bluuebunny/crossref_metadata_2025_split huggingface dataset:
On a conjectured analogue to Clifford's chain in [4]
Authors: Longuet-Higgins M. S. 
Published: November 2008

Clifford's well-known chain of theorems concerning circles in a plane may be stated as follows (the first two links being trivial):(C2) Given any point A, and through it two circles C1 and C2, these meet again in a point A12, say.(C3) Given three circles C1, C2, C3 through A, the three points A23, A13, An lie on a circle C123, say.(C4) Given four circles C1,…,C4 through A, the four circles C234,…,C123 meet in a point A1234 (Wallace(1)).(C5) Given five circles C1,…,C5 through A, the five points A2345,…,A1234 lie on a circle C12345 (Miquel (2)).And so on (Clifford(3)).Grace (4) showed that in three dimensions [3], there is no direct analogue to Clifford's chain, but in [4] he noted the following:(G4) Given any point A in [4], and through it four hyperspheres S1,…, S4, these meet again in a point A1234, say.(G5) Given five hyperspheres Sl,…,S5 through a point A, the five points A2345, A1234 lie on a hypersphere S12345, say.(G6) Given six hyperspheres Sl,…, S7 through A, the six hyperspheres S23456,…, A12345 have in common a point A123456.This leads naturally to the conjecture(G7) Given seven hyperspheres Sl,…, S7 through A, the seven points A234567,…, A123456 lie on a hypersphere S1234567.
The Kahn–Priddy theorem
Authors: Adams J. F. 
Published: November 2008

Let ΦSr(X) be the stable homotopy groupwhere SnX means the n-fold suspension of X. For example, the groups ΦSr(S0) are the stable homotopy groups of spheres. Letbe the ‘infinite-dimensional’ orthogonal group. Then topologists are familiar with the ‘stable J-homomorphism’G. W. Whitehead observed that J factors through an ‘even more stable’ J-homomorphismhe conjectured that J′ is epi (for r > 0).
Internal-inertia waves in a fluid of variable depth
Authors: McKee W. D. 
Published: November 2008

AbstractWaves in a rotating, stratified fluid of variable depth are considered. The perturbation pressure is used throughout as the dependent variable. This proves to have some advantages over the use of the vertical velocity. Some previous three-dimensional solutions for internal waves in a wedge are shown to be incorrect and the correct solutions presented. A WKB analysis is then performed for the general problem and the results compared with the exact solutions for a wedge. The WKB solution is also applied to long surface waves on a rotating ocean.
Open decompositions on ordered convex spaces
Authors: Wong Yau-Chuen 
Published: November 2008

Let (E, ) be a topological vector space with a positive cone C. Jameson (3) says that C given an open decomposition on E if V ∩ C − V ∩ C is a -neighbourhood of 0 whenever V is a -neighbourhood of 0. The concept of open decompositions plays an important rôle in the theory of ordered topological vector spaces; see (3). It is clear that C is generating if C gives an open decomposition on E; the converse is true for Banach spaces with a closed cone, by Andô's theorem (cf. (1) or (9)). Therefore the following question arises naturally:(Q 1) Let (E, ) be a locally convex space with a positive cone C. What condition on is necessary and sufficient for the cone C to give an open decomposition on E?
Some generalizations of the Borsuk–Ulam theorem and applications to realizing homotopy classes by embedded spheres
Authors: Fenn Roger 
Published: November 2008

In this paper, some theorems of the Borsuk-Ulam type (1) are given. One of these can be applied to show that certain homotopy classes in manifolds cannot be realized by embedded spheres. The n-dimensional sphere Sn is the subset of the euclidean spaceRn+l consisting of all points (x1, …,xn+1) satisfying . Let be a piecewise linear (PL) involution on Sn without fixed points.
The reflexion of an acoustic pulse by a plane vortex sheet
Authors: Jones D. S. 
Published: November 2008

AbstractThis paper deals with the influence of a vortex sheet separating two fluids in relative motion on the radiation from a point source of sound. Both the harmonic and impulsive sources are considered and it is found that waves due to Helmholtz instability must be included in order to ensure that there is no field before the source is excited. The instability wave is confined to a finite region and dominates other disturbances in that region. It is suggested that the instabifity wave is initiated by the unrestricted growth of the specularly reflected wave.
On isotropic tensors
Authors: Jeffreys Sir Harold 
Published: November 2008

