 2009

Varieties of topological groups have been investigated in several papers ((2) and (10)-(13)). In this note we investigate the varieties generated by classical Lie groups. In particular we show results of which the following is indicative: The variety generated by the unitary group U(n) contains U(m) if and only if m ≦ n. En route we introduce the notion of a variety of topological Lie algebras which provides a convenient setting in which to answer our questions.
Some inequalities involving the symmetric functions
Authors: Baston V. J.
Published: January 2009

We firstly introduce some notation. Leta(t) ∈Rnthen we denote by α(t) a rearrangement ofa(t) in non-decreasing order. We writea(1)≺a(2)if

and
On groups of Fibonacci type
Authors: Brunner A. M.
Published: January 2009

Let w = w(a0, a1, an–1) be a word in the free group freely generated by a0, a1, …, an–1; let wi, denote the word w(ai, ai+1, …, ai+n–1), where the subscriptsj in aj are reduced modulo n; and let
A characterisation of closed subalgebras of
Authors: Dixon P. G.
Published: January 2009

We show that the class of Banach algebras A isomorphic with normclosed (non-self-adjoint) subalgebras of is characterized by the condition that the norms of polynomials in A be dominated by the norms of the same polynomials in .
The representation of linear operators on spaces of finitely additive set functions
Authors: Cheney C. A. , de Korvin Andre
Published: January 2009

The theory of representing continuous linear operators on function spaces in terms of integrals has had a long and fruitful history, beginning with the Riesz representation theorem in 1909. IfTis such an operator, then the standard representation isT(f) = ∫fdμ, where the integral is denned in diverse ways, depending on the nature of the set of functions and the nature ofT.
Generator conditions on the fitting subgroup of a polycyclic group
Authors: Humphreys J. F.
Published: January 2009

In an earlier paper (3), polycyclic groups in which every subgroup can be generated by d, or fewer, elements were studied. In this paper we investigate the structure of those polycyclic groups G such that every abelian normal subgroup of F(G), the Fitting subgroup of G, can begenerated by at most d elements.
Some applications of double-negation sheafification
Authors: Macnab D. S.
Published: January 2009

In this note we point out that certain algebraic-topological constructions are particular cases of one construction, namely double-negation sheafification. The principal cases we have in mind are concerned with booleanpowers, completions of boolean algebras, and maximal rings of quotients.We conjecture that several other constructions—particularly completion-type constructions—will turn out also to be examples of double-negation sheafification.
Conditions on near-rings which imply that nil N-subgroups are nilpotent
Authors: Oswald A.
Published: January 2009

We assume the reader to be familiar with the basic definitions of near-rings, N-subgroups etc. as presented, for example, in (4). Throughout, N will denote a left near-ring (i.e. a, b, c ∈ N imply a(b + c) = ab + ac) in which 0n = 0 for each n ∈ N. We say that N is strictly semiprime if A2 implies A = (0) where A is an N-subgroup of N. An N-subgroup A is nilpotent if An for some positive integer n and an element a ∈ N is nil if an = 0 for some n. An element a ∈ N is regular if ax = 0 or xa = 0 implies x = 0.
A variant of separability in dual systems
Authors: Popoola J. O. , Tweddle I.
Published: January 2009

In (12) we introduced the concept of essential separability and used it to define two classes of locally convex spaces, δ-barrelled spaces and infra-δ-spaces, which serve as domain and range spaces respectively in certain closed graph theorems (12, Theorems 3 and 7). In this note we continue the study of these ideas. The relevant definitions are reproduced below.
The tensor product of distributive lattices II
Authors: Fraser Grant A.
Published: January 2009

In this paper we continue our study of the tensor product of distributive lattices which was begun in (2). We obtain a representation of the tensor product as a ring of sets and we describe a simple way to construct the tensor product of finite distributive lattices.
Nilpotents in finite symmetric inverse semigroups
Authors: Gomes Gracinda M. S. , Howie John M.
Published: January 2009

In semigroup theory as in other algebraic theories a significant part of the total effort is appropriately applied to the study of certain standard examples occurring, as it were, “in nature”. The most obvious such semigroup is the full transformation semigroup (X) (see [3]) and about this semigroup a great deal is known in both the finite and infinite cases.
Strongly exposed points and a characterization of l1 (Γ) by the Schur property
Authors: Polyrakis Ioannis A.
Published: January 2009

In this paper we study the existence of strongly exposed points in unbounded closed and convex subsets of the positive cone of ordered Banach spaces and we prove the following characterization for the space l1(Γ): A Banach lattice X is order-isomorphic to l1(Γ) iff X has the Schur property and X* has quasi-interior positive elements.
Explicit solutions for a system of coupled Lyapunov differential matrix equations
Authors: Jodar L. , Mariton M.
Published: January 2009

This paper is concerned with the problem of obtaining explicit expressions of solutions of a system of coupled Lyapunov matrix differential equations of the type

where Fi, Ai(t), Bi(t), Ci(t) and Dij(t) are m×m complex matrices (members of ℂm×m), for 1≦i, j≦N, and t in the interval [a,b]. When the coefficient matrices of (1.1) are timeinvariant, Dij are scalar multiples of the identity matrix of the type Dij=dijI, where dij are real positive numbers, for 1≦i, j≦N Ci, is the transposed matrix of Bi and Fi = 0, for 1≦i≦N, the Cauchy problem (1.1) arises in control theory of continuous-time jump linear quadratic systems [9–11]. Algorithms for solving the above particular case can be found in [12]]. These methods yield approximations to the solution. Without knowing the explicit expression of the solutions and in order to avoid the error accumulation it is interesting to know an explicit expression for the exact solution. In Section 2, we obtain an explicit expression of the solution of the Cauchy problem (1.1) and of two-point boundary value problems related to the system arising in (1.1). Stability conditions for the solutions of the system of (1.1) are given. Because of developed techniques this paper can be regarded as a continuation of the sequence [3, 4, 5, 6].
A characterisation of C*-algebras
Authors: Hennings M. A.
Published: January 2009

