bluuebunny/crossref_metadata_2025 huggingface dataset:
Covariant integral equations for Heisenberg operators
Authors: Eden R. J.
Published: November 2008

Sets of rules are obtained for writing down directly the exact integral equations which are satisfied by certain functions of Heisenberg operators in quantum field theory. Three kinds of function are considered: the direct product, the chronological product and the M-product. The matrix elements of the M-product are equal to the Feynman amplitudes studied by Matthews & Salam (1) and the corresponding integral equation is called here the Matthews-Salam (M.-S.) equation. These authors have given a symbolic form of the M.-S. equation and a method of repeated differentiation and integration which can be used to obtain the explicit form of the integral equation in any particular example. In practice their method involves an immense amount of calculation even in quite simple examples. The rules obtained in the present paper make it possible to write down directly the M.-S. equation without any of the tedious calculations implied by the M.-S. method.

So long as the exact theory is used, the three sets of equations (for direct, chronological and M-products) are completely equivalent. When bound state theory is considered by an approximation based on a power series in the coupling constant different results are obtained. The approximation is inapplicable to the direct product equations, and leads to different approximate equations for the amplitudes obtained from the chronological and the M-products even when these amplitudes are identical. This paradox is explained and it is shown that the equation coming from the M-product corresponds to the Bethe-Salpeter equation.
The ehrenfests' wind-wood model in two dimensions
Authors: Haar D. Ter , Green C. D.
Published: November 2008

The consideration of the statistical aspect of the second law of thermodynamics of a previous paper is extended. The influence of fluctuations in the wind-wood model of the Ehrenfests is investigated, and the relative occurrence, mean life and time of recurrence of such fluctuations are calculated. These are found to agree with the results obtained for the simpler models considered previously.
The integration of hypergeometric functions
Authors: Slater L. J.
Published: November 2008

In this paper I consider the problem of integrating a generalized hypergeometric function with respect to one of its parameters, together with some general theorems on hyper-geometric transforms which arise out of this problem, and some interesting examples on these theorems.
Complex homogeneous linear forms
Authors: Rogers K.
Published: November 2008

LetZ, Q, Cdenote respectively the ring of rational integers, the field of rational numbers and the field of complex numbers. Minkowski (4) solved the problem of minimizing

forx, y ∈ Z(i)orZ(ρ), wherea, b, c, d ∈ Chave fixed determinant Δ ≠ 0. Here ρ = exp 2/3πi, andZ(i)andZ(p)are the rings of integers inQ(i)andQ(ρ)respectively. In fact he found the best possible results

forZ(i), and

forZ(ρ), where

while Buchner (1) used Minkowski's method to show that

forZ(i√2). Hlawka(3) has also proved (1·2), and Cassels, Ledermann and Mahler (2) have proved both (1·2) and (1·3). In a paper being prepared jointly by H. P. F. Swinnerton-Dyer and the author, general problems of the geometry of numbers in complex space are discussed and a systematic method given for solving the above problem for all complex quadratic fields Q(ϑ). Here, ϑ is a non-real number satisfying. an irreduc7ible quadratic equation with rational coefficients. The above problem is solved in detail forQ(i√5), for which

and the ‘critical forms’ can be reduced to
Scattering by a semi-infinite resistive strip of dominant-mode propagation in an infinite rectangular wave-guide
Authors: Papadopoulos V. M.
Published: November 2008

The scattering of the dominant transverse electric mode in an infinite perfectly conducting rectangular wave-guide by a semi-infinite resistive strip, centrally placed and parallel to the electric field, is calculated by the use of Laplace transforms. Formulae are derived for the amplitude of the scattered waves, and the numerical results, obtained using a digital computer, are given for various values of the surface resistivity of the strip.
On prime-power groups in which the derived group has two generators
Authors: Blackburn N.
Published: November 2008

Burnside ((1), p. 241) has proved the following theorem:

If G is a non-metabelian p-group, then the centre of the derived group of G cannot be cyclic. In particular, a non-Abelian group of order p3 cannot be the derived group of a p-group.
The asymptotic solution of differential equations with a turning point and singularities
Authors: Thorne R. C.
Published: November 2008

Asymptotic solutions of the differential equation

for large positive values of u, are examined; z is a complex variable in a domain Dz in which P1(z) and z2q(z) are regular and p1(z) does not vanish. In this paper it is shown that there exist Airy-type expansions of the solutions of this equation which are valid uniformly with respect to z in a domain in which z = 0 and z = z0 are interior points. If Dz is unbounded and the equation has a regular singularity at infinity, Airy-type expansions exist which are valid at z = 0, z = z0 and z = δ. If p(z) = constant + O (│z│-1) as │ z │ → ∞ in Dz, similar expansions also exist. The results given here are new.
Thin circular plates supported at several points along the boundary
Authors: Bassali W. A.
Published: November 2008

The deflexion at any point of a thin circular plate subjected to certain loadings along the circumference or over the area of an eccentric circle is obtained when the plate is supported at several points along the boundary, the supports being symmetrically situated with respect to the common diameter of the plate and circle.
A converse theorem on overconvergence of sequences of partial sums
Authors: Noble M. E.
Published: November 2008