(1) Proc. Cambridge Philos. Soc. 73 (1973), page 173Dr E. Leibowitz has called my attention to a mistake in my paper on isotropic tensors. (1) On p. 173, if 12 is annihilated as stated and then we annihilate l13 by the same method, a new contribution to l12 appears. The correct method is to choose α so as to annihilate l21. Then a suitable rotation on the 13 plane removes l31 without re-introducing l21. The rest of the argument stands.
Geometrical aspects of the theory of non-homogeneous Markov chains
Authors: Kingman J. F. C. 
Published: November 2008

AbstractA geometrical representation of the transition matrices of a non-homogeneous chain with N states, in terms of certain convex subsets of , is used to describe aspects of the chain. For example, an important theorem of Cohn on the structure of the tail σ-field is a simple corollary. The embedding problem is shown to be entirely geometrical in character. The representation extends to Markov processes on quite general state spaces, and the tail is then represented by the projective limit of these convex sets.
Proof of the Churchhouse conjecture concerning binary partitions
Authors: Gupta Hansraj 
Published: November 2008

In a recent paper, R. F. Churchhouse has studied the function b(n) giving the number of partitions of n into powers of 2. The generating function of b(n) isso thatUsing this relation, Churchhouse has shown thatandwherewithMoreoverand b(n) is even for n ≥ 2, while b(2n) ≡ (mod 4), for n = 22m−l(2k + 1); and, it is not ≡ (mod 8) for any n. He has conjectured that for k ≥ 1 and n odd,andThe object of this note is to prove these conjectures.
Two-dimensional anisotropic elastic waves emanating from a point source
Authors: Payton Robert G. 
Published: November 2008

AbstractA two (spatial) dimensional initially strained elastic body is excited by a point impulse. Expressions are found for the various displacement components in a form which is readily evaluated by residues. The solid itself is characterized by three parameters which depend on the material properties and the initial deformation. For the case when two of these parameters are equated, explicit expressions for the displacements are given along the Cartesian axes passing through the origin of the point impulse. Wave front singularities and lacunas are identified and discussed. Some typical numerical results are given.
Kernels with only a finite number of characteristic values
Authors: Swann Dale W. 
Published: November 2008

Let K(s, t) be a complex-valued L2 kernel on the square ⋜ s, t ⋜ by which we meanand let {λν}, perhaps empty, be the set of finite characteristic values (f.c.v.) of K(s, t), i.e. complex numbers with which there are associated non-trivial L2 functions øν(s) satisfyingFor such kernels, the iterated kernels,are well-defined (1), as are the higher order tracesCarleman(2) showed that the f.c.v. of K are the zeros of the modified Fredhoim determinantthe latter expression holding only for |λ| sufficiently small (3). The δn in (3) may be calculated, at least in theory, by well-known formulae involving the higher order traces (1). For our later analysis of this case, we define and , respectively, as the minimum and maximum moduli of the zeros of , the nth section of D*(K, λ).
On the Stieltjes transform of generalized functions
Authors: Pandey J. N. 
Published: November 2008

If f(t) belongs to L(0, R) for every positive R and is such that the integralconverges for x > 0, then F(s) exists for complex s(s ╪ 0) not lying on the negative real axis andfor any positive ξ at which f(ξ+) and f(ξ−) both exist.We define an operator Lk, t[F(x)]byUnder the above conditions on f(t), it is known that for all points t of the Lebesgue set for the function f(t),Let Ln, x denote the differentiation operatorSuppose thatconverges for some x¬ 0; then, if f(t) belongs to L(R−1, R) for every R>1,
Uniform summability of power series
Authors: Kurtz J. C. , Sledd W. T. 
Published: November 2008

AbstractIt is shown that for the Cesàro means (C, α) with α > - 1, and for a certain class of more general Nörlund means, summability of the series σan implies uniform summability of the series σan zn in a Stolz angle at z = 1.If B is a normal matrix and (B) denotes the series summability field with the usual Banach space topology, then the vectors {ek} (ek = {0,0,..., 1,0,...}) are said to form a Toplitz basis for (B) relative to a method H if H — Σakek = a for each a = {ak}ε(B). It is shown for example that the above relation holds for B = (C,α), α> − 1 , and H = Abel method; also for B = (C,α) and H = (C,β) with 0 ≤ α ≤ β.Applications are made to theorems on summability factors.
Integrals with a large parameter. Several nearly coincident saddle points
Authors: Ursell F. 
Published: November 2008