It is of some interest to the theory of locally convex *-algebras to know under what conditions such an algebra A is a pre-C*-algebra (the topology of A can be described by a submultiplicative norm such that ‖x*x‖ = ‖x‖2, ∀x∈A). We recall that a locally convex *-algebra is a complex *-algebra A with the structure of a Hausdorff locally convex topological vector space such that the multiplication is separately continuous, and the involution is continuous.
Stability theorems for wedges
Authors: Brown G. , Pryce J.D.
Published: January 2009

We are concerned with the stability properties of uniformly closed wedges in C(E) (resp. C+(E)), the real-valued (resp. non-negative) continuous functionson a compact space E, and solve the following problems in this area:

(a) Let A be a closed semi-algebra in C+(E) such that
Integrals of dynamical systems linear in the velocities
Authors: Collinson C. D.
Published: January 2009

Kilmister (1) has discussed the existence of linear integrals of a dynamical system specified by generalized coordinates qα(α = 1, 2, …, n) and a Lagrangian

repeated indices being summed from 1 to n. He derived covariant conditions for the existence of such an integral, conditions which do not imply the existence of an ignorable coordinate. Boyer (2) discussed the conditions and found the most general Lagrangian satisfying the conditions for the case of two degrees of freedom (n = 2).
Inequalities for certain cyclic sums
Authors: Daykin David E.
Published: January 2009

Let M be a positive integer, let a1, a2, …, aM be non-negative reals, and put aM+i = ai for i = 1, 2, 3. Further let each of v1, v2, v3 and δ1, δ2, δ3 be 0 or 1, giving 26 possibilities. This note is concerned with the problem of finding bounds for each of the non-trivial cases out of the 26 cyclic sums
On uniqueness of flows of a dipolar fluid
Authors: Hills R. N.
Published: January 2009

In 1964 Green and Rivlin (1) introduced a theory of simple force and stress multipoles founded on conventional kinematics. Using a work formula, the force and stress multipoles were defined with the help of the velocity field and its spatial derivatives. More recently, within the framework of this general study, Bleustein and Green (2) examined the theory of the simplest multipolar fluid, the dipolar fluid, and formulated constitutive equations for a homogeneous incompressible linear dipolar fluid.
The asymptotic expansion of the number of tree-like polyhexes
Authors: Wright E. M.
Published: January 2009

We refer to (1) for the definitions of Un and Hn. Our object is to find asymptotic expansions for Un and Hn for large n. This enables us to improve the approximations to Un and Hn for large n found in the last two pages of (1).
On some generalisations of Laguerre polynomials
Authors: Erdélyi A.
Published: January 2009

The sequence of orthogonal functions derived from Laguerre-polynomials is known to be complete, and hence closed, in L2 (0, ∞) if (a) > − 1. In a recent paper Dr Kober introduced a generalisation of this sequence, which enables him to extend the known results also for (a) < − 1. Kober's guiding principle seems to be the following one: The Laguerre orthogonal functions form, for (a) > − 1, a complete system of self and skew reciprocal functions of the Hankel transformation of order a. Now, if (a) < − 1, the ordinary Hankel transform has to be replaced by so-called cut Hankel transform. Hence the system of functions which has to replace-Laguerre orthogonal functions when (a) < − 1, should be a complete system of self and skew reciprocal functions of the cut Hankel transformation of order a, such that it reduces for (a) > − 1 (when the cut Hankel transform reduces to the ordinary one) to the sequence of Laguerre orthogonal functions. This, of course, is by no means a unique definition; nevertheless, together with what one would call the permanence of the Mellin transform, it enabled Kober to find a sequence of functions which (i) reduces to the sequence of Laguerre orthogonal functions when (a) > − 1, m = 0, (ii) is a complete set of self and skew reciprocal functions of the cut Hankel transformation with kernel Ja, m and (iii) has the required qualities of completeness and closedness.
On the Fourier coefficients of a discontinuous function
Authors: Bhatnagar S. P.
Published: January 2009

We suppose throughout that f(t) is periodic with period 2π, and Lebesgue-integrable in (− π, π).

We write

and suppose that the Fourier series of φ(t) and ψ(t) are respectively cos nt and sin nt. Then the Fourier series and allied series of f(t) at the point t = x are respectively and , where A0 = ½a0, An = ancos nx + bnsin nx, Bn = bncos nx − ansin nx and an, bn are the Fourier coefficients of f(t).
Sets of Semi-Commutative Matrices: Part II
Authors: Williamson J.
Published: January 2009

In this section we extend the definition of an E-set, so that it includes sets of the type

where the only restriction on the Ei is that they be non-singular. We now consider matrices of the type

where each ei takes independently the values 0, 1, …, n − 1, while the a(ei) are either complex numbers or else matrices of order r, the product a(ei) E(ei), in the latter case, being interpreted as the direct product of the two matrices a(ei) and E(ei).
On the asymptotic periods of integral functions
Authors: Whittaker J. M.
Published: January 2009

A period of a function f(z) is defined to be a number w(≠0) such that

is identically zero; and it is not a difficult matter to show that an integral function may either have no periods or else a single sequence kλ, k = ±1, ±2, ….
A simple method of establishing relations between Theta-functions of zero argument
Authors: Dalton J. P.
Published: January 2009

In connection with their proof of the fundamental relation

Whittaker and Watson state that “several proofs of this important proposition have been given, but none are simple.”