bluuebunny/crossref_metadata_2025_split huggingface dataset:
Lower-bound energies and the Virial theorem in wave mechanics
Authors: Caldow G. L. , Coulson C. A. 
Published: November 2008

ABSTRACTSeveral forms of the lower-bound variational method for the calculation of the eigenvalues in a wave-mechanical problem are considered, and compared; the particular case of the harmonic oscillator being chosen. All forms have certain unsatisfactory features, but some of them are considerably worse than others. One reason why calculations of lower bounds are in general less satisfactory than Ritz-type calculations of an upper bound is shown to be that whereas, in the presence of a scale factor, this latter wave-function satisfies the virial theorem, in none of the lower-bound wave-functions is this true. Similar calculations are reported for the ground state of the helium atom.
A static spherically symmetric solution of the Einstein—Maxwell—Yukawa field equations
Authors: Stephenson G. 
Published: November 2008

The general relativistic field equations for regions of space containing electromagnetic fields but no matter are,where,and κ = 8ΠG/c4 is the gravitational constant. Here giκ is the metric tensor and Fiκ is the electromagnetic field tensor which satisfies Maxwell's equations ifand,where Ai is the electromagnetic four-potential.
A note on the coefficients of Hilbert characteristic functions in semi-regular local rings
Authors: Narita Masao 
Published: November 2008

Let Q be a semi-regular local ring of dimension d, m be its maximal ideal, and q be an m-primary ideal. Then LQ(Q/qn+1), the length of Q-module Q/qn+1, is equal to the characteristic polynomial PQ(q,n) in n for a sufficiently large value of n:where ei = ei(q), i = 0,1,2,…, d are integers uniquely determined by q, called normalized Hilbert coefficients of q according to (1). It was shown in (1) that e1(q) is a non-negative integer, and is equal to zero if and only if q is generated by a system of parameters. We shall prove, in this paper, that e2(q) is also a non-negative integer, and that this non-negativity is not necessarily true for other coefficients. We shall give an example with negative e3(q).
A note on natural maps of higher extension functors
Authors: Oort F. 
Published: November 2008

Hilton and Rees have proved (cf. (1), Theorem 1·3) that every natural mapis induced by a map from A to B (or, Hom (A, B) → Next1,1 (A, B) is surjective). It follows that Ext1 (B, −) and Ext1 (A, −) are naturally isomorphic if and only if A and B are quasi-isomorphic (loc. cit., Theorem 2·6), i.e. if there exist projective objects P, Q and an isomorphism . One can ask whether these theorems remain true for higher extension functors.
Simple proof of a formula due to Racah
Authors: Griffith J. S. 
Published: November 2008

After some manipulation from the general formula, Racah (1,2) obtained a closed formula for his V coefficient in the special case of all three mj- components zero. He also showed that the integral of the product of three Legendre polynomials followed at once from these special V coefficients. In this case the general formula ((2), eq. 10–14) becomeswhereand a, b, c are all non-negative integers. By changing the parameter of summation to (a+ b−c−p) it follows that is zero unless 2g = a+b+c is even (2).
The numerical solution of the heat conduction equation in one dimension
Authors: Wadsworth M. , Wragg A. 
Published: November 2008

AbstractThe replacement of the second space derivative by finite differences reduces the simplest form of heat conduction equation to a set of first-order ordinary differential equations. These equations can be solved analytically by utilizing the spectral resolution of the matrix formed by their coefficients. For explicit boundary conditions the solution provides a direct numerical method of solving the original partial differential equation and also gives, as limiting forms, analytical solutions which are equivalent to those obtainable by using the Laplace transform. For linear implicit boundary conditions the solution again provides a direct numerical method of solving the original partial differential equation. The procedure can also be used to give an iterative method of solving non-linear equations. Numerical examples of both the direct and iterative methods are given.
On the minimum of a positive quadratic form in n ( ≤ 8) variables (verification of Blichfeldt's calculations)
Authors: Watson G. L. 
Published: November 2008

Let f = f(x1, …, xn) be a positive definite quadratic form in n ( ≤ 8) variables; then we consider the old problem of estimating the minimum of f (its least value for integers xi not all 0) in terms of the determinant Δ(f). Normalizing by supposing the minimum to be 1, the known results may be stated aseach inequality being best possible. Further, for each n, all forms for which equality holds in (1) have integral coefficients and are equivalent to each other.
On the behaviour of systems of coupled dynamos
Authors: Allan D. W. 
Published: November 2008

