bluuebunny/crossref_metadata_2025 huggingface dataset:
Identifiability of countable mixtures of discrete probability distributions using methods of infinite matrices
Authors: Patil G. P. , Bildikar Sheela
Published: November 2008

Introduction. The subject of the mixtures of probability distributions has aroused a fresh interest in the fields of probability and statistics. Several authors, including Blischke, Rider, Robbins, Teicher, Patil and Seshadri, have discussed a variety of the associated probabilistic and statistical problems in varying details in the recent past. This paper attempts to solve by using the methods of infinite matrices some of the problems of the identifiability of countable mixtures of discrete probability distributions. It appears that here is still another area where Mathematics and Statistics can help one another grow. We are extremely thankful to Dr P. Vermes of Birkbeck College of the University of London for developing some theory of infinite matrices relevant to a couple of problems discussed in this paper. While most of the results discussed below are believed to be new, a few of them can be shown to be immediate consequences of a powerful result due to Teicher ((7)) obtained by using a different approach.
Permanent of the product of doubly stochastic matrices
Authors: Brualdi R. A.
Published: November 2008

If A = [aij] is an n × n matrix, the permanent of A is the scalar valued function of A defined by

where the summation extends over all permutations (i1, i2, …, in) of the integers 1, 2, …, n. If we assume that A is a non-negative matrix (that is, that A has non-negative entries) then we come upon an extremely interesting situation. Much work has been done in finding significant upper bounds for the permanent and permanental minors of A (see, e.g. (2, 4–7) in (5) an excellent bibliography is given). If B = [bij] is another n × n non-negative matrix, then we may form the product AB and consider the permanent of this matrix. Unlike the determinant, the permanent is not a multiplicative function. In our circumstances here, however, it is easy to verify and indeed a proof was written down in (1) that
On the gravitational collapse of a cold rotating gas cloud
Authors: Lynden-Bell D.
Published: November 2008

In the course of researches on the formation of galaxies one meets the following idealized problem. What is the form of the collapse under gravitational forces of a uniformly rotating spheroidal gas cloud? In the special case where initially the gas is absolutely cold and of uniform density within the spheroid, we show that the collapse proceeds through a series of uniform, uniformly rotating spheroids until a disk is formed. This demonstration is followed by some remarks and conjectures as to what will happen in the physical case when the initial temperature is not necessarily zero.
A note on complementary spines
Authors: Rourke C. P.
Published: November 2008

This note is concerned with a result about spines of manifolds embedded in a sphere, which proved useful at one point in (2).
Perturbation theory for a linear operator
Authors: Webb J. H.
Published: November 2008

We extend certain results of the theory of closed operators in Banach spaces to general linear operators in normed spaces. A ‘state diagram’ for linear operators is drawn up. We prove some perturbation theorems, improving or correcting certain results of Goldberg.
An inequality involving Radon-Nikodym derivatives
Authors: Kingman J. F. C.
Published: November 2008

In (2) a simple proof was given of the following inequality of Atkinson, Watterson and Moran (l), a slight generalization of a result of Scheuer and Mandel ((4), see also (3)) which is of importance in genetics. If aij, pi, qj (i, j = 1, 2, 3,…) are non-negative numbers with

and if

then

and if then This result is shown in (2) to be a special case of a large class of inequalities involving sets of non-negative numbers depending on several indices, and ‘partial averages’ over subsets of those indices.
Investigations into a new stratified hyperbolic profile
Authors: Heading John
Published: November 2008

The symmetrical Epstein refractive index profile (with free space conditions on each side of the transition) is generalized to a non-symmetrical form including both a barrier and a well by the discovery of a profile that cannot be obtained from the standard Epstein profile merely by adjusting the parameters. The analytical evaluation of the reflexion and transmission properties of the model is more complicated than that which occurs in the study of the Epstein profile.
Solutions of a non-linear differential equation. I
Authors: Billigheimer C. E.
Published: November 2008

We consider in this paper solutions of the equation

where the primes indicate differentiation with respect to s, and a, b, c are constants.
Abstract Köthe spaces. III
Authors: Fremlin D. H.
Published: November 2008

In this paper I shall be considering the space of linear maps between two (perfect) Riesz spaces, and shall show how certain topological properties of these maps are related to the natural order structure of the space. The fundamental result is (e) of section 4, certain special cases of which have been treated in (4) and (5), using a less elliptic method of proof. Probably the most interesting new result in the present paper is section 10 in the special case of both L and M× being L1 spaces (so that |σ| (L, L×) and |σ| (M×, M) are the norm topologies ((2 b), section 7) and Λ(L; M×) is the space of norm-continuous linear maps).
A class of methods for tabular interpolation
Authors: Larkin F. M.
Published: November 2008

A generalization of the Neville–Aitken method is described which allows the construction of interpolating functions, other than polynomials, by means of simple recurrence relations. In particular, simple constructions are given for rational functions and trigonometric series which interpolate prescribed function values at non-equispaced positions of the independent variable.