AbstractIn the contour integralthe functions g( ) and f( ) are analytic functions of their arguments, and N is a large positive parameter. When N tends to ∞, asymptotic expansions can usually be found by the method of steepest descents, which shows that the principal contributions arise from the saddle-points, i.e. the values of z at which ∂f/∂z = 0. The position of the saddle-points varies with the l parameters, denoted by α = (α1,α2,…,αl), and it may happen that two or more saddle-points coincide at a certain value of α. The asymptotic expansions given by the ordinary method of steepest descents are then non-uniform. The case of a single parameter and two nearly coincident saddle- points was studied earlier and leads to Airy functions. Here we are concerned with the case of l parameters and of m saddle-points which are nearly coincident when α is near 0. It is then known that f(z,α) can be transformed locally into a polynomial of degree m + 1, and accordingly we here restrict f(z,α) to be such a polynomial. The form of the resulting asymptotic expansion had already been conjectured (correctly as we now see), and involves certain special functions of m–1 variables. To establish the validity of the expansion we must show that the remainder after any number of terms is of smaller magnitude than the last term retained in the expansion. In other words, we must establish an inequality between integrals which is to hold for sufficiently small α and sufficiently large N, and which involves 1 parameters and m nearly coincident saddle-points.
On the existence of transverse elastic acceleration waves
Authors: Chadwick P. , Currie P. K. 
Published: November 2008

AbstractIt is shown that, at an arbitrary point in a general elastic material and regardless of the prevailing state of deformation, there exists in every plane at least one direction in which a transverse acceleration wave may propagate. The possibility of a transverse wave actually being able to progress in such a specific direction depends upon the associated local wave speed being real and conditions are obtained under which this requirement is satisfied.
Large cycles in large labelled graphs
Authors: Wright E. M. 
Published: November 2008

An (n, q) graph consists of n labelled nodes and q edges, i.e. unordered pairs of different nodes. The maximum possible value of q is N = ½n(n − 1) and the (n, N) graph is the complete graph. We write B(h, k) = h!/{k!(h − k)!}. The number of (n, q) graphs is clearly B(N, q). A k-cycle in a graph consists of a sequenceof k edges, where D1D2,…, Dk are all different nodes. We are concerned here with the probability P = P(n, q, k) that a k-cycle will be present in an (n, q) graph, that is, the proportion of all (n, q) graphs which contain at least one k-cycle. We investigate the limit of P as n → ∞ and in particular try to determine when P → 1 (when we say that almost all (n, q) graphs contain a k-cycle) and when P → 0 (when we say that almost no (n, q) graphs contain a k-cycle).
A characterization of certain compact sets of measure zero
Authors: Connolly D. M. , Williamson J. H. 
Published: November 2008

AbstractWe show that the only compact subsets of [0, 1] of measure zero are the obvious ones, and extend this idea to show that the only compact subsets of [0, 1] that generate semigroups of ℝ of measure zero are the obvious ones. We apply this last result to prove that any proper Raikov system of subsets of ℝ, with a single generator, is contained in a strictly larger proper Raikov system.
The equation A(t, u(t))′ + B(t, u(t)) = 0
Authors: Calvert Bruce 
Published: November 2008

The initial value problem for the equationhas been studied recently as a model for long waves in nonlinear dispersive systems. Benjamin, Bona and Mahony (2) introduced this equation as an alternative to the KdV equation of Korteweg-de Vries. Hence, it is referred to as the BBM equation. They studied solutions u(x, t) of the BBM equation for t ≥ 0 and x∈(− ∞, ∞), satisfying u(x, 0) = g(x). Bona and Bryant(1) carried through the study of the BBM equation for t ≥ 0 and x ∈ [0, ∞), satisfying u(x, 0) = g(x) and u(0, t) = h(t). The aim of this paper is to study the equationwhere At and Bt are mappings defined on subsets of Banach spaces, especially when At is a second order elliptic operator and B is a differential operator of lower order, defined on an unbounded subset Ω of .
A general theory of separation, regularity, and connectedness properties
Authors: McSherry D. M. G. 
Published: November 2008

