bluuebunny/crossref_metadata_2025 huggingface dataset:
Homeomorphisms of non-orientable two-manifolds
Authors: Lickorish W. B. R.
Published: November 2008

In an earlier paper (1) the idea of a c-homeomorphism on a 2-manifold was introduced; it can be described in the following way. Let the 2-manifold M contain an annulus, A, one of the boundary components of which is a simple closed curve c. There is a homeomorphism of A to itself, fixed on the boundary of A, which sends radial arcs onto arcs which spiral once around A (see Fig. 1). This can be extended to

a homeomorphism of M to itself, by the identity on M — A. Intuitively this homeomorphism can be thought of as the process of cutting M along c, twisting one of the now free ends, and then glueing together again. The c-homeomorphism has, in fact, only been defined up to isotopy, but as we shall only be considering homeomorphisms up to isotopy, this is sufficient. It was shown in (1) that any orientation preserving homeomorphism of a closed orientable 2-manifold can be expressed as a product of c-homeomorphisms. In that case, a c-homeomorphism could be associated with any simple closed path on such a manifold, but on a non-orientable 2-manifold a c-homeomorphism can only be associated with a simple closed orientation preserving path. It is this fact that invalidates many of the lemmas of (1) if they are regarded as applying to the non-orientable case. The purpose of this paper is to show that (Theorem 1) c-homeomorphisms do not generate the group of all homeomorphisms of a closed nonorientable 2-manifold, but that if another elementary type of homeomorphism, the Y-homeomorphism, is introduced, then c- and Y-homeomorphisms together do generate this group (Theorem 2). This result leads to a discussion of non-orientable 3-manifolds, to a result analogous to that obtained for orientable 3-manifolds in (1), and to a new proof of the fact that any 3-manifold bounds a 4-manifold. Throughout this paper all manifolds will be considered as possessing an underlying combinatorial structure, all paths will be polygonal with respect to some subdivision of the manifold containing them, and all homeomorphisms and isotopies will be piecewise linear.
The convergence of a sum of independent random variables
Authors: De Barra G. , Slater N. B.
Published: November 2008

Let Xν, ν= l, 2, …, n be n independent random variables in k-dimensional (real) Euclidean space Rk, which have, for each ν, finite fourth moments β4ii = l,…, k. In the case when the Xν are identically distributed, have zero means, and unit covariance matrices, Esseen(1) has discussed the rate of convergence of the distribution of the sums

If denotes the projection of on the ith coordinate axis, Esseen proves that if

and ψ(a) denotes the corresponding normal (radial) distribution function of the same first and second moments as μn(a), then

where and C is a constant depending only on k. (C, without a subscript, will denote everywhere a constant depending only on k.)
On the representation and enumeration of trees
Authors: Glicksman Stephen
Published: November 2008

Scoins(1) has shown that if Π1 = {(1), …, (n)} and Π2 = {(n + 1), …, (n + m)} are two sets of points, there are exactly mn−1nm−1 trees of alternate parity connecting the points of Π1 ∪ Π2, where each tree consists of n + m − 1 segments and each segment joins a point of Π1 to a point of Π2. Another proof based on the three following results is given here.
A combinatorial problem in Abelian groups
Authors: Rogers Kenneth
Published: November 2008

Let α be a prime element of the ring of integers of an algebraic number field, R. Mr C. Sudler verbally raised the question as to how many prime ideal factors α can have. This is equivalent to a problem on the group of ideal classes of R, as we now show. If there is a prime α = p1p2 … Pk, where the prime ideal pi, is in the ideal class Pi, then P1P2…Pk equals the identity class, but no subproduct has this property. The converse holds since every ideal class contains prime ideals. So the problem is equivalent to one on finite Abelian groups, which we now write additively.
On the motion of a sphere through a conducting fluid in the presence of a magnetic field
Authors: Ludford G. S. S. , Singh M. P.
Published: November 2008

Errors in a previous discussion (4) of the title problem for a perfectly conducting sphere are corrected. An alternative method of determining the ultimate motion is described for both a perfectly conducting and a non-conducting sphere moving along the magnetic field.
The isotropic scattering of a concentrated ray pencil from a point source
Authors: Smith M. G.
Published: November 2008

The two-sided Laplace transform is used to obtain a solution of a certain integral equation. The equation is that associated with the transport equation for isotropic scattering, when there is a point source of radiation giving a concentrated ray pencil, in a spherically symmetric homogeneous medium extending to infinity. The solution is then used to find the asymptotic form of the intensity in the forward and backward directions.
Curious behaviour of a proton magnetometer
Authors: Bullard E. C. , Mason C. S. , Mudie J. D.
Published: November 2008

1. Introduction. This Department has made extensive observations with a proton magnetometer towed behind a ship (Hill (2)). The magnetometer consists of a bottle of water surrounded by a coil of wire through which a current flows and produces a polarizing field of about 100 oersted. A small proportion of the protons in the water are polarized along the resultant field. When the current is turned off they precess about the Earth's field and produce in the coil an e.m.f. whose frequency is a measure of the field.
On the Schur product ofH-matrices and non-negative matrices, and related inequalities
Authors: Lynn M. S.
Published: November 2008