Restrictions imposed by requiring the interpolating functions to be invariant under linear transformations of the coordinates are discussed, and application of the technique to the problem of inverse interpolation is also considered.
An expansion theorem applicable to problems of wave propagation in an elastic half-space containing a cavity
Authors: Gregory R. D.
Published: November 2008

This paper is principally concerned with the two-dimensional time harmonic vibrations of an elastic half-space y ≥ 0, containing a submerged cavity in the form of an infinite circular cylinder. Two sequences of line source potentials are obtained which are singular along the axis of the cylinder, satisfy the free surface conditions on y = 0, and represent outgoing waves at infinity. (The radiation condition.) It is proved that any solution of the governing equations which satisfies the free surface conditions and consists of outgoing waves at infinity is expansible as a sum of these fundamental source potentials, with coefficients to be determined from the boundary conditions on the cylinder only.

The requirement of outgoing waves is carefully discussed and it is shown that the conditions taken give rise to, and are satisfied by, potential fields which would be regarded intuitively as representing outgoing waves.
On the coefficient problem for univalent polynomials
Authors: Ławrynowicz Julian
Published: November 2008

This paper is concerned with estimates of the coefficients Cn for normalized polynomials, inverse to univalent functions in the unit disc, i.e. univalent polynomials of the form

considered in the largest domain D such that 0 ∈ D and |pn(w)| < 1 for w ∈ D; this domain will be denoted by . Polynomials of the type (1) have been introduced by Charzyński in (1), where it was proved among other results that the domain is simply connected.
The arithmetic of certain semigroups of positive operators
Authors: Lamperti John
Published: November 2008

Some time ago, S. Bochner gave an interesting analysis of certain positive operators which are associated with the ultraspherical polynomials (1,2). Let {Pn(x)} denote these polynomials, which are orthogonal on [ − 1, 1 ] with respect to the measure

and which are normalized by settigng Pn(1) = 1. (The fixed parameter γ will not be explicitly shown.) A sequence t = {tn} of real numbers is said to be ‘positive definite’, which we will indicate by writing , provided that

Here the coefficients an are real, and the prime on the summation sign means that only a finite number of terms are different from 0. This condition can be rephrased by considering the set of linear operators on the space of real polynomials which have diagonal matrices with respect to the basis {Pn(x)}, and noting that
On a heat flow problem in a hollow circular cylinder
Authors: Ölçer Nuretti̇n Y.
Published: November 2008

Recently, through a repeated application of one-dimensional finite integral transforms, Cinelli(1) gave a solution for the temperature distribution in a hollow circular cylinder of finite length. Since no new ideas or techniques are introduced, the extension claimed in (1) with regard to the finite Hankel transform technique employed in the transformation of the radial space variable in the hollow cylinder problem is trivial, in view of well-known works by Sneddon(2) and Tranter (3), to mention a few. The list of the finite Hankel transforms given in (1) for a variety of boundary conditions at r = a and r = b is the result of routine, algebraic manipulations well known from the general theory of eigenvalue problems specialized for the hollow cylinder. In this list a set of seemingly different series expansions is given for the inverse Hankel transform for each combination of boundary conditions at the two radial surfaces. In each case, the two expressions for inversion can readily be shown to be identical to each other when use is made of the frequency equation. One of the inversion forms is therefore unnecessary once the other is given. Furthermore, the general solution as given by equation (54) of Cinelli(1)does not satisfy his boundary conditions (27), (28), (29) and (30), unless these latter are homogeneous.
On an integral equation arising in the theory of neutron streaming
Authors: Williams M. M. R.
Published: November 2008

It is shown that a singular integral equation, arising in the theory of neutron streaming in cylindrical channels, can be solved by the Wiener-Hopf technique. The singular equation is converted into a non-singular Fredholm one, which may be solved very simply by finite difference methods and matrix inversion.
Influence du tenseur de conductivité électrique sur l'écoulement dans un générateur à électrodes finies
Authors: Dragoş Lazar
Published: November 2008

The paper studies the influence of the electrical conductivity tensor upon the motion of a fluid in a MHD generator, under the conditions shown in our preceding work. The problem is also formulated in terms of the theory of distributions and the system of equations which determines the functions Ψ = ψ + y and φ = ϕ + ϕe depends on two parameters λ and v (= ωτ). Due to the Hall current, the problem is no longer asymmetrical in the variable y, as in the first case, it therefore has to be separated into an asymmetric and a symmetric problem. Developing a method of successive approximations as against two parameters, it can be shown that the exact solution of the problem may be obtained.
Some new generating functions for Jacobi polynomials
Authors: Sharma B. L. , Mittal K. C.
Published: November 2008

The object of this paper is to obtain some new generating functions for Jacobi polynomials. The results obtained, are of general character.
A simple method of summing Bessel series
Authors: Greenwood J. A.
Published: November 2008

A simple formula is derived for transforming the slowly convergent series involving Bessel functions which arise in axially symmetric potential problems into more rapidly convergent series. Using it, sums of series previously found by elaborate special methods may be obtained by direct substitution. The necessary numerical coefficients are given.
The distributions of certain factors occurring in discriminant analysis
Authors: Radcliffe J.
Published: November 2008