A number of authors ((2), (4), (5)) have recently considered problems relating to the convergence of specified sequences of partial sums of a gap Taylor series on its circle of convergence. For convenience in stating such results we shall standardize the Taylor series as Σanzn with the unit circle as circle of convergence, and denote the partial sums and the sum function by sn(z) and f(z) respectively. Following the notation of ((4)) and (5) we shall define φ(x) for 0 ≤ x < ∞ to be the least concave majorant of log (|an| + 2). One of the recent results in this field ((5), Theorem 1, and see also (2)) may be stated as

THEOREM A. Suppose that f(z) is regular on an arc α < arg z < β. Suppose also that there are sequences of integers nk, Nk → ∞ such that Nk – nk → ∞ and am = 0 if nk < m < Nk.
The cooperative behaviour of a two-dimensional defect crystal
Authors: Elcock E. W.
Published: November 2008

The Ising model of a two-dimensional ferromagnetic crystal containing defects is considered. It is shown that the combinatorial method is particularly appropriate to the investigation of the two-dimensional defect crystal and how it may be readily modified to calculate the partition function for such a crystal. It is found that the cooperative behaviour of the infinite perfect crystal, as indicated by the occurrence of a logarithmic singularity in the specific heat as a function of temperature, is not destroyed in the defect crystal; the singularity persists, though at a progressively lower temperature as the density of defects is increased, until a high density of defects is reached. The value of this limiting density of defects, above which the cooperative behaviour is destroyed, is calculated.
A method for the numerical integration of the linear diffusion equation
Authors: Hartree D. R.
Published: November 2008

In one numerical method for integration of the diffusion equation in one dimension, the time derivative is replaced by a finite difference in a time interval, and the space derivative by the mean of its values at the beginning and end of the interval. This leads to a set of ordinary differential equations, one for each interval, which have to be solved in succession. Each of these equations is second-order with two-point boundary conditions; the process of integration from one end is severely unstable, the more so the smaller the tune interval. This paper is concerned with a practical, direct and stable method for solving them by integration of two first-order equations, one being integrated inwards and the other outwards, one boundary condition being satisfied in each integration. The extension to axially symmetrical diffusion is briefly considered.
The existence of Bartlett-Rajalakshman goodness of fitG-tests for multivariate autoregressive processes with finitely dependent residuals
Authors: Walker A. M. , Lindley D. V.
Published: November 2008

Bartlett and Rajalakshman (3) have derived two types of large sample goodness of fit tests for the hypothesis that a multivariate (or vector) time series {X(t)},t= 0, ± 1, ± 2, …, is generated by a linear autoregressive process. This may be defined as the stationary solution of an equation of the form

where theAiare square matrices such that the roots of the determinantal equation ‖ zp+A1zp−1+ … +Ap‖ = 0 have moduli less than unity, and {U(t)} is a sequence of independent random vector variables with a common distribution.
A sufficient condition for the differentiation of certain classes of set functions
Authors: Hayes Charles A.
Published: November 2008

We shall be concerned with the differentiation of an arbitrary set function ψ from certain classes of set functions with respect to another fixed set function φ. In (1), § 2·5, a halo evanescence condition is shown to be necessary and sufficient for the differentiation of a fixed measure ψ with respect to φ. Here we formulate a type of halo condition which ensures the differentiability of entire classes of set functions. Although the necessity of this condition is not shown, we demonstrate by means of an example that the results cannot be improved upon in certain respects.
On the decomposition of spaces in Cartesian products and unions
Authors: Ganea Tudor , Hilton Peter J.
Published: November 2008

The present paper is concerned with particular cases, obtained by suitably restricting the spaces involved, of the following general problem.

Given a topological space X, we ask whether there exist integers n ≥ 2 and non-contractible spaces X1, …, Xn such that X has the homotopy type of the Cartesian product X1, × … × Xn or of the union X1, v … v Xn.
Cumulants of some distributions arising from a two-state Markoff chain
Authors: Iyer P. V. Krishna , Shakuntala N. S.
Published: November 2008

The asymptotic values of the variances and covariances for the distribution of the transition numbersAA, AB, BAandBBfor a two-state Markoff chain have been given by Bartlett (l). Whittle (3) obtained the probability distribution and their factorial moments for thek-state chain under the restriction that the first and the last observations belong to therth and thesth state. But the expressions for the factorial moments according to Whittle himself are of little immediate use. He obtains useful results for the low order factorial moments by approximating in a grosser fashion, and the evaluation of the cumulants from these factorial moments is rather cumbersome. The object of this note is to give the first four cumulants and product cumulants which are exact to the ordernr(p11–p21)n−rwhereris the order of the cumulant, for the distribution of the transition numbers of a two-state Markoff chain. They have been calculated by using the method developed by Iyer and Kapur(2). This method can equally be used for evaluating the asymptotic values of the cumulants for thek-state also. This is done by differentiating directly the determinantal characteristic equation.
A lower bound for the critical probability in a certain percolation process
Authors: Harris T. E.
Published: November 2008