ABSTRACTThe behaviour of a system of two coupled disk dynamos is studied in considerable detail. It is shown that in several cases the stability criteria and general behaviour are closely related to those for the single dynamo studied by Bullard, which are resumed briefly. Extended numerical investigations of the non-linear behaviour of the system are presented: these show that a true ‘reversal of field’ may occur.
The least extension of a closure operator on a sublattice
Authors: Davies Roy O. 
Published: November 2008

Introduction. We recall (see Morgan Ward (3), Birkhoff (1), p. 49) that a closure operator on a complete lattice ℒ is a mapping f from ℒ into ℒ satisfying the conditionsfor all X, Y ∈ ℒ. The set of all closure operators on ℒ is itself a complete lattice with respect to the partial ordering in which f1 ≤ f2 if and only if f1(X) ≥ f2(X) for all X ∈ ℒ for every subset ℱ of and every element X ∈ ℒ we have.
Bending of a simply-supported shallow spherical shell under a uniform line load on a part of a parallel circle
Authors: Mitra D. N. 
Published: November 2008

AbstractThe two fourth-order partial differential equations giving the transverse displacement and the stress functions for shallow shells are reduced to a sixth-order differential equation in one variable which has been solved, and the values of displacements, stress resultants and couples are all expressed in terms of Bessel Functions of imaginary argument. Numerical values of the displacement presented for points on the shell on different meridian lines, showed that on the symmetrical line deflexion is maximum at the point of loading but the point of maximum deflexion shifts gradually towards the pole.
A Banach space that cannot be made into a BIP space
Authors: Olagunju P. A. 
Published: November 2008

1. A Banach space E over the complex field C is said to be a Banach inner-product (BIP) space if there exists a mapping 〈.,.〉 of E × E into C satisfying:(i) 〈x, x〉 ≥ 0 (x ∈ E) with equality only if x = 0;(ii) 〈x, y〉 = 〈y, x〉 (x, y ∈ E);(iii) 〈x + λy, z〉 = 〈x, z〉 + λ〈y, z〉 (x, y, z ∈ E, λ ∈ C);(iv) 〈x, x〉 ≤ k2 ‖x‖2 (x ∈ E),where k is a fixed positive number. Thus 〈.,.〉 is an inner product on E, which induces a norm ‖·‖1 by the relation
The low frequency scalar diffraction by an elliptic disc
Authors: Sleeman B. D. 
Published: November 2008

SummaryThe problem of scalar Dirichlet diffraction of a plane wave by an elliptic disc is discussed. A scheme is given whereby the low frequency expansion of the scattered field may be readily obtained. Series expansions are obtained for the far-field amplitude up to and including the second order in the wave number. The first two terms of the scattering cross-section are also derived.
A quotient ordered space
Authors: McCartan S. D. 
Published: November 2008

It is well known that, in the study of quotient spaces it suffices to consider a topological space (X, ), an equivalence relation R on X and the projection mapping p: X → X/R (where X/R is the family of R-classes of X) defined by p(x) = Rx (where Rx is the R-class to which x belongs) for each x ∈ X. A topology may be defined for the set X/R by agreeing that U ⊆ X/R is -open if and only if p-1 (U) is -open in X. The topological space is known as the quotient space relative to the space ) and projection p. If (or simply ) since the symbol ≤ denotes all partial orders and no confusion arises) is a topological ordered space (that is, X is a set for which both a topology and a partial order ≤ is defined) then, providing the projection p satisfies the propertya partial order may be defined in X/R by agreeing that p(x) < p(y) if and only if x < y in x. The topological ordered space is known as the quotient ordered space relative to the ordered space and projection p.
Path representations of the quantum-mechanical energy-level density
Authors: Bezák Viktor 
Published: November 2008

AbstractThe quantum-mechanical energy-level density g(E) is given as a functional of the quantum-mechanical kernel K(q″, q′, t″ −t′). On taking the kernel K in the Feynman's form, one obtains the function g(E), without solving a Schrödinger equation. As an example, the embedding of a particle in the one-dimensional square well with infinitely high walls is analysed. The functions K(x″, x′, t−t′) and g(E) are represented as sums of terms corresponding to classical paths of different types. By an adequate choice of some terms due to the ‘most important’ paths, one may construct partial sums giving approximations of the function g(E). The utilization of such approximations for estimation of energy levels is demonstrated.
The effect of surface tension on the waves produced by a heaving circular cylinder
Authors: Evans D. V. 
Published: November 2008

