bluuebunny/crossref_metadata_2025_split 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 toa 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 sumsIf denotes the projection of on the ith coordinate axis, Esseen proves that ifand ψ(a) denotes the corresponding normal (radial) distribution function of the same first and second moments as μn(a), thenwhere 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

AbstractScoins(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.
The isotropic scattering of a concentrated ray pencil from a point source
Authors: Smith M. G. 
Published: November 2008

AbstractThe 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.
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 matrixTheSchur Product(Schur (14)) of two matricesA, B∈ ℛnis denned bywhereA= (aij),B= (bij),C= (cij) andLet ϕ be the mapping of ℛninto the real line defined byfor allA∈ ℛn, where, as in the sequel,.
Quantum theory of unstable particles
Authors: Martin J. L. 
Published: November 2008

AbstractA 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 equationfor 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.
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, thenis 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 numberWe observe two easy results.
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 existsuch 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 equationscan be defined, following Perron ((12)), aswhere ℒ denotes lim sup as t → + ∞. When |A(t)| is bounded in (0,∞), μ is finite; in other cases it could be ± ∞.
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 setsand assume thatIt 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

AbstractIn 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.
Quadratic forms overR(t)
Authors: Knight James T. 
Published: November 2008

AbstractA 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.
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 iswhereand 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

AbstractA 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.
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

SummaryThe 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.
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).
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 equationHere 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 equationon 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
Spherically symmetric space-time of class one and electromagnetism
Authors: Pandey S. N. , Kansal I. D 
Published: November 2008

AbstractIt 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 thatand 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 establishedwhere the symbol means that in the expression following it i is to be replaced by –i and the two expressions are to be added.
On the short-wave asymptotic solution of a problem in acoustic radiation
Authors: Leppington F. G. 
Published: November 2008

AbstractA 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

AbstractIn an attempt to give extensions of certain results in the theory of Mac-Robert's E-function and Meijer's (G-function, the integralsandare 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 formulagives 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 totalsand 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.
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 congruencesbe
Quadratic Lyapunov functions for linear systems
Authors: Venkatesh Y. V. 
Published: November 2008

AbstractThe 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).
DS-spaces of Robinson and Trautman: topological identifications and asymptotic symmetry
Authors: Foster J. 
Published: November 2008

AbstractSome 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.
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

AbstractThis 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 sequenceis 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) homomorphismsone 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 isThe 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.
The theory of trailing Taylor columns
Authors: Lighthill M. J. 
Published: November 2008

AbstractWhen 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.