Consider a lattice L in the Cartesian plane consisting of all points (x, y) such that either x or y is an integer. Points with integer coordinates (positive, negative, or zero) are called vertices and the sides of the unit squares (including endpoints) are called links. Each link of L is assigned the designation active with probability p or passive with probability 1 − p, independently of all other links. To avoid trivial cases, we shall always assume 0 < p < 1. The lattice L, with the designations active or passive attached to the links, is called a random maze. A set of links is called connected if the points comprising the links (including endpoints) form a connected point set in the plane.
On the Equation ax − by = 1. II
Authors: Cassels J. W. S.
Published: November 2008

The following conjecture was apparently first enunciated by Catalan (3) in 1844 but has never been proved.
On the existence of periodic solutions of a certain third-order differential equation
Authors: Ezeilo J. O. C.
Published: November 2008

In this paper we shall be concerned with the differential equation

in which a and b are constants, p(t) is a continuous periodic function of t with a least period ω, and dots indicate differentiation with respect to t. The function h(x) is assumed continuous for all x considered, so that solutions of (1) exist satisfying any assigned initial conditions. In an earlier paper (2) explicit hypotheses on (1) were established, in the two distinct cases:

under which every solution x(t) of (1) satisfies

where t0 depends on the particular x chosen, and D is a constant depending only on a, b, h and p. These hypotheses are, in the case (2),

or, in the case (3),

In what follows here we shall refer to (2) and (H1) collectively as the (boundedness) hypotheses (BH1), and to (3) and (H2) as the hypotheses (BH2). Our object is to examine whether periodic solutions of (1) exist under the hypotheses (BH1), (BH2).
On the 1-dimensional category of Cartesian products
Authors: Berstein I.
Published: November 2008

The following example arose out of an unsuccessful attempt to improve a result by Ganea and Hilton (5), according to which

implies that for each field of coefficients F one at least of the factors Xi is acyclic; here cat X is the Lusternik-Schnirelmann category of X. It was natural to ask whether this assumption also implies that one at least of the Xi is simply connected. This could be proved if it were possible to obtain spaces of arbitrarily large 1-dimensional category ((3), (4)) by forming Cartesian products of sufficiently many 2-dimensional pseudoprojective spaces (1) corresponding to different primes. However, we prove that this is impossible:

Theorem. If P1, …, Pn are 2-dimensional pseudo-projective spaces corresponding to different primes p1, …, pn, then

Before proving this theorem, we recall the definitions involved. The 1-dimensional category cat1X of a space X is the least number of open sets covering X and such that each closed path in any of them is contractible in X. A 2-dimensional pseudo-projective space P corresponding to the prime p is the complex obtained from the disk |z| ≤ 1 by identifying the points ei[θ+(2mπ|p)], m = 0, 1, …, p − 1. We have (1)

for k ≥ 2 and cat1P = 3 since π1(P) is not free and cat1P ≤ 1 + dim P (4).
Stability of a Column of Rotating Viscous Liquid
Authors: Gillis J.
Published: November 2008

A long column of viscous liquid of radius a and uniform density p is rotating about its axis with angular velocity ω. It is shown that this motion is stable to plane perturbations of wave number 8 provided that the surface tension T satisfies . This critical value is higher than that required for stability of the similar motion of a nonviscous liquid, but is otherwise independent of the coefficient of viscosity. The rate of development of instability when T is less than the critical value is also studied. Some numerical results are given.

The condition has recently been obtained by Hocking (2) for the special cases of very high and very low Reynolds numbers.
The survival of a mutant under general conditions
Authors: Moran P. A. P.
Published: November 2008

In genetics it is of considerable interest to be able to calculate the probabilities that a mutant gene will either be lost to the population or that its allele will be lost, since, in general, if no further mutation occurs, one of these two events must ultimately take place in any finite population. A good deal of research has been done on this problem by Fisher (l) and Kimura (3) and the purpose of this paper is to extend their results to more general situations and to discuss the validity of their approximations.
Representations of the n-dimensional rotation group
Authors: Stone A. P.
Published: November 2008

The commutators of the infinitesimal operators of the n-dimensional rotation group Rn with vector operators under Rn are expressed in a vectorial notation. The infinitesimal operators for the representations (l 0…0) are treated in detail. Shift operators for l are constructed and are used to derive the branching rule for these representations. The energy levels and degeneracy of bound states of a particle under an inverse square force in n dimensions are found by wave mechanics and by expressing the Hamiltonian in terms of Casimir's operator for Rn+1. Differential operators which transform one radial wave function into another are obtained.
Simple reduction theorems for extension and torsion functors
Authors: Northcott D. G.
Published: November 2008

Let Λ be a commutative ring with an identity element and c an element of Λ which is not a zero divisor Denote by Ω the residue class ring Λ/Λc. If now M is a Λ-module for which c is not a zero divisor, and A is an Ω-module, then a theorem of Rees (2) asserts that, for every non-negative integer n, we have a Λ-isomorphism

This reduction theorem has found a number of useful and interesting applications.
On the connexion between Hausdorff measures and generalized capacity
Authors: Taylor S. J.
Published: November 2008

