bluuebunny/crossref_metadata_2025 huggingface dataset:
The Parseval formulae for monotonic functions. II
Authors: Edmonds Sheila M.
Published: November 2008

It is clear from § 1 that we shall have quite a large number of theorems to consider, as we may distribute the conditions of monotony and existence in a good many different ways. It is therefore convenient to collect together the various results for series and for transforms in tabular form, for reference and comparison; this is done at the end of this section.

The transform table is very simple. The theorems contained in it have been discussed at length in M.F.(I); they may be summarized by saying that (1) holds in all the cases considered, except perhaps in the consine case of section [B] where the truth is not known. The series results are apparently more complicated as well as more numerous, since there are extra conditions to be imposed in [A, 1], [B, 1], [C′, 1] and [C, 2]. However, these are certainly satisfied if we suppose f and g to be positive whenever they are given to be monotonic (a condition which automatically holds in the transform theorems, since in these the monotonic functions tend to zero at infinity). If we confine our attention to this case, the series results are what we should expect from the transform ones, except that the difficulty in section [B] for transforms is not reproduced for series in [B, 1] or [B′, 1]; it is reproduced in [B, 2].
Supersonic flow over an inclined wing of zero aspect ratio
Authors: Stewartson K.
Published: November 2008

An asymptotic expression is found for the lift distribution on a long, narrow, laminar wing, at incidence in a supersonic stream. The approximations of the linearized potential theory are used.
High-speed flow of a gas past an approximately elliptic cylinder
Authors: Levey H. C.
Published: November 2008

In this paper, a family of exact solutions of the problem of two-dimensional flow of a compressible perfect fluid about a cylinder is found, the solutions being generalized from those for the flow of an incompressible fluid about an elliptic cylinder of arbitrary eccentricity and angle of attack. The circulation is taken to be zero and the speed of the fluid at infinity subsonic. This analysis is an application of the general theory given by T. M. Cherry (1, 2); it was done to exhibit the details of the analysis for a flow other than that corresponding to the low-speed flow past a circular cylinder.
The characterization of quantum-mechanical operators
Authors: Cooper J. L. B.
Published: November 2008

It is well known that the operators mainly employed in quantum theory are hermitian; it is less well known amongst physicists that they are required, in addition, to be self-adjoint. This is essential for the validity of the result known in quantum theory as the representation theorem and in the mathematical theory as the resolution of the identity. The purpose of this paper is to show that the self-adjoint operators can be characterized by a condition which is nearer to having a physical significance than those given in the literature.
A boundedness theorem for non-linear differential equations of the second order
Authors: Reuter G. E. H.
Published: November 2008

1. This paper deals with the differential equation

(dots denoting derivatives with respect to t), where for large x the ‘restoring force’ term g(x) has the sign of x and the ‘damping factor’ kf(x) is positive on the average. It will be shown that every solution of (1) ultimately (for sufficiently large t) satisfies

with B independent of k. The conditions on f(x), g(x) and p(t) (stated in §§ 2, 3) are rather milder than those assumed by Cartwright and Littlewood (1, 2) and Newman (3) in proving similar results.
On the propagation of disturbances from moving sources
Authors: Walters A. G.
Published: November 2008

The concept of the Green's vibrational function given in an earlier paper by the author is used to obtain a general expression for the disturbance from a point source. The potential due to transient sources of sound moving with subsonic and supersonic velocities is derived from this. It is found that the Doppler effect for a supersonic source differs from that for a subsonic source. In the former case it is found that two frequencies are heard simultaneously from a source emitting a note of one frequency.