AbstractThe notion of basic topological binary relations is introduced, and corresponding to each such relation we define -regularity, -connectedness, and a separation axiom . The family of separation axioms so obtained is shown to include all the standard axioms T0, T1, …, T5, while familiar properties such as hyperconnectedness and ultraconnectedness are among the class of connectedness conditions. The regularity conditions include R0, R1, and regularity itself.
Complex potential equations I. A technique for solution
Authors: Collins C. B. 
Published: November 2008

AbstractThis work amalgamates and solves certain problems arising in differential equation theory and in classical differential geometry.It describes a novel technique for solving systems of non-linear partial differential equations of the formwhere f(γ) and g(γ) are arbitrarily assigned functions. The circumstances are determined under which compatible solutions exist, not only when γ is real, but also when γ is complex, and all of the corresponding solutions are found. This is done by using a geometric technique that incorporates the equipotential surfaces of constant γ. In general, these surfaces are imaginary, and a fairly extensive treatment of such surfaces in (complexified) 3-dimensional Euclidean space is included. A close association is discovered between the set of equipotential surfaces and the class of surfaces of constant radii of principal normal curvature.
Some more characterizations of Banach spaces containing l1
Authors: Haydon Richard 
Published: November 2008

In a series of recent papers ((10), (9) and (11)) Rosenthal and Odell have given a number of characterizations of Banach spaces that contain subspaces isomorphic (that is, linearly homeomorphic) to the space l1 of absolutely summable series. The methods of (9) and (11) are applicable only in the case of separable Banach spaces and some of the results there were established only in this case. We demonstrate here, without the separability assumption, one of these characterizations:a Banach space B contains no subspace isomorphic to l1 if and only if every weak* compact convex subset of B* is the norm closed convex hull of its extreme points.
Perturbations by operators converging compactly to zero
Authors: Martens Eric 
Published: November 2008

AbstractThis paper considers perturbations in locally convex spaces of semi Fredholm operators T by a sequence of operators {Kn} converging compactly to zero in a sense extending that from (2). It is shown that, under suitable conditions, α(T + Kn) ≤ α(T), β(T + Kn) ≤ β;(T) and κ(T + Kn) = κ(T) for large n.
The eigenvalue problem for infinite systems of linear equations
Authors: Sayer F. P. 
Published: November 2008

Given an infinite system of linear equationswhere the aij depend on a parameter λ, the eigenvalue problem is to determine values of λ for which xj (j = 1, 2, …) are not all zero. This problem (Taylor (3) and Vaughan (4)) can arise in the vibration of rectangular plates. Little theoretical work, however, appears to have been done concerning the existence and determination of the eigenvalues. The usual procedure (see (3) and (4)) is to consider a truncated or reduced system of N equations and find the values of λ for which the determinant of the N × N matrix [aij] vanishes. If a particular λ tends to a constant value as N is increased then this value is assumed to be an eigenvalue. The question therefore arises as to what happens if no limit exists. Can we assert that there are no eigenvalues? By constructing an appropriate example we show that the non-existence of a limit does not imply the non-existence of eigenvalues. In order to construct our example we first establish a result concerning the Legendre polynomials.
An algorithm for the total, or partial, factorization of a polynomial
Authors: Farmer M. R. , Loizou G. 
Published: November 2008

AbstractA globally convergent algorithm is presented for the total, or partial, factorization of a polynomial. Firstly, a circle is found containing all the zeros. Secondly, a search procedure locates smaller circles, each containing a zero, and the multiplicities are then calculated. Thirdly, a simultaneous Iteration Function is used to accelerate convergence. The Iteration Function is chosen from a class of such functions derived herein to deal with the general case of multiple zeros; various properties of these functions are also discussed. Finally, sample numerical results are given which demon-strate the effectiveness of the algorithm.
Transport theory in anisotropic media
Authors: Williams M. M. R. 
Published: November 2008

AbstractWe develop a theory of particle scattering in anisotropic media. That is, a medium in which the microstructure causes the mean free paths of the particles to become dependent on their direction of motion with respect to some fixed axis. The equation which results is similar to the normal, one-speed Boltzmann transport equation but has cross-sections which are functions of direction. This equation is solved for arbitrary cross-sectional dependence on direction in plane geometry.
Eberlein compacts and spaces of continuous functions
Authors: Hunter Richard J. , Lloyd J. W. 
Published: November 2008