Significance tests for several hypothetical discriminant functions have been developed by Williams (7,8) and considered further by the author (6). The test criteria consist of the factors in certain factorizations of the residual likelihood criterion, when the effect of the hypothetical discriminant functions has been eliminated. The independence and distributions of the factors can be seen by geometrical considerations, to be a consequence of the manner in which the factors are constructed in the sample space. In the case of a single hypothetical discriminant function Kshirsagar (5) has produced analytic proofs, by means of matrix transformations, of the independence and distributions of the factors. In this paper we shall give analytic proofs of the independence and distributions of the factors, given in sections 4 and 5 of the authors' paper (6), by extending Kshirsagar's proof to the case of several hypothetical discriminant functions.
The Dirac spinor in six dimensions
Authors: Lord E. A.
Published: November 2008

The spinor representations of the rotation group in a six-dimensional space with indefinite metric are shown to be four-component spinors, which become the usual Dirac spinors when the formalism is restricted to a four-dimensional subspace. Eriksson's work on the five-dimensional Lorentz group is found to result from a restriction of the six-dimensional treatment to a five-dimensional subspace, and the algebraic significance of Eriksson's work is thereby clarified.
On product sets in a unimodular group
Authors: McCrudden M.
Published: November 2008

Let G be a locally compact Hausdorff topological group, with µ the left Haar measure on G, and µ* the corresponding inner measure. If R denotes the real numbers, ℬ(G) denotes the Borel† subsets of G of finite measure, and VG = {µ(E):E∈ℬ(G)}, then, following Macbeath(4), we define ΦG: VG × VG→ R by

where AB denotes the product set of A and B in G.
On Nörlund summability of derived Fourier series
Authors: Sharma R. M
Published: November 2008

Let f(θ) be integrable L in (−π, π) and periodic with period 2π and let
On the hyperplanes of a matroid
Authors: Welsh D. J. A.
Published: November 2008

Matroid theory was first studied by Whitney (1) as an abstract theory of linear independence in vector spaces. Recently its importance in graph theory has been noticed by Tutte (2), Edmonds (3) and Nash-Williams (4,5). Less interest has been shown in the extremely close relationship between matroids and incidence geometries. In this note we develop the more geometrical aspects of matroid theory, paying particular attention to the fundamental role of the hyperplanes of a matroid in this theory.
Ultrabarrelled groups and the closed graph theorem
Authors: Iyahen S. O.
Published: November 2008

Suppose that t is a group homomorphism from a topological group E into a topological group F. When is it true, that the closure of t−1(V) in E is a neighbourhood of the identity in E for every neighbourhood V of the identity in F? This question arises naturally in the study of the closed graph theorem in the context of topological groups; for example, see (1) and ((3), p. 213). The concept of a g-ultrabarrelled space introduced in this paper is the result of an investigation aimed at answering this question.
On a Tauberian theorem for absolute harmonic summability
Authors: Varshney O. P. , Prasad Govind
Published: November 2008

Let Σan be a given infinite series with the sequence of partial sums {Sn}. Let the sequence be defined by

where is given by
Extension of a theorem of Sunouchi
Authors: Srivastava V. P.
Published: November 2008

1·1. Definitions and notations. Let ∑an be a given infinite series, and let sn be its nth partial sum. We denote by and the nth Cesàro means of order α(α > − 1) of the sequences {sn} and {n. an} respectively.
Subalgebras of modular annihilator algebras
Authors: Barnes B. A.
Published: November 2008

Throughout this paper we deal only with complex and semi-simple algebras. Let B be such an algebra. We denote the socle of B as SB. B is a modular annihilator algebra if B/SB is a radical algebra, i.e. if every element of B is quasi-regular modulo the socle of B; see (1) or (12). Now assume that B is a modular annihilator algebra and a Banach algebra. Then any semi-simple closed subalgebra of B is a modular annihilator algebra by ((4), Cor. to Theorem 4·2,). It is not true, however, that any semi-simple subalgebra A of B is a modular annihilator algebra, even when A is a Banach algebra in some norm. We give a simple example to illustrate this. Let A be the algebra of all complex functions f, continuous on the closed unit disk D in the complex plane, analytic in the interior of D, and such that f(0) = 0. A is a Banach algebra in the usual sup norm over D. Now consider the norm on A defined by

Let B be the completion of A in this norm. A has an involution * defined by and also ‖ff*‖ = ‖f‖2 for all f ∈ A. Therefore B is a B*-algebra. It is not difficult to verify that the only non-zero multiplicative linear functionals on A which are continuous with respect to the norm ‖·‖, are the point evaluations at 1/n, n = 1, 2 …. It follows that every non-zero multiplicative linear functional on B is an extension of one of these point evaluations to B. Thus B can be identified with the algebra of all complex sequences which converge to zero.
Exact solutions for transverse electromagnetic wave propagation in a cylindrically stratified axially magnetized plasma
Authors: Westcott B. S.
Published: November 2008