The theory is applied to determine some solutions of the two dimensional equation of supersonic, irrotational compressible flow, corresponding to the flow around an aerofoil taking into consideration the entropy changes at the shock wave.
Spectral representation of linear transformations in ω
Authors: Williamson J. H.
Published: November 2008

1. In this paper we consider the following problems, for continuous linear transformations in the space ω, whose points are arbitrary sequences of complex numbers, with the usual topology:

(a) the establishment of a spectral representation theorem; and

(b) the general description of the spectrum.
Wave reflexion near a wall
Authors: Robinson A.
Published: November 2008

The field of flow due to a shock wave or expansion wave undergoes a considerable modification in the neighbourhood of a rigid wall. It has been suggested that the resulting propagation of the disturbance upstream is largely due to the fact that the main flow in the boundary layer is subsonic. Simple models were produced by Howarth, and Tsien and Finston, to test this suggestion, assuming the co-existence of layers of uniform supersonic and subsonic main-stream velocities. The analysis developed in the present paper is designed to cope with any arbitrary continuous velocity profile which varies from zero at the wall to a constant supersonic velocity in the main stream. Numerical examples are calculated, and it is concluded that a simple inviscid theory is incapable of giving an adequate theoretical account of the phenomenon. The analysis includes a detailed discussion of the process of continuous wave reflexion in a supersonic shear layer.
Non-measurable interpolation sets
Authors: Noble M. E.
Published: November 2008

In an earlier paper the problem of the interpolation of an integral function f(z) in terms of the values taken on a sequence zn was studied by means of generalized σ-functions with zeros at zn. The purpose of the present paper is to show that the methods used in I can be modified to deal with functions of finite order and type in the angle Sα, | arg z | ≤ α. There is even a considerable saving since, for an Sα with α < π, we can escape from the awkward convergence condition Dρ (ii) of I. Most of our results, and to a great extent also their proofs, are direct generalizations of well-known results for the lattice points m + in (cf. the references at the end of I, particularly (2), (5), (6), (14), (18)).
Parametric surfaces
Authors: Reifenberg E. R.
Published: November 2008

1. In this paper I investigate the tangential properties of parametric surfaces, leading to the main result that the Lebesgue area is the Hausdorff two-dimensional measure of the set of points of the surface where an approximate tangential plane exists.
The minimum of a non-homogeneous binary cubic form
Authors: Chalk J. H. H.
Published: November 2008

1. Let f(x, y) = ax3 + bx2y + cxy2 + dy3 be a binary cubic form with real coefficients. According as the discriminant D = b2c2+ 18abcd − 4ac3 − 4db3 − 27a2d2 satisfies D ≥ 0 or D < 0, the form f(x, y) can be factorized into either three real, or one real and two conjugate complex, linear factors. A form f1(x, y) is said to be equivalent to f2(x, y) if there is an integral unimodular substitution on the variables x, y transforming f1(x, y) into f2(x, y). Thus f2(x, y) is equivalent to f1(x, y) and, in fact, the set of all forms with given discriminant can be divided into classes of equivalent forms. Davenport (3) has specified a member of each class which satisfies
Note on non-homogeneous linear forms
Authors: Barnes E. S.
Published: November 2008

Let θ be an algebraic number of degree n, with conjugates θ(1), …, θ(n), where θ(1), …, θ(r) are real and θ(r+j), θ(r+s+j) are complex conjugates for j = 1, …, s. [Here r ≥ 0, s ≥ 0, r + 2s = n.] Let ω1, …, ωn be a basis for the integers of k(θ), and set
On the decrease of velocity with depth in an irrotational water wave
Authors: Longuet-Higgins M. S.
Published: November 2008

The following theorems are proved for irrotational surface waves of finite amplitude in a uniform, incompressible fluid:

(a) In any space-periodic motion (progressive or otherwise) in uniform depth, the mean square of the velocity is a decreasing function of the mean depth z below the surface. Hence the fluctuations in the mean pressure increase with z.

(b) In any space-periodic motion in infinite depth, the particle motion tends to zero exponentially as z tends to infinity. The pressure fluctuations at great depths are therefore simultaneous, but they do not in general tend to zero.

(c) In a progressive periodic wave in uniform depth the mass-transport velocity is a decreasing function of the mean depth of a particle below the free surface, and the tangent to the velocity profile is vertical at the bottom. This result conflicts with observations in wave tanks, and shows that the waves cannot be wholly irrotational.