For any real function h(t) which is continuous and monotonic increasing for t > 0 with , Hausdorff (10) in 1918 denned a Carathéodory measure with respect to h(t) which has subsequently been known as Hausdorff measure. For analysing sets in Euclidean space, these measures have proved both useful and interesting. Given a real function Φ(t) which is continuous and monotonic decreasing for t > 0 with , Frostman(9) in 1935 denned capacity with respect to Φ(t). Lebesgue measure in Euclidean k-space is a special case of Hausdorff measure, and capacity with respect to Φ(t) becomes logarithmic capacity or Newtonian capacity in the cases , Φ(t)=1/t, respectively. The interrelationship between h-measure and Φ-capacity has been of interest in both directions: (i) in applications to function theory one may be able to determine whether or not a set has positive capacity by examining the h-measure for suitable h(t) (see, for example, (5)); (ii) it may be possible to determine the measure properties of a set from knowledge of its capacity (see, for example, (7) and (17)).
Equations for stochastic path integrals
Authors: Bartlett M. S.
Published: November 2008

Apart from the study of the integral

where {X(u)} is a stationary Gaussian process with autocorrelation function ρ(t), by Kac and Siegert(1), most stochastic functionals of the general type

have been considered for {X(u)} either additive or Markovian (see, for example, (2), (3)), and in the Markovian case only for diffusion-type processes (Darling and Siegert (4)). More general approaches exist (e.g. (5), (6)), but seem less concerned with the investigation of specific problems. Some preliminary remarks here are therefore aimed at examining the structure of integrals of type (2), or such further extensions of the formal Riemann sum type

that would be expected to have well-behaved distributional properties for {X(t)} Markovian, and associated equations for studying these properties. As an example, the sum

say, is considered (i) for the normal linear Markovian process, (ii) for simple birth-and-death and emigration—immigration processes.
Expansions of the Poiseuille and the irregular Coulomb functions
Authors: Meligy A. S.
Published: November 2008

An expansion of the Whittaker function Mk, m(z) in terms of functions of the same kind is obtained. It generalizes previous results and is used to derive expansions of the second Whittaker function Wk,m (z) in terms of the M functions for the cases m = 0 and ½. The case m = 0 provides expansions for the Poiseuille and the exponential integral functions. The case m = ½ provides an expansion for the irregular Coulomb wave function having angular momentum zero in terms of the regular Coulomb functions.
Two inequalities for parabolic cylinder functions
Authors: Olver F. W. J.
Published: November 2008

In this paper upper bounds are established for the principal solution of the differential equation

and its derivative, for unrestricted values of the complex variable t and the complex parameter μ. The results may have little interest in their own right, but they are of great value in developing the asymptotic theory of linear second-order differential equations in a domain containing two turning points. Equation (1·1) is the simplest example of a differential equation of this type.
Correlation problems in evolutionary stochastic processess
Authors: Ramakrishnan Alladi , Radha T. K.
Published: November 2008

In a previous contribution to theseProceedings(Ramakrishnan(1)) the concept of product density was introduced to describe the statistical distribution of a discrete number of particles in a continuous spaceE, corresponding to a single pointt, wheretis the parameter with respect to which the stochastic process evolves. This is extended to densities corresponding to n points on thetaxis and correlation problems associated with these density functions are studied with particular reference to electron-photon cascades.
On queues in which customers are served in random order
Authors: Kingman J. F. C.
Published: November 2008

The queue M |G| l is considered in the case in which customers are served in random order. A formal solution is obtained for the waiting time distribution in the steady state, and is used to consider the exponential decay of the distribution. The moments of the waiting time are examined, and the asymptotic form of the distribution in heavy traffic is found. Finally, the problem is related to those of the busy period and the approach to the steady state.
On rearrangement of series
Authors: Diananda P. H.
Published: November 2008

Throughout this note the series is a rearrangement (in the order of its terms) of the series of complex terms a1 + a2 + ….
Generalization of the Fundamental Theorem on Subadditive Functions
Authors: Hammersley J. M.
Published: November 2008

Statement of results. Let g(x) be a given real finite function of a real variable x. Let F(g) denote the class of all real Lebesgue-measurable and finite (though possibly unbounded) functions f(x) which satisfy where η may depend upon f Let F* denote the class of all real f(x) such that exists and satisfies − ∞ ≤l<+∞.
Asymptotic Cumulants for Distributions ofk-State Markov Chains
Authors: Krishna Iyer P. V. , Shakuntala N. S.
Published: November 2008

The expectation, variance and covariance for different states of ak-state Markov chain have been given by Patankar (6), Whittle (7), Good (3) and Bhat (l). Patankar's results involve theklatent roots of the determinantal equation. As it is not easy to determine the latent roots whenk> 2, the actual asymptotic values of variances and covariances cannot be readily evaluated. Whittle gives exact probability distributions for the transitions, but the moments have been obtained after some gross approximations. Good (3) and Bhat(l) have given the first two moments and product moment for the frequency of different states. By using certain methods developed by Iyer and Kapur(4), the first four cumulants and product cumulants for the transition numbers of a two-state Markov chain were calculated and presented in an earlier publication(5).
On the Number of Real Roots of a Random Algebraic Equation
Authors: Samal G.
Published: November 2008

Let Nn be the number of real roots of a random algebraic equation The coefficients ξν are independent random variables identically distributed with expectation zero; the variance and third absolute moment are finite and non-zero. It is proved that

where εν tends to zero, but εν log n tends to infinity. The measure of the exceptional set tends to zero as n tends to infinity.
Concerning nilpotent wreath products
Authors: Liebeck Hans
Published: November 2008