The equations describing transverse propagation of electromagnetic waves in a cylindrically stratified plasma, in which a static magnetic field is imposed parallel to the axis of cylindrical symmetry, are examined. A systematic method developed previously for isotropic media, Heading (3), Westcott (6) is adapted to yield generalized proffles relating to the plasma properties. These proffles are capable of producing exact solutions for the field distribution in terms of standard transcendental functions. Previously known proffles appear as special cases of the analysis.
Propagation of torsional disturbance in an infinite solid of piezoelectric β-quartz
Authors: Paul H. S. , Rao B. Srinivasa
Published: November 2008

A systematic study of the stress distribution in an infinite elastic solid when time dependent and time independent body forces act on certain regions of the solid is made by Eason, Fulton and Sneddon(l). They have stated the importance of studying problems concerning moving point charges in electrodynamics. Mukhopadhyay(2) has discussed the disturbance created by the propagation of torsional disturbance in an infinite elastic solid due to time dependent rotating body forces.
A note on rank and direct decompositions of torsion-free Abelian groups. II
Authors: Corner A. L. S.
Published: November 2008

According to well-known theorems of Kaplansky and Baer–Kulikov–Kapla nsky–Fuchs (4, 2), the class of direct sums of countable Abelian groups and the class of direct sums of torsion-free Abelian groups of rank 1 are both closed under the formation of direct summands. In this note I give an example to show that the class of direct sums of torsion-free Abelian groups of finite rank does not share this closure property: more precisely, there exists a torsion-free Abelian group G which can be written both as a direct sum G = A⊕B of 2 indecomposable groups A, B of rank ℵ0 and as a direct sum G = ⊕n ε zCn of ℵ0 indecomposable groups Cn (nεZ) of rank 2, where Z is the set of all integers.
On the Feynman path integral in q, , p space
Authors: Balazs N. L.
Published: November 2008

An alternative definition is proposed for the kernel used by Feynman. This definition involves a functional integration in a q, , p space, treating these variables as independent. The equivalence of this definition to the Feynman one and to the one using the variables q, p is exhibited.
Extremum principles for electrostatic and magnetostatic problems
Authors: Arthurs A. M. , Robinson P. D.
Published: November 2008

Complementary variational principles are derived for a class of electrostatic and magnetostatic problems using the pairs of adjoint operators (grad, – div) and (curl, curl). This theory unifies the principles of Dirichlet and Thomson in electrostatics and of Schrader in magnetostatics. The results are illustrated by deriving upper and lower bounds for the capacity of a surface, and it is shown how such bounds can be systematically improved by Ritz procedures.
Square Banach spaces
Authors: Cunningham F.
Published: November 2008

In (2) I described a canonical isometric representation of an arbitrary real Banach space X by vector-valued functions (with the uniform norm) on a compact Hausdorif space ω with the following properties: (1) the representing function space is invariant under multiplications by continuous real functions on ω; (2) the norm of each representing function, as a real non-negative function on ω, is upper semicontinuous; and (3) this decomposition of X is maximally fine. I called attention to the class of spaces X for which at every point of ω the component space of this representation is one-dimensional or 0, so that the representing functions are in effect real valued. I propose to call such Banach spaces square, because of the shape of the unit ball in the two-dimensional case. In (2) I stated without proof, erroneously as it turns out, that the class of square spaces coincides with what Lindenstrauss in (4) called G-spaces. The primary purpose of this paper is to show that the class of square spaces is actually properly contained in that of G-spaces. It is known ((2), p. 620, Example 1) that it contains properly the class of continuous function spaces C(ω). Among G-spaces are the M-spaces treated by Kakutani as Banach lattices (3). I shall show further that neither class, square spaces or M-spaces (regarded now purely as Banach spaces), contains the other.
On the Dirichiet product of Cesàro-summable series
Authors: Jukes K. A.
Published: November 2008

Segal(1) in his paper ‘Summability by Dirichlet convolutions’ makes direct use (in the proofs of Theorems 4 and 6) of a result on the Dirichlet product of two series which may be stated as follows:

is (C, k)-summable to the sum α (k≥0) and is (C, l)-summable to the sum β, (l ≥ 0), then the Dirichlet product

is (C, k + l + 1)-summable to the sum αβ.’
Invertible powers of ideals over orders in commutative separable alǵebras
Authors: Singer Michael
Published: November 2008

The purpose of this paper is to obtain quantitative results on invertible powers of (fractional) ideals in commutative separable algebras over dedekind domains. This is connected with the work of Dade, Taussky and Zassenhaus(2) on ideals in noetherian domains. We do not, however, make use of their paper, but rather draw on the general theorems on ideals in commutative separable algebras established by Fröhlich (3), in particular his qualitative result that some power of any given ideal is invertible. Our basic result (Theorem 1) concerns the invertibility of powers of a particular type of ideal, the componentwise dedekind ideals defined below. From this we deduce a general result (Theorem 2), which includes as a special case Theorem C of (2) for the case of separable field extensions; specifically, the (n – 1)th power of any ideal is invertible, where n is the dimension of the algebra. Although, as we show, it is possible to deduce Theorem 2 from (2), we have here an independent proof of one of the main results of (2) based entirely on the results in (3). As a further application of Theorem 1 we obtain a new result on ideals over the group ring of an abelian group over the ring of rational integers; the (t – 1)th power of such an ideal is invertible, where t is the maximum number of simple components of the rational group algebra of any Sylow subgroup. We also show that this is the best possible result when some Sylow subgroup whose rational group algebra has t components is cyclic.
Summability (L) of Fourier series
Authors: Nanda M. M. , Das G.
Published: November 2008