1.Introduction. Let ℛndenote the set of alln×nmatrices with real elements, and letdenote the subset of ℛnconsisting of all real,n×n, symmetric positive-definite matrices. We shall use the notationto denote that minor of the matrixA= (aij) ∈ ℛnwhich is the determinant of the matrix

TheSchur Product(Schur (14)) of two matricesA, B∈ ℛnis denned by

whereA= (aij),B= (bij),C= (cij) and

Let ϕ be the mapping of ℛninto the real line defined by

for allA∈ ℛn, where, as in the sequel,.
Quantum theory of unstable particles
Authors: Martin J. L.
Published: November 2008

A quantum model of the scattering of a particle via absorption and re-emission at the origin is described. The model is capable of exhibiting an arbitrary number of resonances, and the associated unstable states are discussed. The emphasis is on obtaining exact results. It is found that the exponential decay law is of limited application, and that in all cases of non-zero coupling the decay goes ultimately as t−3. Finally, it is pointed out that the treatment of unstable states in this type of model depends only on a knowledge of the phase shift as a function of radial momentum, and a possible phenomenological method is suggested.
A note on the geometrical optics of diffraction by an interface
Authors: Grimshaw R. H. J.
Published: November 2008

1. It is well known that solutions of the Cauchy problem for the wave equation represent disturbances obeying the laws of geometrical optics. Specifically a solution ψ of the wave equation

for which ψ = δψ/δt = 0 initially outside a surface C0, vanishes at time t in the exterior of a surface Ct parallel to and at a normal distance ct from C0 (see e.g. (l), page 643). Analogous results hold for the solutions of any linear hyperbolic second-order partial differential equation with boundary-value conditions of the Cauchy type. Boundary conditions of the type representing reflexion have been treated by Friedlander(2). He showed that as well as the incident and reflected wavefronts, there sometimes exists a ‘shadow’ where diffraction occurs, and that the diffracted wave fronts are normal to the reflecting surface, the corresponding rays travelling along the surface and leaving it tangentially. The purpose of this paper is to extend these results to refraction, where instead of a purely reflecting surface we have an interface between two different homogeneous media.
Mrs Perkins's quilt
Authors: Trustrum G. B.
Published: November 2008

In a recent paper of the same title (2), Conway considered the problem of dividing a square of side N, into squares of integral side. The case when we demand that all these squares are of unequal size has been considered in some detail by various authors ((l),(4),(6),(7)). Conway considered the case with the condition relaxed. Dudeney ((3), Problem 173) has considered the same problem for the special case N = 13; hence the title of this paper.
A note on B*-algebras
Authors: Allan G. R.
Published: November 2008

A normed algebra A is a pre-B*-algebra if its norm satisfies ∥x*x∥= ∥x∥2 for all elements x ∈ A; if A is also complete in its norm, then A is a B*-algebra (see (l), page 180). In the study of certain locally convex algebras, the problem arose of expressing the condition that an algebra be a pre-B*-algebra in terms of its properties as a locally convex algebra, rather than in terms of the norm. A solution to this problem is presented in this note; the application to the theory of locally convex algebras will appear elsewhere.
A lemma in homological algebra
Authors: Kelly G. M.
Published: November 2008

In the development of homological algebra, one has to prove at some point that, in defining the derived functors of ⊗ and of Hom, it makes no difference whether we resolve both variables or only one of them. Taking ⊗ ( = ⊗R) as a typical example, what has to be proved is

(A) If the complex F is projective, or even flat, as a right R-module, and if f: P → A is a projective resolution of the left R-module A, then

is an isomorphism.
On cardinal numbers associated with locally compact Abelian groups
Authors: Reade J. B.
Published: November 2008

We are concerned in this work with the following question:

Suppose that i is a continuous algebraic isomorphism from the topological group H onto a subgroup of the topological group G and suppose that the image i(H) is not closed in G; then what can we say about the cardinal number

We observe two easy results.
Some fields associated with a moving point, and weak solutions of the wave equation
Authors: Ward G. N.
Published: November 2008

If P is a point in space-time, and P′ is a point whose space coordinates are continuous functions of time and whose speed in space is always less than the speed of light, then the time, T, at which the square of the interval between P and P′ is R, is a two-valued real function of R and the space-time coordinates of P for a range of R that includes R = 0. For fixed R in this range, each branch of T defines a scalar field over the whole of space-time, and, as R → 0, these fields reduce to the retarded and advanced times of P′ at P. Any function of T is also a field that reduces to a retarded or advanced field as R → 0. An interesting property of these fields is that there are identical relations between their derivatives with respect to R and the space-time variables, which enable the space-time derivatives to be expressed in terms of the R-derivatives; moreover, these relations are linear in the derivatives and take the same form for every such field.
On a functional equation
Authors: Rajagopal C. T. , Reddy A. R.
Published: November 2008

Introduction. The object of this note is to give a proof of the following theorem on the solution of an important functional equation.
On the differentials in the Adams spectral sequence for the stable homotopy groups of spheres. II
Authors: Maunder C. R. F.
Published: November 2008

In (7), we investigated some elements in the Adams spectral sequence for the stable homotopy groups of spheres, and proved that they were never boundaries, for any differential. This paper extends and generalizes these results: we consider more elements than in (7), and also prove that many of them do not survive to the E∞ term, so that they fail to be cycles for some dr, and this differential is therefore non-zero.
Notes on general topology
Authors: Ward A. J.
Published: November 2008