AbstractIn this paper the effect of surface tension on water waves is considered. The usual assumptions of the linearized theory are made. A uniqueness theorem is derived for the waves at infinity for a general class of bounded two-dimensional obstacles in a free surface by means of an energy argument. It is shown how the wave amplitude at infinity depends on the prescribed angle at which the free surface meets the normal to the obstacle. The particular case of a heaving half-immersed circular cylinder is considered in detail, and an expression obtained for the velocity potential in terms of a convergent infinite series, the coefficients of which may be computed.
Couette flow of dipolar fluids
Authors: Shirkot K. 
Published: November 2008

AbstractDipolar fluids have been discussed by Bleustein and Green in quite a recent paper (3). They have obtained explicit constitutive equations for homogeneous incompressible linear dipolar fluids. An application of this theory is made here to the problem of Couette flow. Monopolar and dipolar stress fields are determined. Numerical values for velocity and rate of shear for different values of radius in the annulus are given. Graphs are also drawn to compare our results with the classical ones. It is found that the rate of shear does not change sign within the annulus.
Changes of signs of restrictions of Fourier–Stieltjes transforms
Authors: Gaudry G. I. 
Published: November 2008

AbstractOur principal result deals with a function φ defined on the ‘non-negative half’ X+ of a discrete ordered group X, the dual of a compact Abelian connected group G. Suppose φ vanishes outside a countable set S = {χ1, χ2 …} and has the following property: for every t in a set 0 ⊂ (0, 1) of positive Lebesgue measure, the function on X+ defined to be 0 off S and to be rn(t)φ(χn) for χ = χn, where rn is the nth Rademacher function, is the restriction to X+ of a Fourier–Stieltjes transform. We conclude that φ ∈ ℓ2(X+).
The least limit point of the spectrum associated with sinǵular differential operators
Authors: Eastham M. S. P. 
Published: November 2008

Let τ be the formally self-adjoint differential operator denned bywhere the pr(x) are real-valued, , and p0(x) > 0. Then τ determines a real symmetric linear operator T0, given by T0f = τf, whose domain D(T0) consists of those functions f in the complex space L2(0, ∞) which have compact support and 2n continuous derivatives in (0, ∞) and vanish in some right neighbourhood of x = 0 ((7), p. 27–8). Since D(T0) is dense in L2(0, ∞), T0 has a self-adjoint extension T. We denote by μ the least limit point of the spectrum of T. The operator T may not be unique, but all such T have the same essential spectrum ((7), p. 28) and therefore μ does not depend on the choice of T.
A second anticommutant theorem for symmetric ternary algebras
Authors: Joseph A. 
Published: November 2008

AbstractIt is pointed out that many axiomatic systems include a condition of closure under binary composition. In the present development a study is made of the consequences of the weaker requirement of closure under ternary, or three-fold composition. As a preliminary step in the investigation, subsets of groups closed under n-fold multiplication are examined and characterized abstractly. Attention is then focused upon ternary algebras defined as linear subspaces of ℬ (ℋ) (bounded operators on some Hilbert space ℋ) closed under three-fold multiplication. These are shown to satisfy, under suitable conditions, a secondanticommutanttheorem, the analogue of the second commutant theorem which holds for (binary) algebras. It is a result which provides a useful technique for studying such ternary algebras, the structure of which are examined in some detail.
Singularities of two-dimensional exterior solutions of the Helmholtz equation
Authors: Millar R. F. 
Published: November 2008

AbstractA technique for locating possible singularities of two-dimensional ex-terior harmonic functions was discussed in a previous paper. In the present work, the method is generalized to exterior solutions of the Helmholtz equation. Although the procedure deviates in some of its details from the earlier exposition, the conclusions are similar. In particular, it is verified that solutions of the Laplace and Helmholtz equations that satisfy the same Dirichlet boundary condition on the same boundary, possess the same convex hull of singularities. The possibility of extending the method to more general equations is raised.
A synthetic proof of de Longchamps' chain
Authors: Longuet-Higgins M. S. 
Published: November 2008