In a recent paper, (2), Baumslag showed that a wreath product of A by B is nilpotent if and only if A and B are nilpotent p-groups for the same prime p, with A of finite exponent and B finite. We shall calculate the (nilpotency) class of such groups when A and B are Abelian. This provides a lower bound for the class in the general case. We give a simple construction for a set of non-nilpotent metabelian groups which satisfy a finite Engel condition. With Engel class defined by Definition 6·1, we show that there are nilpotent groups of arbitrarily large nilpotency class for which the nilpotency class is equal to the Engel class.
The stability of a queue with non-independent inter-arrival and service times
Authors: Loynes R. M.
Published: November 2008

Introduction

Here we shall mention only the results referring to stability. The definitions of the various quantities Tn, Sn, SNn, and the basic hypotheses made concerning their structure will be found in §§ 2·1, 3·1 or 4·1. For convenience we shall introduce some further terminology in this section. The single-server queues {SNn, Tn} arising in connexion with queues in series will be called the component queues, and the queue {Sn, sTn} implicit in the discussion of many-server queues will be called the consolidated queue. We have already in § 2.33 called the single-server queue {Sn, Tn} critical if E(S0-T0) = 0. We shall now call it subcritical if E(S0 − To) > 0 and supercritical if E(S0 − T0) < 0. A system of queues in series is subcritical if each component queue is subcritical, critical if (at least) one component queue is critical and the rest are subcritical, and supercritical if (at least) one component queue is supercritical. A many-server queue will be described in these terms according to the character of its consolidated queue. Finally, a single-server queue {Sn, Tn} will be said to be of type M if it has the property considered in Corollary 1 to Theorem 5: the sequences {Sn} and {Tn} are independent of each other, and one is composed of mutually independent non-constant random variables.

Single-server queues:

(i) Subcritical: stable (Theorem 3).

(ii) Supercritical: unstable (Theorem 2).

(iii) Critical: stable, properly substable, or unstable (examples in §2·33, including one due to Lindley); unstable if type M (Theorem 5, Corollary 1).

Queues in series:

(i) Subcritical: stable (Theorem 7).

(ii) Supercritical: unstable (Theorem 7).

(iii) Critical: stable, properly substable, or unstable, if the component queues are substable (examples in § 3·2); unstable if any component queue is unstable (Theorem 7), and in particular if any critical component queue is of type M (Theorem 7, Corollary).

Many-server queues:

(i) Subcritical: stable or properly substable (Theorem 8, and example in § 4·3).

(ii) Supercritical: unstable (Theorem 8).

(iii) Critical: stable, properly substable, or unstable, if consolidated queue is substable (examples in § 4·3); unstable if consolidated queue unstable (Theorem 8), and in particular if this is of type M (Theorem 8, Corollary).

From Lemma 1 it follows that none of these queues can be properly substable if all the servers are initially unoccupied.
A storage model with continuous infinitely divisible inputs
Authors: Gani† J. , Prabhu N. U.
Published: November 2008

This paper presents a general theory of storage for dams subject to a steady release, and with continuous inputs having infinitely divisible distributions. The paper extends some earlier work by Downton, Lindley, Smith and Kendall ((10), see discussion), and gives a detailed account of results previously sketched by the authors (6).
Isometric flows in Hilbert space
Authors: Sz.-Nagy Béla
Published: November 2008

1. Let {Vi}i≥0 be a weakly (hence also strongly) continuous semigroup of (linear) contraction operators on a Hilbert spaceH, i.e. |Vt| ≤ 1 (t≥ 0). LetZandWdenote the corresponding infinitesimal generator and cogenerator, i.e.

Zis in general non-bounded, but closed and densely defined, andWis a contraction operator (everywhere defined inH), such that 1 is not a proper value ofW. Conversely, every contraction operatorWnot having the proper value 1 is the infinitesimal cogenerator of exactly one semigroup {Vi} of the above type; one has namely

in the sense of the functional calculus for contraction operators (4).
A note on blocks of modular representations
Authors: Srinivasan Bhama
Published: November 2008

1. The notation used will be that of (2).

G is a finite group,

is a ring which is either

(1·1) a field of characteristic p, or

(1·2) the ring of valuation integers of a field of characteristic 0 with a complete discrete valuation. In this case, is the maximal ideal of .
Diffraction of sound pulses by a fluid cylinder
Authors: Mishra S. K.
Published: November 2008

In this paper, we consider the problem of diffraction of two-dimensional sound pulses by a homogeneous fluid circular cylinder contained in another homogeneous fluid. The line source is situated outside the cylinder and is parallel to its axis. It is supposed that the velocity of sound inside the cylinder is less than the velocity of sound in the surrounding medium. We investigate the problem by the method of dual transformation as developed by Friedlander. The pulse propagation modes both inside and outside the cylinder are obtained, We interpret the modes as diffracted pulses in terms of Keller's Geometrical Theory of Diffraction. The results agree with Friedlander's conjecture.
Imbedding and isotopy of spheres in manifolds
Authors: Levine J.
Published: November 2008

1. The following results are special cases of theorems of Irwin and Zeeman, respectively (see (4), (9), (10)).

(a) Let V be a (2n − m + 1)−connected piecewise-linear m−manifold (bounded or not), where m − n ≥ 3. Then any element of πn(V) can be represented by a piecewise-linear imbedding of Sn in V.