There is a close parallelism between the theories of convergence of directed nets and of filters, in which ‘subnet’ corresponds, in general, to ‘refinement’. With the standard definitions, however (1), pages 65 et seq., this correspondence is not exact, as there is no coarsest net converging to x0 of which all other nets with the same limit are subnets. (Suppose, for example, that a net X = {xj,: j ∈ J} in R1 has both the sequence-net S = {n−1; n = 1, 2,…} and the singleton-net {0} as subnets. Then (with an obvious notation), there exist

such that j0 ≥ jn for all n, while jn ≥ j0 for all n ≥ n0 say. But, given any j ∈ J, there exists n with jn ≥ j: it follows that jn ∈ j for all n ≥ n0 (independent of j); thus X cannot converge to 0. Even if nets with a last member are excluded, a similar result can be obtained by considering the net Y = {yθ; Θ an ordinal less than ω1}, where yθ = 0 for all Θ. If X has both Y and S as subnets we can show that (with a similar notation) there exists Θ0 such that Θ ≥ Θ0 implies jθ ≥ all jn, but also n0 such that n ≥ n0 implies ; the rest is as before.) Moreover, the theory of convergence classes, (l), pages 73 et seq., contains a condition (Kelley's condition (c)) whose analogue need not be separately stated for filters. These differences can be removed by adopting a wider definition of subnet, a course which does not seem unnatural, inasmuch as the standard definition is already wider than the ‘obvious’ one, and our proposed definition is equivalent to the standard one in the special case of sequences.
Bounds for the characteristic exponents of linear systems
Authors: Smith R. A.
Published: November 2008

For an n-vector x = (xi) and n × n matrix A = (aij) with complex elements, let |x|2 = Σi|xi|2,|A|2 = ΣiΣj|aij|2. Also, M(A), m(A) denoteℜλ1,ℜλn, respectively, where λ1,…,λA are the eigenvalues of A arranged so that ℜλ1 ≥ … ≥ ℜλn. Throughout this paper A(t) denotes a matrix whose elements aij(t) are complex valued Lebesgue integrable functions of t in (0, T) for all T > 0. Then M(A(t)), m(A(t)) are also Lebesgue integrable in (0, T) for all T > 0. The characteristic exponent μ of a non-zero solution x(t) of the n × n system of differential equations

can be defined, following Perron ((12)), as

where ℒ denotes lim sup as t → + ∞. When |A(t)| is bounded in (0,∞), μ is finite; in other cases it could be ± ∞.
Bundle structures on manifolds
Authors: Wagoner J. B.
Published: November 2008

This note gives homotopy-theoretic criteria in the metastable range for an (n + l)-manifold with boundary to be diffeomorphic to the total space of an l-disc bundle over a closed n-manifold and for two such structures to be equivalent. The results are similar to some theorems of (8) and the main technique used in the proofs is surgery on a map ((l), pp. 42–46, (7), (10)). All manifolds will be smooth.
An interpretation of negative probabilities
Authors: Halton John H.
Published: November 2008

1. Let (S, S,μ) be a measure-space, and let f(x) and ø(x) be two μ-integrable functions defined in S. We shall assume that f(x) is a ‘difficult’ function, of which all that is known is how to calculate f(x), generally by a rather complicated process, for any given x∈S; and that ø(x) is an ‘easy’ function, whose value for any x, and whose integral over any subset T of S belonging to the class S of μ-measurable sets, are relatively simple to obtain. We shall define the sets

and assume that

It will be of interest to consider those functions ø(x) which may be regarded as approximations to f(x) in S.
Circular inclusion in an infinite elastic medium with a circular inhomogeneity
Authors: Bhargava R. D. , Kapoor O. P.
Published: November 2008

In a previous paper ((l)) the authors gave the solution for the two-dimensional circular inclusion problem in a medium containing a circular cavity. This paper seeks to solve the more general problem of a similar inclusion when the cavity is replaced by an inhomogeneity which could be of a different elastic material. The solution consists in finding three sets of suitable complex potential functions ø(z) and ψ(z) for three regions: the inhomogeneity, the inclusion and the rest of the material. The solution depends upon the evaluation of the complex potentials for a material containing the inhomogeneity when on the former a finite force is acting at some fixed point. It may be noted that two sets of ø(z) and ψ(z) have to be found in this case: one for the inhomogeneity and the other for the rest of the material. This may be taken as an auxiliary problem.
Addendum to a paper of Conner and Floyd
Authors: Wall C. T. C.
Published: November 2008

In two recent papers of Conner and Floyd ((2)) and ((3)), the additive structure of the SU-cobordism (or bordism) ring was completely determined. The object of this note is to point out that their results can also be used to determine the somewhat complicated multiplicative structure.
Quadratic forms overR(t)
Authors: Knight James T.
Published: November 2008

A theory of quadratic forms is developed overR(t), the field of rational functions with real coefficients, instead of over an algebraic number field: some results are analogous to the classical ones, but many, in particular the non-finiteness of the class number, whose more detailed treatment will be the subject of a later paper, are not analogous.
Local maxima of stationary processes
Authors: Leadbetter M. R.
Published: November 2008