Let f(t) be Lebesgue integrable in (-π, π) and be periodic with period 2π; let the Fourier series of f(t) at t = y be
The order-bound topology on Riesz spaces
Authors: Wong Yau-Chuen
Published: November 2008

1. Introduction. Let (X, C) be a Riesz space (or vector lattice) with positive cone C. A subset B of X is said to be solid if it follows from |x| ≤ |b| with b in B that x is in B (where |x| denotes the supremum of x and − x). The solid hull of B (absolute envelope of B in the terminology of Roberts (2)) is denoted to be the smallest solid set containing B, and is denoted by SB. A locally convex Hausdorff topology on (X, C) is called a locally solid topology if admits a neighbourhood-base of 0 consisting of solid and convex sets in X; and (X, C, ), where is a locally solid topology, is called a locally convex Riesz space.
A note on the number of solutions Np of the congruence y2 ≡ x3 − Dx(mod p)
Authors: Rajwade A. R.
Published: November 2008

The object of this note is the following theorem together with a table of the (1 + i)n division points for n = 1, 2, 3, 4, 5 on the elliptic curve y2 = x3 − Dx which has complex multiplication by i = √(−1).
On the construction of G-spaces and applications to homogeneous spaces
Authors: Gottlieb Daniel Henry
Published: November 2008

In (3), the author defined the notion of a G-space. A G-space is a weaker notion than that of an H-space. The main purpose of this paper is to present various means of constructing G-spaces. As an application of some of the techniques of (3) and of this paper (though not an application of the concept of G-space) we shall prove the following theorem:

Theorem. Let G be a connected compact Lie group and let H be a connected subgroup of maximal rank. Then H3(G/H; Z) = 0. In fact, the Hurewicz homomorphism is trivial for odd dimensions.
The period equation for primes p congruent to 1 (mod 5)
Authors: Rajwade A. R.
Published: November 2008

Let p = 1 + 5n be a rational prime congruent to 1 (mod 5). Let ζ = e2πi/p and let g be a primitive root mod p. Let the non-zero residues g, g2, …, gp-1 (mod) p be divided into five classes , ℬ, , , ℰ, where gν ∈ , ℬ, , , ℰ according as ν ≡ 0, 1, 2, 3, 4 (mod 5). Let

be the 5-nomial periods. Then it is well known (see (3)) that they are the roots of a monic polynomial with integral coefficients. Our object is to determine these coefficients in terms of the quantities A, B, C, D, E, Y, Z considered in a previous paper (2), p. 65. A large number of relations connecting these quantities have been obtained in the above-mentioned paper and we shall use these relations to simplify the coefficients and get them in a reasonably compact and symmetrical form.
An extension of Lukacs's result
Authors: Shanbhag D. N.
Published: November 2008

Consider X and Y to be independent and non-degenerate random variables. If X and Y are non-negative then it follows from (6) that for all given values z of X + Y, the expected values of X and X2 are of the type λz and λ′ z2, respectively, if and only if X and Y have the gamma distributions with the same scale parameter. This is an improvement over Lukacs's result (3). Since the characterization problems for the Poisson, binomial, negative binomial and normal distributions resemble the corresponding problem for the gamma distribution it is quite reasonable to expect the characterizations similar to the above for these distributions. This is established in what follows.
Quantum mechanics and the gravitational red shift
Authors: Stoeckli H. F.
Published: November 2008

It is shown that the formula for the gravitational red shift predicted by the theory of general relativity can also be derived by classical quantum mechanics combined with relativistic arguments. The agreement between the two derivations is a consequence of the separability of the time-dependent wave function, and of the first-order time differential in the wave equation.
The effects of wall conductivity in magnetohydrodynamic duct flow at high Hartmann numbers
Authors: Temperley D. J. , Todd L.
Published: November 2008

Laminar motion of a conducting fluid in a rectangular duct is discussed. The applied magnetic field is uniform and parallel to one pair of sides of the duct. Classical theory is used and it is shown that the two successive limiting processes, lim (σwall → ∞; hσ wall → a finite, constant limit) and lim (M → ∞) are not always freely interchangeable; M being the Hartmann number, σwall the electrical conductivity of the duct wall and h the typical ratio of (wall thickness/duct width). A general expansion procedure for M ≫ 1, valid for all types of wall conductivities, is devised. A critical discussion of the deficiencies in the classical model is given.
The combinatorial structure of the Hughes plane. II
Authors: Room T. G.
Published: November 2008