(b) Let M be a (2n − m + l)-connected closed piecewise-linear (m − 1)-manifold, where m − n ≥ 3. Then two piecewise linear imbeddings of Sn−1 in V are isotopic if and only if they are homotopic.
On the one-centre expansion of Meijer' G-function
Authors: Edelstein L. A.
Published: November 2008

1. Introduction and summary. In many problems in theoretical physics one encounters situations in which it is desirable to expand an orbital centred at one point about another point.
Conditions equivalent to central symmetry of convex curves
Authors: Hammer P. C. , Smith T. Jefferson
Published: November 2008

In this paper we establish that a convex planar body C is centrally symmetric provided either one of the following conditions hold:

(1) Each line halving the circumference of the boundary γ of C is a diametral line.(A diametral line is a line intersecting C in a chord of maximal length in the family of parallel chords.)

(2) Each line halving the area of C is a diametral line.
The spectra of Fredholm operators in locally convex spaces
Authors: Olagunju P. A. , West T. T.
Published: November 2008

1.Notation and definitions. In this paper necessary and sufficient conditions are found for the spectrum of a Fredholm operator in a locally convex space (always taken to be Hausdorff) to lie on the non-negative real axis of the complex plane. Some results of Grothendieck(2) allow us to obtain the results in this general form; an interesting special case is the trace-class of operators in a general Banach space. We also deal with the case of Hilbert–Schmidt operators in a Hilbert space.
Cobordism of combinatorialn– manifolds forn≤ 8
Authors: Wall C. T. C.
Published: November 2008

The object of this paper is two-fold: first to collect together the known facts about combinatorial cobordism in general, and then to calculate the groups for the first 8 dimensions. As in (29), we shall denote the unoriented and oriented cobordism groups in dimensionnbyand Ωn, and will distinguish the combinatorial from the differential case by affixesc, d.
On the least quadratic non-residue
Authors: Godwin H. J.
Published: November 2008

If n(p) is the least quadratic non-residue of a given prime p then it is known ((1)) that n(p) = O(pα) for any α > 1/(4 √e). LeVeque ((2), page 122) gives the following bound with an explicit constant: n( p) < √ p for p ≠ 2, 3, 7, 23. In the present paper an elementary and self-contained proof is given of a result slightly stronger than LeVeque's. Some numerical results, which indicate the extent to which the result proved falls short of what actually obtains, are appended.
Abelian curves of 2-power conductor
Authors: Ogg A. P.
Published: November 2008

Let Q be the field of rational numbers, and let A be an Abelian curve (an Abelian variety of dimension one) defined over Q. Following Weil, the conductor of A is

where p runs over all primes, and the exponent is explained below; the primes dividing N are exactly those for which A has degenerate reduction modulo p.
On tubular neighbourhoods of manifolds. I
Authors: Hirsch Morris W.
Published: November 2008

Introduction. LetXbe a submanifold ofY, in either the topological, smooth, or piecewise linear ( =PL) categories. Anormal cell bundleonXinYis a bundle ξ = (p, E, X) in the category whose fibre is a closed cell, and such thatEis a neighbourhood ofXinYandp: E → Xis a retraction. The triple (Y, X, ξ) is atubular neighbourhood, or briefly, atube. For convenience we may refer to a tube by its cell bundle.
The commuting inverses of a square matrix
Authors: Englefield M. J.
Published: November 2008

An inverse AI for an arbitrary matrix A was first given by Moore (4). Since the application to solution of linear equations only depended on the property A AI A = A, Bjerhammar (2) used this equation to define the set of generalized inverses AI. If A is regular, then only the regular inverse A−1 satisfies this definition. If A is a generalized inverse of AI, so that AI = AI AAI, then AI is a reciprocal inverse.
The Evaluation of Eigenvalues of a Differential Equation Arising in a Problem in Genetics
Authors: Miller G. F. , Goodwin E. T.
Published: November 2008

This paper concerns the determination of the smallest eigenvalue of a second order differential equation containing two parameters which arises in problems concerning genic selection under random drift in a population of finite size. A table of values is given, the method of computation is described, and the asymptotic behaviour for large values of one of the parameters is investigated.
On absolute summability factors of infinite series and their application to Fourier series
Authors: Mohapatra R. N. , Das G. , Srivastava V. P.
Published: November 2008

Definition. Let {sn} be the n-th partial sum of a given infinite series. If the transformation

where

is a sequence of bounded variation, we say that εanis summable |C, α|.
A method for improving iteration procedures
Authors: Manning Irwin
Published: November 2008

A technique is developed for improving the speed and range of convergence of iteration procedures. The method is shown to include, as special cases, Newton's method and Aitken's δ2-acceleration process.
Some bilinear generating functions for Jacobi polynomials
Authors: Manocha H. L.
Published: November 2008