Two natural definitions for the distribution function of the height of an ‘arbitrary local maximum’ of a stationary process are given and shown to be equivalent. It is further shown that the distribution function so defined has the correct frequency interpretation, for an ergodic process. Explicit results are obtained in the normal case.
Overstability in the magnetohydrodynamic Bénard problem at large Hartmann numbers
Authors: Gibson R. D.
Published: November 2008

The possibility of instability arising as overstability in the magnetohydrodynamic Bénard problem is discussed. Results are derived for large Hartmann numbers when the regions outside the fluid layer are conducting or insulating, the interfaces being rigid or free.
The two-dimensional jet
Authors: Conway William E.
Published: November 2008

Consider the two-dimensional steady flow of an incompressible inviscid fluid under no body forces between two infinite horizontal planes. When an aperture is made in the lower plane, the fluid will issue as a jet bounded by free streamlines along which the pressure is constant. At infinity the flow in the jet will be uniform and parallel.
A note on the Stokes phenomenon
Authors: Duty R. L.
Published: November 2008

For a certain class of differential equations it is shown that the Stokes phenomenon can be related to a basic change in the character of the differential equation.
Bundles of Grassmannians and integrality theorems
Authors: Roberts R. S.
Published: November 2008

Introduction. For applications in differential topology it is desirable to be able to find restrictions on the possible real and complex vector bundles over a given manifold. These restrictions usually state that the top component of some specific rational multinomial in the various characteristic classes is an integral multiple of the fundamental cocycle. For example, in proving non-embedding or non-immersion theorems (compare (1); (10)) one could test whether the normal bundle is consistent with these integrality conditions.
A pair of complementary theorems on convergence rates in the law of large numbers
Authors: Heyde C. C. , Rohatgi V. K.
Published: November 2008

Introduction. Let Xi (i= 1, 2, 3,…) be a sequence of independent and identically distributed random variables with law ℒ(X) and write The Kolmogorov-Marcinkiewicz strong law of large numbers (Loève(6), p. 243) has the following statement:

If E|X|r < ∞, then with cr = 0 or EX according as r 1 or r ≥ 1.
On Riesz logarithmic summability of the ultraspherical series
Authors: Pandey G. S.
Published: November 2008

Let f{θ, φ) be a function defined for the range 0 ≤ φ π, 0 ≤ φ ≤ 2π on a sphere S. The ultraspherical series associated with this function is

where

and the ultraspherical polynomials are defined by the following expansion

.
On the pseudoprimes of the form ax + b
Authors: Rotkiewicz A.
Published: November 2008

A composite number n is called a pseudoprime if n|2n− 2.

Theorem 1. If a and b are natural numbers such that (a, b) = 1, then there exist infinitely many pseudoprimes of the form ax + b, where x is a natural number.

The proof of this theorem is given by the author in (5). This proof is based on the following two lemmas.
Convergence in lower bound formulae
Authors: Walmsley Mary
Published: November 2008

A convergence criterion is applied to the eigenvalue problems, which occur in lower bound calculations for , when either the energy or the nuclear charge is taken as the eigenvalue. In the former problem the lower bounds do not converge to the exact energy and the continuum causes a larger error for σu (1) than for σg (1). However, when the nuclear charge is taken as the eigenvalue the lower bound procedure converges.
The asymptotic discretization error of a class of methods for solving ordinary differential equations
Authors: Watt J. M.
Published: November 2008

The order and asymptotic form of the error of a general class of numerical method for solving the initial value problem for systems of ordinary differential equations is considered. Previously only the convergence of the methods, which include Runge-Kutta and linear multistep methods, has been discussed.
On the laminar compressible boundary layer induced by the passage of a plane shock wave over a flat wall
Authors: Ackroyd J. A. D.
Published: November 2008

The laminar compressible boundary layer induced by the passage of a plane shock wave over a flat wall is examined in detail. Use is made of the empirical viscosity-temperature relationship μ ∝ Tω. The boundary-layer equations are solved numerically for various values of the index ω, Prandtl number and shock strengths. The resulting solutions are then used to construct simple semi-empirical relationships for some of the more important boundary-layer parameters.
Diffraction of an incomplete expansion wave by a corner
Authors: Anderson W. M.
Published: November 2008

The solution by Powell (6) of the diffraction of a complete expansion wave by a corner is extended to an incomplete wave. It is shown that the pressure is discontinuous across one of the boundaries and becomes logarithmically singular at a point on it. Graphs of the pressure distribution on the fixed boundary are given.
On Perron's method of summation
Authors: Kwee B.
Published: November 2008

Let λ > 0 and let

If , we say that the series is summable (P, λ) to s.
A new generating function for Jacobi polynomials
Authors: Sharma B. L.
Published: November 2008

In this paper we give a new generating function for the Jacobi polynomials. The result obtained is of general character and includes as particular cases some of the results given earlier by Carlitz(2), Salam(1), Manocha and Sharma (3,4).
Fourier series for H-functions
Authors: Parashar B. P.
Published: November 2008

In this note the author has derived three Fourier series expansions for H-function which is more general than Meijer's G-function. Some integrals and a recurrence relation for H-function have been derived as corollaries.
On a conjecture of Ilieff
Authors: Brannan D. A.
Published: November 2008