A famous chain of theorems, due originally to de Longchamps (l) and afterwards rediscovered by Pesci(2), Morley(3) and Grace(4), goes as follows:(1) Given four lines in a plane, the four circumcentres O3 of the triangles formed by omitting each one of the lines in turn lie all on the same circle C4 with centre O4 say.(2)Given five lines in a plane, the centres O4 of the five circles C4 obtained by omitting each of the five lines in turn lie all on the same circle C6 with centre O5 say.And in general(3) Given (n+1) lines in a plane, the (n+1) centres On of the circles Cn+1 formed by omitting each of the lines in turn lie all on the same circle Cn+1 with centre On+1.
Stratified laminar flow of two immiscible fluids
Authors: Packham B. A. , Shall R. 
Published: November 2008

AbstractWe consider the steady co-current flow of two immiscible viscous liquids in a horizontal pipe, the fluid interface being ripple-free and plane. It is shown that if the cross-section of the duct is symmetric with respect to the interface, the velocity distribution may be expressed in terms of two separate pipe-flow solutions. One corresponds to the flow of a single fluid occupying the whole of the pipe, and the second to a similar flow in a pipe whose cross-section coincides with that of the region occupied by one fluid in the two-phase motion. Flow rates and other quantities of engineering interest are evaluated, and several particular flows are discussed.
The product of the distributions x−r and δ(r−1)(x)
Authors: Fisher B. 
Published: November 2008

The product of two distributions f and g on the open interval (a, b), where −∞ ≤ a < b ∞, was defined in (1) as the limit of the sequence {fn·gn} provided this sequence is regular in (a, b), wherefor n = 1, 2, … and ρ is a fixed infinitely differentiable function having the following properties:
The fractional parts of an additive form
Authors: Cook R. J. 
Published: November 2008

Heilbronn (6) proved that for every ε ≥ 0 and N ≥ 1 and every real θ there is an integer x such that,where C(ε) depends only on ε and ∥α∥ is the difference between α and the nearest integer, taken positively. Danicic(1) obtained an analogous result for the fractional parts of nkθ, the proof of this is more readily accessible in Davenport(4). Danicic(2) also obtained an estimate for the fractional parts of a real quadratic form in n variables, and in order to extend this result to forms of higher degree it is desirable to first obtain results for additive forms.
Some extremum principles for magnetohydrodynamic flow in conducting pipes
Authors: Smith P. 
Published: November 2008

AbstractExtremum principles are obtained for the laminar flow of a conducting liquid in a straight pipe of arbitrary cross-section with conducting walls of arbitrary thickness, which is subject to a uniform applied magnetic field. These extrema provide upper and lower bounds for the mass-flow rate in the pipe. These bounds are particularly useful in the construction of asymptotic estimates for the flow rate at high Hartmann number. The method is illustrated by its application to the pipe of circular cross-section and to pipes with thin conducting walls.
The variational derivative of degenerate Lagrange densities
Authors: Pavelle Richard 
Published: November 2008

AbstractThe variational derivative of Lagrange densities which are functions of the metric tensor and its first and second derivatives is considered. This tensor is generally of fourth order in the derivatives of the metric tensor. If derivatives of any order are not present in the variational derivative then the Lagrange density is said to be degenerate in these derivatives. An explicit expression for the variational derivative of Lagrange densities which are degenerate in third and/or fourth derivatives is displayed in tensor form.
Saturated systems of symmetric convex domains; results of Eggleston, Bambah and Woods
Authors: Dumir Vishwa Chander , Khassa Dharam Singh 
Published: November 2008

Let K be a closed, bounded, symmetric convex domain with centre at the origin O and gauge function F(x). By a homothetic translate of K with centre a and radius r we mean the set {x: F(x−a) ≤ r}. A family ℳ of homothetic translates of K is called a saturated family or a saturated system if (i) the infimum r of the radii of sets in ℳ is positive and (ii) every homothetic translate of K of radius r intersects some member of ℳ. For a saturated family ℳ of homothetic translates of K, let S denote the point-set union of the interiors of members of ℳ and S(l), the set S ∪ {x: F(x) ≤ l}. The lower density ρℳ(K) of the saturated system ℳ is defined bywhere V(S(l)) denotes the Lebesgue measure of the set S(l). The problem is to find the greatest lower bound ρK of ρℳ(K) over all saturated systems ℳ of homothetic translates of K. In case K is a circle, Fejes Tóth(9) conjectured thatwhere ϑ(K) denotes the density of the thinnest coverings of the plane by translates of K. In part I, we state results already known in this direction. In part II, we prove that ρK = (¼) ϑ(K) when K is strictly convex and in part III, we prove that ρK = (¼) ϑ(K) for all symmetric convex domains.
The (11, 3)-arcs of the Galois plane of order 5
Authors: Bramwell D. L. , Wilson B. J. 
Published: November 2008