In a previous paper (3) the writer has proved

where is a Jacobi polynomial defined as ((4) p.255)
A short proof of the existence of additive inverses of topological microbundles
Authors: West R. W.
Published: November 2008

In this note we prove the following result due to Milnor ((5), Theorem 4·1).
On a conjecture of Hughes
Authors: Cofman Judita
Published: November 2008

D. R. Hughes stated the following conjecture: If π is a finite projective plane satisfying the condition: (C)π contains a collineation group δ inducing a doubly transitive permutation group δ* on the points of a line g, fixed under δ, then the corresponding affine plane πg is a translation plane.
On the almost everywhere convergence of Fourier series
Authors: Yadav B. S.
Published: November 2008

Let f be a 2π-periodic function of the class L(−π,π). Put

We call, with Žuk(6), the quantity L(p)(h, f) the L-modulus of smoothness of order p of the function f. Žuk has recently obtained, in (5) and (6), generalizations of a number of classical results on the absolute convergence of Fourier series, as also on the order of Fourier coefficients by employing the concept of the L-modulus of smoothness which is obviously a more general concept than that of the modulus of continuity. It is the purpose of this note to prove a theorem on the almost everywhere convergence of Fourier series of f involving the concept of L(p)(h, f).
Mixed boundary-value problems for an elastic half-space
Authors: Keer L. M.
Published: November 2008

In this paper the mixed problem for an isotropic, elastic half-space is considered. Boundary conditions are prescribed interior and exterior to a circular region of unit radius, and the state of stress is assumed to be axially symmetric. Several authors have treated this problem. Mossakovskii(1) considered a punch adhering to and indenting an elastic half-space. Has solution was obtained by introducing certain operators that transformed the half-space problem into a problem in plane potential theory. The method of linear relationship was used to solve this auxiliary problem and inverse operators returned the plane to the half-space. The general case of a circular, rigid punch adhering to a half-space was treated by Ufliand (2,3) and a solution was obtained through the use of toroidal coordinates and the Mehler-Fok integral transforms.
Cohomology operations and duality
Authors: Maunder C. R. F.
Published: November 2008

Since Thom first introduced the notion of the ‘dual’ of a Steenrod square, in (12), it has become apparent that calculation with such duals in the cohomology of, say, a simplicial complex X will often yield information about the impossibility of embedding X in Sn, for certain values of n. For example, the celebrated theorem that cannot be embedded in can easily be proved in this way. In this paper, we seek to generalize this method to any pair of extraordinary cohomology theories h* and k*, and natural stable cohomology operation θ: h* → k*. We show in section 3 that a simplicial embeddingf: X → Sn gives rise via the Alexander duality isomorphism to a homology operation

which is independent of n, the particular embedding f, and even the particular triangulations of X and Sn. If h* and k* are multiplicative cohomology theories, there are Kronecker products

if h0(S0) = k0(S0) = G, a field, and the Kronecker products make h*, h* and k*, k* into dual vector spaces over G, then can be turned into a cohomology operation c(θ): k*(X)→h*(X), by using this duality. This is certainly true if h* = k* = H*(;Zp), p prime, and in this case we have the original situation considered by Thom, who showed, for example, that
On the right lower circular density of two-dimensional Jordan curves in the plane
Authors: Larman D. G.
Published: November 2008

We use J(a, b) to denote a Jordan curve of positive two-dimensional measure in the plane, with end-points a and b. If υ is a point of J(a, b), we define the right lower arc density at υ by

where J( υ, υ′) is the largest arc, whose left-end point is υ, which is contained in the disc c(υ, r).
The weak convergence of the empirical process with random sample size
Authors: Pyke Ronald
Published: November 2008

In many applied probability models, one is concerned with a sequence {Xn: n > 1} of independent random variables (r.v.'s) with a common distribution function (d.f.), F say. When making statistical inferences within such a model, one frequently must do so on the basis of observations X1, X2,…, XN where the sample size N is a r.v. For example, N might be the number of observations that it was possible to take within a given period of time or within a fixed cost of experimentation. In cases such as these it is not uncommon for statisticians to use fixed-sample-size techniques, even though the random sample size, N, is not independent of the sample. It is therefore important to investigate the operating characteristics of these techniques under random sample sizes. Much work has been done since 1952 on this problem for techniques based on the sum, X1 + … + XN (see, for example, the references in (3)). Also, for techniques based on max(X1, X2, …, XN), results have been obtained independently by Barndorff-Nielsen(2) and Lamperti(9).
A generalization of the chromatic number
Authors: Chartrand G. , Geller D. P. , Hedetniemi S.
Published: November 2008

One of the most widely studied concepts in graph theory is that of the colouring of graphs. Much of the research in this area has dealt with the chromatic number χ(G) of a graph G, defined as the minimum number of colours which can be assigned to the points of G so that adjacent points are coloured differently. One can obtain a natural generalization of this concept as follows: we define the n-chromatic number χn (G) to be the smallest number of colours which one can associate with the points of G so that not all points on any path of length n are coloured the same. The 1-chromatic number is then simply the chromatic number.
A note on the construction of confidence intervals for the coefficients of a second canonical variable
Authors: Radcliffe J.
Published: November 2008