We will here prove the following special case of an (unpublished) conjecture for a polynomial of degree ndue to L. Ilieff:

Theorem. If all zeros of the cubic polynomial p3(z) lie in |z| ≤ 1, then at least one zero of p3′(z) lies in or on the boundary of a circle of radius unity around each zero of p′3(z).
Some formulae for generalized Rice polynomials
Authors: Manocha H. L.
Published: November 2008

The aim of this note is to obtain some formulae for generalized Rice polynomials as defined by (2)
On a singular eigenvalue problem
Authors: Naylor D. , Dennis S. C. R.
Published: November 2008

Sears and Titchmarsh (1) have formulated an expansion in eigenfunctions which requires a knowledge of the s-zeros of the equation

Here ka > 0 is supposed given and β is a real constant such that 0 ≤ β < π. The above equation is encountered when one seeks the eigenfunctions of the differential equation

on the interval 0 < α ≤ r < ∞ subject to the condition of vanishing at r = α. Solutions of (2) are the Bessel functions J±is(kr) and every solution w of (2) is such that r−½w(r) belongs to L2 (α, ∞). Since the problem is of the limit circle type at infinity it is necessary to prescribe a suitable asymptotic condition there to make the eigenfunctions determinate. In the present instance this condition is
Two dimensional mixed boundary-value problems in a wedge-shaped region
Authors: Williams W. E.
Published: November 2008

In a recent paper Srivastav (2) considered the solution of certain two-dimensional mixed boundary-value problems in a wedge-shaped region. The problems were formulated as dual integral equations involving Mellin transforms and were reduced to the solution of a Fredholm integral equation of the second kind. In this paper it will be shown that a closed form solution to the problems treated in (2) may be obtained by elementary means.
Sums of three roots of unity
Authors: Jones A. J.
Published: November 2008

The object of this paper is to prove a conjecture made by Robinson (1) concerning cyclotomic integers, that is, integers that are rationally expressible in terms of roots of unity.
Spherically symmetric space-time of class one and electromagnetism
Authors: Pandey S. N. , Kansal I. D
Published: November 2008

It is well known that a spherically symmetric space-time is, in general, of class two. A necessary and sufficient condition for a spherically symmetric space-time to be of class one has been obtained in terms of the Riemann curvature tensor. By means of a transformation property of s.s. space-time, three distinct cases are shown to exist. The incompatibility of class one spherically symmetric space-times with Rainich algebraic conditions is established in these three cases. It is concluded that spherically symmetric electromagnetic fields cannot be embedded in a flat space of 5-dimensions.
A note on the homology of ideals ǵenerated by three elements in local rings
Authors: Burch Lindsay
Published: November 2008

In this note, I show that, if Q is a local ring and s an integer, 1 < s < codim Q, then there is an ideal As of Q, generated by three or fewer elements of Q, such that

and that, if Q is not regular, then there is an ideal B of Q, generated by three or fewer elements and of infinite homological dimension over Q.
Definite integrals involving Whittaker functions
Authors: Ragab F. M. , Simary M. A.
Published: November 2008

Little is known about definite integrals for the Whittaker functions. The object of this paper is to establish such results and the following formulae are established

where the symbol means that in the expression following it i is to be replaced by –i and the two expressions are to be added.
Use of Gauss' hypergeometric function and Meijer's G-function in the production of heat in a cylinder
Authors: Bajpai S. D.
Published: November 2008

Cases in which heat is produced in solids are becoming increasingly important in technical applications ((3), pp. 11–12).
On the short-wave asymptotic solution of a problem in acoustic radiation
Authors: Leppington F. G.
Published: November 2008

A study is made of a problem in acoustic radiation, with the object of developing a rigorous theory of short-wave asymptotic solutions of the Helmholtz wave equation. The particular example under consideration is that of the radiation of sound waves by a smooth convex cylinder pulsating with prescribed normal velocity V(s) exp (−iωt), where s denotes arc length of the curve C bounding the cylinder cross-section, and ω is the radian frequency.
Integration of certain products associated with a generalized Meijer function
Authors: Srivastava H. M. , Joshi C. M.
Published: November 2008

In an attempt to give extensions of certain results in the theory of Mac-Robert's E-function and Meijer's (G-function, the integrals

and

are evaluated, for positive integral values of n, in terms of Agarwal's and their numerous interesting particular cases are deduced. The scope of a further generalization of (i) and (ii), with the aid of the Mellin inversion formula, is also discussed.

It is observed that the integral (ii) provides an elegant generalization of some of the recent results of Srivastava which, in turn, incorporate a well-known formula due to Slater.
A rigorous derivation of the exact contingency formula
Authors: Halton John H.
Published: November 2008

The formula

gives the conditional probability of an (r1 × r2 × … × rk) k-variate contingency table (where is the entry in the cell identified by the k indices im), given all the marginal totals

and the hypothesis that the k random variables tabulated are mutually independent. This formula appears in (1) and more restricted forms can be found elsewhere in the literature. Certain criticisms have been made of the logic and clarity of the available proofs of (1); it is hoped that the alternative derivation given here will dispel any residual doubts, and will provide better insight into the nature of the problem.
Lipschitz classes on finite dimensional groups
Authors: Walker P. L.
Published: November 2008