1. It was shown by Barlotti(1) that the number, k, of points on a (k, n)-arc in a Galois plane S2, q, of order q, where n and q are coprime, satisfiesRegular arcs, in which all the points are of the same type have been studied by Basile and Brutti(2) and, for n = 3 by d'Orgeval(4). By means of an electronic computer Lunelli and Sce(5) have enumerated many arcs in Galois planes of low order. The object of this note is to show how the (11, 3)-arcs of S2, 5, none of which is regular, may be described using only geometrical properties.
The asymptotic analysis of canonical problems in high-frequency scattering theory
Authors: Boyd W. G. C. 
Published: November 2008

AbstractThis is the second part of a two-part paper in which a systematic method is advanced for treating separation of variables problems asymptotically as the frequency becomes large. The method assumes an integral representation in which the contour of integration is the real axis. The contour is then deformed in the neighbourhood of the real axis to derive rigorous asymptotic expansions of the field. In this paper the method is applied to scattering in homogeneous media by the circular and parabolic cylinders.
Completely stretched sets
Authors: Lillington J. N. 
Published: November 2008

Introduction. All the sets X considered in this paper will be compact convex subsets of the Euclidean plane R. We shall require the following definitions. For x, y, z ∈ R let L(x, y, x) denote the minimum of all the lengths of all connected sets containing x, y and z. Define the stretch of x with respect to X to beand the stretch of X to be
On a separation theorem of Rogers
Authors: Davies Roy O. 
Published: November 2008

In (1) (where references and terminology may be found), C. A. Rogers has proved the following surprising theorem.Letbe a class of sets, all Souslin-in a topological space X. If A is a Souslin-set and B is an analytic set that does not meet A, then there is a Borelian-set C with A ⊂ C and C ∩ B = Ø
Differential games with optional stopping
Authors: Kalton N. J. 
Published: November 2008

Consider a differential game of survival governed by the differential equationin , with pay-offwhere tF is the entry time of the trajectory (t, x(t)) into a given terminal set F. Under suitable conditions on f, g, h and the terminal set F, it was shown in (3) that the question of existence of value of such a game can be approached by considering a certain pair of partial differential equations called the Isaacs-Bellman equations.
Bounds for surface solitary waves
Authors: Keady G. , Pritchard W. G. 
Published: November 2008

In these notes we give proofs of some properties of surface solitary waves. Assuming the existence of solitary-wave solutions to the nonlinear boundary-value problem (P) defined below, it is shown (i) that the wave is a wave of elevation alone, and (ii) that at large distances it is asymptotic to a uniformsupercriticalstream (i.e. the Froude number, wherecis the speed of the stream,his its depth, andgis the gravity constant).We also deduce a number of inequalities relatingF2to a/h, whereais the maximum displacement of the free surface from its value at infinity. In particular, it is shown for the wave of greatest height that 1·480 <F2< 2.
Cesàro iterations of Hausdorff matrices
Authors: Kuttner B. 
Published: November 2008

1. Jurkat and Peyerimhoff(3) (or see (4), pages 50–56) have given a definition of a function, f(A), of an infinite matrix A = (ank). In particular, they considered Aα, where a is any real constant. This is defined wherever A is normal; that is to say, when ank = 0 for k > n, and when, for all n, ann ±0. They also defined the ‘Cesàro iteration’ Aα. This is obtained by defining a matrix B = (bnn) by(so that every diagonal element of B is 1), forming Bα, and then dividing each row by the row sum; that is to say, if Bα = (b(α)nk), we takewhereThe definition requires, of course, that ≠ 0.
A theorem on bounded synthesis
Authors: Galanis E. 
Published: November 2008

LetGbe a locally compact Abelian group.DEFINITION 1. A compact subset K ⊂ G is called Kroneclcer set if for every continuous function f on K of modulus identically one (|f(x)| = 1, ∀x ∈ K) and for every ε 0 there exists x ∈ Ĝ such that