AbstractLet X be a Hausdorff topological space. We consider various locally convex spaces of continuous real valued functions on X and give necessary and sufficient conditions in order that (i) they contain an absolutely convex weakly compact total subset and (ii) they contain an absolutely convex total subset which is an Eberlein compact, when given the weak topology.
Phase-integral treatment of wave reflexion by real potentials
Authors: Lundborg Bengt 
Published: November 2008

AbstractApproximate analytical expressions for the wave reflexion amplitude for transmission above one-dimensional real smooth and slowly varying potentials in the absence of classical turning points are obtained in two different ways by means of the phase-integral method developed by N. Fröman and P. O. Fröman. The wave reflexion is shown to be governed by the complex transition points for the problem. The great accuracy of the formulae obtained, as well as their respective ranges of applicability, is illustrated by applying them, up to the eleventh order phase-integral approximation, to an exactly solvable model, for which the potential represents a monotonic step.
Multi-indexing and multiple clustering
Authors: Melzak Z. A. 
Published: November 2008

1. Simple linear clustering. One of the simplest clustering problems is the following: if n points are taken independently and uniformly at random on the interval I = [0, L], what is the probability Q = Q(n, a, L) that no two points are closer than a? The well-known answerif (n − 1) a ≤ L, and Q = 0 otherwise, can be obtained in a variety of ways. Perhaps the simplest one is the original method of E.C.Molina(5). If x1,…,xn are the abscissas of the n points, then there are n! equiprobable orderings of the xi's consistent with the condition that no two points are closer than a. The probability of each one isLet yi = xi − (i − 1) a for i = 1,…,n, then the above becomeswhich is
Proof of the antithetic-variates theorem for unbounded functions
Authors: Wilson James R. 
Published: November 2008

The method of antithetic variates introduced by Hammersley and Morton (2) is one of the most widely used Monte Carlo techniques for estimating an unknown parameter θ. The basis for this method was established by Hammersley and Mauldon(l).in the case of unbiased estimators with the formwhere each of the variates ξj is required to have a uniform marginal distribution over the unit interval [0,1]. By assuming that n = 2 and that the gj are bounded Borel functions, Hammersley and Mauldon showed that the greatest lower bound of var (t) over all admissible joint distributions for the variates ξj can be approached simply by arranging an appropriate strict functional dependence between the ξj. Handscomb(3) extended this result to the case of n > 2 bounded antithetic variates gj(ξj). In many experiments involving distribution sampling or the simulation of some stochastic process over time, the response functions gry are unbounded. This paper further extends the antithetic-variates theorem to include the case of n ≥ 2 unbounded antithetic variates gj(ξj) each with finite variance.
Hamiltonian and algebro-geometric integrals of stationary equations of KdV type
Authors: Wilson George 
Published: November 2008

In this paper we shall generalize a theorem of Bogoyavl'enskii (2) showing the equivalence of two apparently different families of integrals of the ‘higher stationary Korteweg–de Vries (KdV)’ equations. We recall that the KdV equations form a hierarchy of evolution equationsfor an unknown function u(x, t). The equations can be written in ‘Lax form’ (7)where L is the Schrödinger operator (δ2/δx2) + u, and P+ runs through all the (ordinary) differential operators such that the commutator on the right of (1·1) has order zero.
The attenuation of surface waves by scattering
Authors: Hudson J. A. 
Published: November 2008

AbstractSmall variations in the elastic parameters of a solid medium give rise to scattering of elastic waves and hence to a reduction in their amplitudes. The attenuation coefficients associated with this process have been calculated for surface waves.Expressions have been derived for waves with wavelengths long compared with the mean wavelength of the variation of the elastic parameters in space. The attenuation coefficients were found to be the same as if the attenuating mechanism were a particular type of visco-elasticity.
A quadratic recurrence of Faltung type
Authors: Wright E. M. 
Published: November 2008

We write and x1 = 1. In a recent paper (3), Stein and Everett consider the sequence defined byand investigate whethertends to a limit as n → ∞. The case b = 0 has a combinatorial interpretation (see (1)) and, in (2), they use this to prove that xn → e−1 in this case. Even for positive integral b, the number Sn has no known combinatorial interpretation, but they prove (3) that xn → x under the hypothesis that Sn+1Sn−1 ≥ Sn i.e. that Sn is convex. By computation and induction, they prove this hypothesis for b = k/5, where k = 1, 2,…, 50.