In this paper G will denote a compact metric Abelian group having finite topological dimension, dim G. The discrete dual group of G will be denoted by Γ. The metric on G will be denoted by d(x, y), or |x-y| since translation-invariance is assumed.
On rational primes p congruent to 1 (mod 3 or 5)
Authors: Rajwade A. R.
Published: November 2008

Let p ≡ 1 (mod 3) or ≡ 1 (mod 5) be a rational prime and g a primitive root mod p. The non-zero residues g, g2,…,gp−1 (mod p) can then be divided into 3 or 5 classes , , , or , , , , respectively, by letting gν ∈ , , (respectively , , , , ) according as ν ≡ 0, 1, 2, (mod 3) (respectively 0, 1, 2, 3, 4 (mod 5)). Problems regarding the distribution of 1, 2, …, p − 1 amongst the 3 (respectively 5) classes are many. In this paper we consider the following problem in some detail (we shall state it here for the case p ≡ 1 (mod 3)). Let α, β, γ be typical members of , , respectively. Let the number of solutions of the congruences

be
Bilateral generating functions involving Jacobi polynomials
Authors: Manocha H. L.
Published: November 2008

In paper(1) it has been proved that

where the Jacobi polynomial is denned as ((3), p. 255)
A note on the summation of divergent power series
Authors: Scraton R. E.
Published: November 2008

Many mathematical problems which do not yield a closed-form solution admit of a solution in the form of a power series; differential equations are an obvious example. The direct use of this power series is limited to the interior of its circle of convergence, and this places a restriction—often a severe restriction—on its usefulness. The method described in this paper enables this restriction to be alleviated in many cases; it also enables the convergence of a power series within its circle of convergence to be improved. The method is based on the Euler transformation.
Quadratic Lyapunov functions for linear systems
Authors: Venkatesh Y. V.
Published: November 2008

The paper deals with the existence of a quadratic Lyapunov function V = x′P(t)x for an exponentially stable linear system with varying coefficients described by the vector differential equation The derivative dV/dt is allowed to be strictly semi-(F) and the locus dV/dt = 0 does not contain any arc of the system trajectory. It is then shown that the coefficient matrix A(t) of the exponentially stable system is not identically equal to a unit matrix multiplied by a scalar. The result subsumes that of Lehnigk(1).
On integral functions with distinct asymptotic values
Authors: Hayman W. K.
Published: November 2008

Suppose that p > 1 and let Γ1, Γ2, …, Γp be Jordan arcs going from 0 to ∞ and having no other common point and numbered in anti-clockwise order around the origin. Let Γp+1 = Γ1 and let Dv be the domain bounded by Γv and Γv+1 for v = 1 to p.
Special functions and forced symmetrical vibrations of a circular elastic plate
Authors: Bajpai S. D.
Published: November 2008

In this paper we have employed Gauss's hypergeometric function, Legendre function and Jacobi polynomial to obtain the solutions of a fundamental differential equation of certain types of forced symmetrical vibrations of a circular elastic plate of finite radius.
DS-spaces of Robinson and Trautman: topological identifications and asymptotic symmetry
Authors: Foster J.
Published: November 2008

Some topological identifications are introduced into the DS-spaces of Robinson and Trautman, and the symmetry and asymptotic symmetry groups of the resulting spaces are calculated. The latter have structures similar to the GBM group, each being the semi-direct product of a group of supertranslations by a group of classical type.
A cosmoloǵical basis for hyperbolic velocity space
Authors: Prokhovnik S. J.
Published: November 2008

It is shown that a cosmological model of light propagation associated with a uniformly expanding universe provides a physical significance to the hyperbolic velocity space by which this model can be described. Some implications of this result are discussed.
On the short-wave asymptotic motion due to a cylinder heavinǵ on water of finite depth. I
Authors: Rhodes-Robinson P. F.
Published: November 2008

This paper is a first investigation into the short-wave asymptotic motion due to a cylinder heaving on water of finite constant depth, and we present a non-rigorous method for an arbitrary smooth cylinder which intersects the free surface normally. The reduction of the potential problem by the use of two auxifiary potentials, introduced by the subtraction of formal solutions constructed from the limit potential, enables us to find (i) an integral expression, by using a subsidiary approximate Green's function, for the potential in the far field which is evaluated after making plausible assumptions about the asymptotic value of the first auxiliary potential on the cylinder and below the free surface; and (ii) a form for the virtual mass, using the second auxiliary potential whose value on the cylinder is deduced from that of the first to which it is related. Thus we obtain the asymptotic evaluation of the coefficients describing the wave-making and virtual mass of the heaving cylinder, which depend on the limit potential and have no formal dependence on the depth of the bottom.
Trace in additive categories
Authors: Thomas Alan
Published: November 2008

1. Let ℳ be an additive category. (We refer to ((1), Ch. IX) for the definition of an additive category and associated terms. In particular a sequence

is exact if i = kerp and p = coker i.) We write End M for Hom(M, M). A trace on ℳ with values in an abelian group G is a collection of (abelian group) homomorphisms

one for each M ∈ ℳ, satisfying the following two conditions:

(i) Exactness. Given a commutative diagram with exact rows,

then tA(f) + tC(h) = tB(g).
An algebraic system of equations with element (n − m)−1, m and n positive unequal integers
Authors: Billington A. E.
Published: November 2008