In the first part of this paper, tests were described for determining which points of any line in a Galois plane II of order q2 are to be transferred to the conjugate line in order to transmute II into the corresponding Hughes plane Ω. In this part of the paper the tests are refined to provide, in relation to some fixed point in the central subplane δ0 of Ω (i) a simple geometrical condition of transfer for a certain set of ½q(q2−1) points of II and (ii) a simple aglebraic condition for the remaining points of II – δ0. These tests eliminate from the computation (for a given value of q) the necessity of calculating the third coordinates of ½q2 (q2−1) points in order to determine which are not-squares.
Coupled fields in inhomogeneous warm plasmas with static pressure gradients. I
Authors: Burman R.
Published: November 2008

.This paper deals with small amplitude waves in inhomogeneous warm electron plasmas. The waves are coupled electromagnetic and electron-acoustic waves, and are described by Maxwell's equations together with single-fluid hydrodynamical equations. Here, previous work is generalized by including the effect of a static pressure gradient. Coupled wave equations are obtained and specialized to the case of a planar stratified plasma. Then, as a preliminary to a treatment of wave coupling, the behaviour of the solutions of the uncoupled wave equations in a coupling region is investigated. The static pressure gradient complicates the behaviour of the uncoupled field components; singularities occur at two points which coalesce as the static pressure gradient is allowed to tend to zero.
Entropy and simple waves in multidimensional gas flow
Authors: Levine Lawrence E.
Published: November 2008

It is shown that the n-dimensional simple wave flow of a perfect gas, that is, flow in which the velocity components, pressure and density depend on only one generating function, must be isentropic. This result is then employed to explain why two independent investigations of apparently different types of simple waves have led to the same results.
Uniqueness of an independent transversal of a family of sets
Authors: Brualdi Richard A.
Published: November 2008

In his book Transversal Theory (4) Mirsky introduces the problem of finding a necessary and sufficient condition in order that a finite family of sets have a unique independent transversal. The purpose of this note is to give one necessary and sufficient condition. More generally, we give a necessary and sufficient condition in order that a common basis of two rank-finite matroids be unique. We then specialize this result to give an answer to the above question.
Cayley maps and symmetrical maps
Authors: Biggs Norman
Published: November 2008

In this paper we shall show how combinatorial methods can be applied to the study of maps on orientable surfaces. Our main concern is with maps which possess a certain kind of symmetry, called vertex-transitivity. We show how an extension of the well-known method of Cayley can be used to construct such maps, and we give conditions which suffice for the automorphism groups of these maps to have non trivial vertex-stabilizers. Finally, we investigate the special case when the skeleton of the map is a complete graph; a classical theorem of Frobenius then implies that all vertex-transitive maps are given by our extension of Cayley's construction.
Inclusion relations for Riesz typical means
Authors: Jakimovski A. , Tzimbalario J.
Published: November 2008

Necessary and sufficient conditions for sequence-to-sequence or sequence-to-function summability method to include (R, λ, α), when 1 < α ≤ 2, are given. Also, for suitably restricted sequences λ, necessary and sufficient conditions for a series-to-sequence or series-to-function summability method to include (R, λ, α) for 1 < α ≤ 2 are given. These results are obtained by showing that a certain sequence {δj} (j ≥ 0) is a Schauder-basis in Rλα(N) for each α, 1 < α ≤ 2.
The instability of a vortex sheet on a subsonic stream under acoustic radiation
Authors: Jones D. S. , Morgan J. P.
Published: November 2008

This paper is concerned with the linear theory of the transmission of sound through a vortex sheet separating two fluids in relative motion, but with the same density and sound speed. For harmonic excitation asolution is determined, with particular attention to transition regions where large effects might be expected, and it is found that Helmholtz instability plays no role in this solution. However, this harmonic field does not lead to a solution of the initial value problem which satisfies causality. When causality is complied with an additional field must be superimposed which gives waves growing exponentially in space in the harmonic case and a singularity, which is more severe than has been previously encountered, in the time-dependent problem. The consequent effects of this instability are discussed.
Kolmogorov's differential equations for non-stationary, countable state Markov processes with uniformly continuous transition probabilities
Authors: Goodman Gerald S. , Johansen S.
Published: November 2008

1. Summary

We shall consider a non-stationary Markov chain on a countable state space E. The transition probabilities {P(s, t), 0 ≤ s ≤ t <t0 ≤ ∞} are assumed to be continuous in (s, t) uniformly in the state i ε E.
On the three-body problem
Authors: Lekner J.
Published: November 2008

We consider the ground state of a system of three interacting particles of equal mass. An integro-differential equation is obtained for the optimum pair function f in the product wavefunction Ψ(123) = f(12)f(13)f(23). The solution for harmonic forces reproduces the known exact ground state. Approximate analytic solutions are obtained for inverse-square forces, and for a general force law in the semiclassical limit.
On the structure of graded formal groups of finite characteristic
Authors: Schochet Claude
Published: November 2008

The structure of a wide range of Hopf algebras over fields is well known. Corresponding theorems for Hopf algebras over graded rings do not exist. This note is devoted to a Decomposition Theorem for certain Hopf algebras over rings (of finite characteristic) of the types which occur in topology, and a study of homological properties of the factors which therein arise.
A characterization of the circle and its applications to Hermitian operators
Authors: Bollobás Béla
Published: November 2008