(d) Analogous results are proved for the solitary wave.
On approximate solutions of linear differential equations
Authors: Jeffreys Harold
Published: November 2008

Asymptotic approximations of Green's type to solutions of differential equations are studied, with special reference to the uniformity of the approximation given by the first term. In extension to the complex variable this is found to require substantial restrictions on the region considered. An anomaly previously noticed is traced to non-uniformity of approximation. The case where the coefficient χ0 has a simple zero and χ1 is not zero is treated by a simple method.
On Bose-Einstein Condensation
Authors: Landsberg P. T.
Published: November 2008

This paper contains a proof that the description of the phenomenon of Bose-Einstein condensation is the same whether (1) an open system is contemplated and treated on the basis of the grand canonical ensemble, or (2) a closed system is contemplated and treated on the basis of the canonical ensemble without recourse to the method of steepest descents, or (3) a closed system is contemplated and treated on the basis of the canonical ensemble using the method of steepest descents. Contrary to what is usually believed, it is shown that the crucial factor governing the incidence of the condensation phenomenon of a system (open or closed) having an infinity of energy levels is the density of statesN(E) ∝Enfor high quantum numbers, a condition for condensation beingn> 0. These results are obtained on the basis of the following assumptions: (i) For large volumesV(a) all energy levels behave likeV−θ, and (b) there exists a finite integerMsuch that it is justifiable to put for thejth energy levelEj=c V−θand to use the continuous spectrum approximation, wheneverj ≥ Mc θ τ are positive constants, (ii) All results are evaluated in the limit in which the volume of the gas is allowed to tend to infinity, keeping the volume density of particles a finite and non-zero constant. The present paper also serves to coordinate much of previously published work, and corrects a current misconception regarding the method of steepest descents.
The inhomogeneous minima of complex cubic norm forms
Authors: Swinnerton-Dyer H. P. F.
Published: November 2008

1. Let K1, K2, K3 be three conjugate* cubic algebraic number fields, of which we shall take K1 to be real and K2, K3 to be complex conjugate. Let ωi1, ωi2, ωi3 the conjugate basis for the integers of Ki. Write

so that ξi runs through the integers of Ki as the xj run independently through the rational integers, and the ξi (i = 1, 2, 3) are conjugate. The determinant of the ξi, regarded as linear forms in the xj, is ± √d, where d is the common discriminant of the fields.
An extension of the Ahlfors distortion theorem
Authors: Huckemann F.
Published: November 2008

1. The conformal mapping of a strip domain in the z-plane on to a parallel strip— parallel, say, to the real axis of the ζ ( = ξ + iμ)-plane—brings about a certain distortion. More precisely: consider a cross-cut on the line ℜz = c joining the two sides of the frontier of the strip domain (in these introductory remarks we suppose for simplicity that there is only one such cross-cut on that line), and denote by ξ1(c) and ξ2(c) the lower and upper bounds of ξ on the image in the ζ-plane. The theorem of Ahlfors (1), now classical, states that

provided that

where a is the width of the parallel strip and θ(c) the length of the cross-cut.
A remark on quantum-mechanical perturbation theory
Authors: Kemmer N.
Published: November 2008

In the course of preparing lectures on elementary quantum theory the author developed a way of presenting time-independent perturbation theory that seems to have some advantages over more familiar presentations. There is nothing fundamentally new in the method, and if any claim of greater power can be made for the new version, it is at best for situations in which high-order perturbations are of interest. Quantum-field theorists will no doubt recognize the relation of this presentation to much that has been done in their subject recently.
Compact linear operators of volterra type
Authors: Ringrose J. R.
Published: November 2008

1·1. Throughout this paper, all functions considered are assumed to be measurable, and an equation between two functions indicates equality almost everywhere. When the space Lp(a, b) is considered, the interval (a, b) may be finite, semi-infinite, or infinite.
The edge conditions and field representation theorems in the theory of electromagnetic diffraction
Authors: Heins A. E. , Silver S.
Published: November 2008