The system of equations here to be analysed is

The method of solution is by way of appeal to the theory of singular integral equations of Cauchy type. Apart from the self-reciprocal series of Titchmarsh(l) no comparable systems appear to have been discussed in the literature, the order of the matrix element falling foul of the restrictions otherwise there imposed.
On the geometric structure of classical field theory in Lagrangian formulation
Authors: Śniatycki Jędrzej
Published: November 2008

Geometric structure of classical field theory in Lagrangian formulation is investigated. Symmetry transformations with generators depending on higher-order derivatives are considered and the corresponding conservation laws are obtained.
The theory of trailing Taylor columns
Authors: Lighthill M. J.
Published: November 2008

When Rossby number is small but Ekman number is very much smaller, study of the flow field far from a body moving at right angles to the axis of rotation of a large body of fluid indicates that the region of influence should not be a Taylor column parallel to the axis, but a trailing Taylor column, bent backwards on both sides of the body at a small angle (proportional to Rossby number) to the axis. The paper reviews the physical significance of, and experimental evidence for, this conclusion.
An application of the Thue–Siegel–Roth theorem to elliptic functions
Authors: Coates J.
Published: November 2008

Let α1, …, αn be n ≥ 2 algebraic numbers such that log α1,…, log αn and 2πi are linearly independent over the field of rational numbers Q. It is well known (see (6), Ch. 1) that the Thue–Siegel–Roth theorem implies that, for each positive number δ, there are only finitely many integers b1,…, bn satisfying

where H denotes the maximum of the absolute values of b1, …, bn. However, such an argument cannot provide an explicit upper bound for the solutions of (1), because of the non-effective nature of the theorem of Thue–Siegel–Roth. An effective proof that (1) has only a finite number of solutions was given by Gelfond (6) in the case n = 2, and by Baker(1) for arbitrary n. The work of both these authors is based on arguments from the theory of transcendental numbers. Baker's effective proof of (1) has important applications to other problems in number theory; in particular, it provides an algorithm for solving a wide class of diophantine equations in two variables (2).
Linear theory of Cosserat surface and elastic plates of variable thickness
Authors: Green A. E. , Naghdi P. M. , Wenner M. L.
Published: November 2008

Within the scope of the linear isothermal theory of an elastic Cosserat surface, constitutive equations are derived for an initially flat Cosserat surface in which the initial director (along the normal to the initial surface) is allowed to depend on the surface coordinates. These constitutive equations correspond to those for bending and stretching of a transversely isotropic three-dimensional plate. Special attention is given to the relevance and applicability of the results to bending of (three-dimensional) plates of variable thickness and comparison is made with a set of equations for elastic plates of variable thickness obtained, by an approximation procedure, from the three-dimensional equations.
Massey products in K-theory. II
Authors: Snaith V. P.
Published: November 2008

0. Introduction: In (14), higher order operations in K-theory, called Massey products, were introduced. These were motivated by the construction, in (7), of a spectral sequence in equivariant K-theory
Notes on existence of eigenvalues
Authors: Davis A. M. J.
Published: November 2008

The following work arises from the study of a rotating fluid contained in a rectangular vessel whose depth depends on one coordinate only.
Classification of crystal lattices
Authors: Schwarzenberger R. L. E.
Published: November 2008

There are two natural ways in which to attempt a classification of the lattices which occur in crystallography or, more generally, of n-dimensional lattices. The first method is algebraic and proceeds by classifying the symmetry groups of the lattices. Thus an n-dimensional lattice is a discrete subgroup

of Rn and determines a symmetry group

which is a subgroup of the orthogonal group On. Two lattices T1 and T2 are said to determine the same crystal system if the symmetry groups G(T1) and G(T2) are conjugate: that is there exists a linear isomorphism ø: Rn → Rn such that
On the uniformization of plane direction fields, and of second-order partial differential operators
Authors: Finn Robert
Published: November 2008

One of the topics covered by most textbooks on partial differential equations is the classification problem for second-order equations in the plane. In a typical treatment, it is shown that under some smoothness conditions on the coefficients, hyperbolic and parabolic equations can be reduced locally to normal form (uniformized) by an elementary procedure. For elliptic equations, the procedure fails unless an extraneous hypothesis (analyticity of the coefficients) is introduced. It is then pointed out that a different and much deeper method (essentially the general uniformization theorem) is effective for the elliptic case and even yields a global result. It is striking, but not surprising in view of recent developments on generalized solutions, that the alternate (global) procedure for elliptic equations requires much less smoothness of the coefficients than is needed for a sensible local result in the other cases.
On weak convergence of empirical processes for random number of independent stochastic vectors
Authors: Sen Pranab Kumar
Published: November 2008

By the use of a semi-martingale property of the Kolmogorov supremum, the results of Pyke (6) on the weak convergence of the empirical process with random sample size are simplified and extended to the case of p(≥1)-dimensional stochastic vectors.
A convolution involving Bell polynomials
Authors: Heselden G. P. M.
Published: November 2008

A convolution formula is established for Bell polynomials. This is expressed in seven equivalent ways and used to derive further properties of these polynomials. The application of these results to some twenty-seven special polynomial sets is shown and illustrated in the case of binomial, Hermite, Gegenbauer and generalized Bernoulli sets.
Quadratic Langrangians and static gravitational fields
Authors: Buchdahl H. A.
Published: November 2008

It is shown that the solutions of the field equations generated by the most general effective quadratic LagrangianaR2+bRklRklwhich are static, singularity-free, of classC4and asymptotically flat (of rankA3) are (i) conformally flat when 3a+b= 0, (ii) flat when 3a+b≠ 0 (b≠ 0), and (iii) have zero scalar curvature whenb= 0.
The combinatorial structure of the Hughes–Zassenhaus plane of order 25
Authors: Room T. G.
Published: November 2008

The Hughes–Zassenhaus plane, , of order 25 is the simplest of the exceptional Hughes planes (see, e.g. (1), p. 391).