Let X be a complex normed space and let T be a Hermitian linear operator on X (i.e. the numerical range of T is real). It was proved by Bonsall (see (4); Theorem 10.13) that then

for every u ∈ X. This inequality was improved by the author (3) to

Inequality (1) is closely related (see (3)) to some inequalities of Hadamard (6) and Kolmogorov (7) about the successive derivatives of functions in L∞(−∞, ∞). It was also shown in (3) (and was, in fact, shown already in (7)) that the constant 2 is best possible in (1). However, as we shall see, inequality (1) can be sharpened considerably if T attains its norm on u, i.e. if ‖Tu‖ = ‖T‖‖u‖.
Simultaneous approximation of two additive forms
Authors: Liu Ming-Chit
Published: November 2008

1. Introduction. In 1948, H. Heibronn (6) improved Vinogradov's result (11) and obtained that for any ε > 0 there exists some positive constant C(ε) which depends on ε only such that, for any real number θ and any integer N ≥ 1, there is an integer x satisfying

Here ∥t∥ means the distance from t to the nearest integer.
Martin's axiom and Hausdorif measures
Authors: Ostaszewski A. J.
Published: November 2008

A theorem of Besicovitch, namely that, assuming the continuum hypothesis, there exists in any uncountable complete separable metric space a set of cardinality the continuum all of whose Hausdorif h-measures are zero, is here deduced by appeal to Martin's Axiom. It is also shown that for measures λ of Hausdorff type the union of fewer than 2ℵ0 sets of λ-measure zero is also of λ-measure zero; furthermore, the union of fewer than 2ℵ0 λ-measurable sets is λ-measurable.
General moment functions and a density version of the central limit theorem
Authors: Smith Walter L. , Basu Sujit K.
Published: November 2008

1. Introduction. We shall employ the customary abbreviations: p.d.f. for probability density function; d.f. for distribution function; c.f. for characteristic function.
On continuous functions with graphs of σ-finite linear measure
Authors: Foran James
Published: November 2008

In this paper comparisons are made between the class of continuous functions of generalized bounded variation and the class of continuous functions with graphs having σ-finite length i.e. linear measure. An investigationof the differentiability and approximate differentiability of such functions discloses the fact that the latter class is considerably more extensivethan the former one. The following definitions will be needed:

(1) A function f is said to be of bounded variation (VB) on a set E if

where the supremum is taken over all sequences {[ai, bi]} of non-overlapping intervals with endpoints in E.
On the Uryson type of stochastic integral equations
Authors: Hardiman S. T. , Tsokos Chris P.
Published: November 2008

An investigation of the random or stochastic integral equations of the form

and

is presented, where ω ∈ Ω, the supporting set of the probability measure space (Ω, A, P). The existence and uniqueness of a random solution, a second-order stochastic process, of the equations is considered. Several theorems utilizing fixed point theorems and successive stochastic approximations give sufficient conditions for the existence of a random solution.
An examination of Whitham's exact averaged variational principle
Authors: Gilbert A. D.
Published: November 2008

Whitham's exact averaged variational principle is found by a new method. Comparisons are drawn between this method and the method advanced by Whitham. Further results about the Euler equations from this variational principle are obtained.
Exact analysis of simply supported rhombic plates under uniform pressure
Authors: Rajaiah K. , Rao Akella Kameswara
Published: November 2008

The simply supported rhombic plate under transverse load has received extensive attention from elasticians, applied mathematicians and engineers. All known solutions are based on approximate procedures. Now, an exact solution in a fast converging explicit series form is derived for this problem, by applying Stevenson's tentative approach with complex variables. Numerical values for the central deflexion and moments are obtained for various corner angles. The present solution provides a basis for assessing the accuracy of approximate methods for analysing problems of skew plates or domains.
Some geometry of constant curvature surfaces
Authors: Lowe P. G. , Melchers R. E.
Published: November 2008

1. Introduction. The optimum design of plane reinforced concrete slabs frequently requires consideration of deflected shapes for the slab such that through every point there is at least one principal curvature (κ1) direction on the deflected surface which is a constant curvature direction. The most interesting cases are those where only one principal direction has this property.
Some properties of the correlation function of quantum stochastic processes
Authors: Pulé J. V.
Published: November 2008

It is found that the correlation functions of quantum stochastic processes are very similar to the corresponding functions for classical stochastic processes. It is shown also that if the state for the process satisfies the KMS boundary condition then this is reflected in the analyticity of the continuation of the correlation.
A uniformly asymptotic solution for incompressible flow past thin sharp-edged aerofoils at zero incidence
Authors: Sheer A. F.
Published: November 2008

Expansions obtained from classical subsonic thin-aerofoil theory break down in the neighbourhood of the aerofoil edges. At sharp edges the method of matched asymptotic expansions fails to remedy this. Here this failure is explained, and in the case of incompressible flow past a symmetric aerofoil at zero incidence it is shown that by proper choice of the dependent variable an expansion may be obtained which is uniformly asymptotic. Finally, the case of a circular-arc aerofoil is considered in more detail.
The distribution of current on a cylindrical antenna
Authors: Harvey C. C.
Published: November 2008