The goodness of fit of hypothetical discriminant functions has been discussed by Bartlett(1), Kshirsagar(2, 3), Radcliffe(4) and Williams (5–8). The associated problem of obtaining confidence intervals for the coefficients of a single discriminant function or canonical variable has been dealt with by Bartlett (1).
Arithmetic on curves with complex multiplication by √ − 2
Authors: Rajwade A. R.
Published: November 2008

This paper is a contribution to the verification of conjectures of Birch and Swinnerton-Dyer about elliptic curves (1). The evidence that they produce is largely derived from curves with complex multiplication byi. It is natural to consider other kinds of complex multiplications and here we shall make a start on the case when the ring of complex multiplications is isomorphic to the ring Z[σ], where σ2= − 2.
Finite time stability and comparison principles
Authors: Kayande A. A. , Wong J. S. W.
Published: November 2008

Motivated by discussion on practical stability in LaSalle and Lefschetz (3), Weiss and Infante (5), have discussed various notions of stability over finite time interval of a given differential system. This theory of stability differs from the usual stability theory mainly by the preassigned limits to which any given solution of the differential system must adhere. Sufficient conditions for these notions of stability are given in (5) in terms of certain Lyapunov-like functions satisfying some appropriate differential inequalities. The purpose of this article is to introduce some complementary notions of finite time stability and weaken the conditions on the differential inequalities involving Lyapunov-like functions by the use of comparison principles.
The influence of surface tension on the reflection of water waves by a plane vertical barrier
Authors: Evans D. V.
Published: November 2008

In this paper the effect of surface tension is included in a well-known problem in the theory of two-dimensional infinitesimal water waves. The problem is that of the reflection of waves from a fixed vertical barrier immersed to a depth a into deep water. It is shown how the solution for the velocity potential may be determined uniquely when simple assumptions are made concerning the behaviour of the free surface near the barrier. In particular, expressions are derived for the reflection coefficient, defined as the ratio of the amplitude of the reflected wave to that of the incident wave, at infinity, and the transmission coefficient, defined similarly. It is shown how these coefficients, for small values of the surface tension force, tend to the values obtained by Ursell (4) when surface tension is ignored. The related problem of a completely immersed vertical barrier extending to a distance a from the surface may be solved in a similar manner. Expressions for the reflection and transmission coefficients for this case are given.
Linear operators with closed range
Authors: Webb J. H.
Published: November 2008

This paper continues the investigation begun in (6), and uses the definitions and notation of that paper.
On the scattering of short surface waves by a finite dock
Authors: Leppington F. G.
Published: November 2008

A sinusoidal travelling wave-train is at normal incidence upon a two-dimensional finite dock fixed on the surface of a body of water of great depth, and the problem investigated herein is that of finding the limiting form of the induced velocity potential for short waves. Of particular interest are the amplitudes of the wave-trains reflected and transmitted towards infinity by such an obstacle. The potential is expressed as a sum of coupled semi-infinite dock potentials, whence results a pair of weakly coupled integral equations for the solution. This formulation of the problem is shown to be amenable to an approximate solution for large wave-numbers, and the first few terms are derived in formal expansions for the reflection and transmission coefficients.
Arithmetic on curves with complex multiplication by the Eisenstein integers
Authors: Rajwade A. R.
Published: November 2008

This paper is a contribution to the verification of conjectures of Birch and Swinnerton-Dyer about elliptic curves (1). The evidence that they produce is largely derived from curves with complex multiplication byi.In a previous paper (8), we had considered curves with complex multiplication by √ − 2. Here we shall look at the case when the ring of complex multiplications is isomorphic to the ring Z[ω], where ω3= 1, ω ≠ 1.
A uniqueness theorem for magnetohydrodynamic duct flows
Authors: Hunt J. C. R.
Published: November 2008

We analyse the steady, fully developed, laminar flows of uniform, electrically conducting, incompressible fluids along arbitrarily shaped ducts of constant cross-section under the action of transverse magnetic fields to show that these flows are uniquely specified by certain parameters, namely: either the pressure drop along the duct per unit length or the volume flow rate and either the electric current leaving the walls of the duct at each point where the duct is connected to an external electrical circuit or the electric potential at these points. The most important, and novel, feature of the analysis is the proof that no secondary flow can exist in these fully developed flows. The analysis includes the special cases where the walls of the duct are non-conducting, and where the fluid is non-conducting (e.g. Poiseuille flow); in the latter case the uniqueness of the flow has not previously been demonstrated, at least in print.
An integral involving the product of a G-function and a generalized hypergeometric function
Authors: Chhabra S. P. , Singh F.
Published: November 2008

1. In this note a definite integral involving the product of Meijer's G-function and a generalized hypergeometric function is evaluated with the help of finite difference operators E. As the generalized hypergeometric function and Meijer's G-function may be converted into a number of polynomials and higher transcendental functions, the integral leads to a generalization of many results. The results given by Agrawal ((l), integral 4), Sharma ((7), integral 1) in these proceedings, Bhonsle ((2), p. 188) and Saxena ((6), p. 198) follow as particular cases of our formula. In what follows v, μ, m and h are positive integers and either of v and μ may be zero. The symbol Δ(m, a) denotes a set of m parameters a/m, (a + 1 )/m, …, (a + m − 1 )/m.