In this paper, an incidence table is constructed analogous to tables for the field plane and the regular Hughes plane, and the following rather surprising properties of the plane emerge:

(i) The table has much greater internal symmetry than either of the analogous tables,

(ii) any polarity with regard to a conic in the central subplane Δ0 of extends to two pairs of polarities in one pair with 30 and the other with 60 singular points each in addition to the six in Δ0.
Convergence of measurable operators
Authors: Yeadon F. J.
Published: November 2008

Segal(4) defines the algebra of measurable operators affiliated with a von Neumann algebra, and convergence nearly everywhere of a sequence of measurable operators, and shows that addition is jointly sequentially continuous and multiplication separately sequentially continuous in the star topology associated with convergence nearly everywhere.
Local systems and Sylow subgroups in locally finite groups. II
Authors: Rae A.
Published: November 2008

1.1. Introduction. In this paper, we continue with the theme of (1): the relationships holding between the Sπ (i.e. maximal π) subgroups of a locally finite group and the various local systems of that group. In (1), we were mainly concerned with ‘good’ Sπ subgroups – those which reduce into some local system (and are said to be good with respect to that system). Here, on the other hand, we are concerned with a very much more special sort of Sπ subgroup.
On sums of determinants of intersection matrices of Petrie matricest
Authors: Gordon M. , Tutte W. T.
Published: November 2008

A Petrie matrix (Kendall (1), (2)) is a matrix of O's and l's such that the l's in each column occur consecutively. An example follows.
Uniformly bounded projections and semi-groups of operators
Authors: Okikiolu G. O.
Published: November 2008

Classes of operators called uniformly bounded projections defined on Banach spaces have been formally introduced (see for example (2), 8·9·28; 8·9·92), and their connexions with semi-groups of operators considered. Notable cases of such operators include the Dirichlet integrals which are connected with the semi-group of Poisson operators in Lp In the main conclusions of this paper, there is an exposition of the connexion between classes of uniformly bounded projections and semi-groups of operators. In particular, it is shown that certain uniformly bounded analytic semi-groups of operators give rise to classes of uniformly bounded projections. Various of the known special cases in Lp are also considered as Fourier trans form multiplier operators with characteristic functions of intervals as multipliers.
‘Line-cobordism’
Authors: Gate M. , Whiston G.
Published: November 2008

The following extension problem is of considerable geometrical interest. Given a compact manifold X = ∂Z the boundary of a compact manifold Z, does the normal bundle of X in Z extend to a line bundle on Z? A related question involving foliations is if X is a compact boundary, is X a leaf of a codimension one foliation on any compact manifold bounded by X? Obviously if it were such a leaf, its normal bundle in the bounded manifold would extend. Reinhart posed both these problems in the language of cobordism theories(1), (2) defining a vector cobordism theory to a deal with questions of the first genre and a foliated-cobordism theory to deal with the second. He was able to obtain complete results in his vector-cobordism theory, whilst the foliated-cobordism theory remains an unsolved problem. Our aim in this article is to extend Reinhart's results on vector-cobordism theory to unoriented line bundles. Whereas in vector-cobordism the Euler number is a cobordism invariant, in the cobordism theory that we shall describe, the modulus of the Euler number becomes a cobordism invariant.
Classical electrodynamics: the equation of motion
Authors: Hogan P. A.
Published: November 2008

In this paper we derive the Lorentz-Dirac equation of motion for a charged particle moving in an external electromagnetic field. We use Maxwell's electromagnetic field equations together with the assumptions (1) that all fields are retarded and (2) that the 4-force acting on the charged particle is a Lorentz 4-force. To define the self-field on the world-line of the charge we utilize a contour integral representation for the field due to A. W. Conway. This by-passes the need to define an ‘average field’. In an appendix the case of a scalar field is briefly discussed.
Products of positive operators
Authors: Bernau S. J.
Published: November 2008

Let H be a complex Hilbert space. Recall that a bounded linear operator A, on H, is positive if (Ax, x) ≥ 0 (x ∈ H) (so that A = A* necessarily) and positive definite if A is positive and invertible.
Convex fundamental domains of Fuchsian groups
Authors: Beardon A. F.
Published: November 2008

In this paper a Fuchsian group G shall be a discrete group of Möbius transformations each of which maps the unit disc △ in the complex plane onto itself. We shall also assume throughout this paper that G is both finitely generated and of the first kind.
Remark on the theorem of Carathéodory
Authors: Buchdahl H. A.
Published: November 2008

A proof of the Theorem of Carathéodory is exhibited in a covariant formalism.