The iterative solution of Hallen's integral equation for the current distribution on a lossless dipole transmitting antenna is studied from first principles, and a general solution in terms of an expansion function is derived. In this new solution even the zero-order approximation gives a reasonable representation of both the quadrature and the in-phase current distributions. The choice of an expansion function is discussed. At each stage the approximations which must be made in order to obtain earlier iterative solutions are clearly described. Three low order solutions are derived in detail: the zero-order solution using both a constant expansion parameter and a simple expansion function, and the rst-order solution using a constant expansion parameter. The results are compared and special attention is paid to the very short antenna, although the results are good for lengths up to and including the half-wave dipole. The results are presented in such a way that the first-order admittance of any given antenna may be readily calculated.
The transmission of surface waves through a gap in a vertical barrier
Authors: Porter D.
Published: November 2008

The two-dimensional configuration is considered of a fixed, semi-infinite, vertical barrier extending downwards from a fluid surface and having, at some depth, a gap of arbitrary width. A train of surface waves, incident on the barrier, is partly transmitted and partly reflected. The velocity potential of the resulting fluid motion is determined by a reduction procedure and also by an integral equation formulation. It is shown that the two methods lead to the same Riemann–Hilbert problem. Transmission and reflexion coefficients are calculated for several values of the ratio gap width/mean gap depth.
Some special operators and new classes of locally convex spaces
Authors: Chilana Ajit Kaur
Published: November 2008

We consider some special operators on a locally convex Hausdorff space to itself, which have neat spectral theories and prove some perturbation results. This leads us to define and study a few special classes of locally convex spaces in which various subsets of the algebra of continuous linear operators either coincide or are closely related with each other. These are then compared to the classes of barrelled, infrabarrelled and DF-spaces and examples are given to distinguish them from one another.
On a summability theorem of Berg, Crawford and Whitley
Authors: Garling D. J. H. , Wilansky A.
Published: November 2008

We recall that a matrix A is said to sum a sequence x if Axε c, the space of all convergent sequences, and that A is conservative if it sums every convergent sequence. If A is conservative, A defines a continuous linear operator on c. Berg (2), Crawford (3)and Whitley (9) have proved the following theorem:

Theorem 1. A conservative matrix sums no bounded divergent sequence if and only if,considered as an operator on c, it is range closed and has finite-dimensional null-spac
Resonances in isotropic active media
Authors: Westcott B. S.
Published: November 2008

The propagation of electromagnetic waves through isotropic media containing a continuous energy-gain mechanism is considered and the existence of perfect resonance properties is investigated for both homogeneous and inhomogeneous layers.
On the involution classes of the linear groupsGLn(K), SLn(K), PGLn(K), PSLn(K)over fields of characteristic two
Authors: Dye R. H.
Published: November 2008

1.Introduction. 1.1 In (5), (6) and (7) I have catalogued the classes of involutions of the orthogonal, symplectic and unitary groups over perfect fields of characteristic 2. So, too, were obtained the involution classes of the well-known relatives of these groups. For each group the relevant classes have simple and explicit descriptions in terms of the geometry of its setting, and an overall pattern can be seen.
Embedding piecewise linear manifolds with boundary
Authors: Gordon C. Mca.
Published: November 2008

1. Introduction. It is known (6) that the connectivity conditions on the manifolds in Irwin's embedding theorem (7) can be replaced by a connectivity condition on the map. This has led to the suggestion that in Hudson's analogue of Irwin's theorem for bounded manifolds (5), (6), the connectivity conditions on the manifolds modulo boundary can be replaced by the corresponding connectivity condition on the map modulo boundary. The main purpose of the present note is to show that this is in fact false, as is the formal analogue of Irwin's and Hudson's theorems for manifold triads. Its other purpose is to announce a related embedding theorem for manifolds with boundary which appeared in (3).
A singularity in an oceanic boundary layer
Authors: Creegan Aileen , Dyke P. P. G. , Johnson J. A.
Published: November 2008

A β-plane ocean is considered, and dissipative processes are modelled by using a coefficient of lateral friction. Attention is focused on the singularity which occurs if the angle that the western boundary makes with the north approaches ½π. The dimensions of the singular region are found using boundary layer techniques, and the solution for the transport stream function valid within it is obtained in closed form.
The critical coupling of modes in a tapered earth-ionosphere wave guide
Authors: Budden K. G.
Published: November 2008

In a stratified wave guiding system, it is often possible to adjust parameters, such as the wave frequency or some dimension of the guide, so that the propagation constants of two modes become equal. These modes are then said to be degenerate. In a tapered guide some of the parameters are slowly varying functions of distance x measured along the guide, and the conditions of degeneracy can be satisfied for some value x0 of x, possibly complex. Here the two modes are said to be critically coupled, and x0 is called a ‘coupling point’ in the complex x-plane. Cases are known where two modes in the earth-ionosphere wave guide could approach critical coupling in the twilight region of the earth, where the guide is tapered. The x dependence of the composite field for the two modes can then be expressed in terms of an Airy Integral function. The theory is closely analogous to that used for studying critical coupling of Ordinary and Extraordinary radio waves within a horizontally stratified ionosphere.