In 1897 Rayleigh(14) pointed out that it is possible to obtain a family of solutions to certain problems in diffraction theory, notably those involving plane obstacles, by the simple operation of differentiation of some one solution. This, of course, alters the nature of the solution, especially so in the neighbourhood of a sharp edge. For, at an edge the parent solution may be finite and yet its derivative can become infinite. Further, such a differentiated solution may produce an infinite total energy in the neighbourhood of the edge. It is, therefore, natural to ask what conditions are required to make the solutions of these problems physically and mathematically acceptable and thus define the boundary-value problem uniquely.
A note on valuations associated with a local domain
Authors: Rees D.
Published: November 2008

The purpose of the present note is to prove the following two theorems:

Theorem 1. Let Q be an equicharacteristic local domain with maximal ideal m. Let a be any ideal of Q. Then the intersection of all integrally closed m-primary ideals of Q which contain a is the integral closure ā of a.

Theorem 2. If Q is as above, and if S denotes the set of valuations on the field of fractions F of Q which are associated with Q, then the intersection of the valuation rings belonging to valuations in S is the integral closure of Q.
On relations between certain intrinsic topologies in partially ordered sets
Authors: Ward A. J.
Published: November 2008

In the first part of this paper we consider partially ordered sets for which the intrinsic ‘interval topology’ ((1), p. 60) is Hausdorffian. The main result of this section is the following: the interval topology of a lattice is Hausdorffian if and only if convergence with respect to the interval topology is equivalent to strong o*-convergence. This may be regarded as an answer either to Birkhoff's problem 23 ((1), p. 62) or to his problem 25 ((1), p. 64). In the case of a complete lattice, we have an alternative formulation. The interval topology of a complete lattice is Hausdorffian if and only if every filter (or, alternatively, every directed net) in the lattice has an o-convergent refinement. This condition may be regarded as a strong type of compactness.
The laminar boundary-layer equation: a method of solution by means of an automatic computer
Authors: Leigh D. C. F.
Published: November 2008

A method, very suitable for use with an automatic computer, of solving the Hartree-Womersley approximation to the incompressible boundary-layer equation is developed. It is based on an iterative process and the Choleski method of solving a simultaneous set of linear algebraic equations. The programming of this method for an automatic computer is discussed. Tables of a solution of the boundary-layer equation in a region upstream of the separation point are given. In the upstream neighbourhood of separation this solution is compared with Goldstein's asymptotic solution and the agreement is good.
The correction of observational data for instrumental band width
Authors: Kahn F. D.
Published: November 2008

The readings Φ(x) of an instrument with the apparatus functiong(x) are related to the true valuesf(x)of the observed function by the equation

A three-point interpolation operator can be chosen to simulate the spreading action of a class of functionsg(x). Under certain conditions the interpolation operator has an exact inverse which, when applied to Φ(x), gives a function which is a better approximation tof(x). The accuracy obtained is discussed, and the effect of the method on the standard errors in the measurement is found.
The use of economized polynomials in mathematical tables
Authors: Clenshaw C. W. , Olver F. W. J.
Published: November 2008

The advantages of using polynomial approximations for the purpose of constructing interpolable numerical tables on punched cards have been pointed out by Sadler (7). The object of this paper is to demonstrate the value of this method for ordinary published tables.
The ergodic properties of non-homogeneous finite Markov chains
Authors: Hajnal J. , Bartlett M. S.
Published: November 2008

The paper deals with the behaviour of finite non-homogeneous Markov chains having regular transition matrices. Two types of ergodic behaviour are distinguished, and sufficient conditions are given for each type. Chains with commutative transition matrices are specially investigated. By means of expansions in terms of the characteristic roots several types of behaviour of chains which are not ergodic are studied.
A characterization of frontier
Authors: Gravett K. A. H. , Scott D. B.
Published: November 2008

Let X be a set and let f be a mapping of P(X), the set of all subsets of X, into P(X). Conditions are given which are necessary and sufficient for there to exist a topology on the set X with respect to which, for all S ∈ P(X), the frontier of S is fS.
The anomaly of convex bodies
Authors: Woods A. C.
Published: November 2008

1. Let (x1, x2, …, xn) be Cartesian coordinates in Rn. We identify the point X = (x1, x2, …, xn) with these coordinates with the vector X of components x1, x2, …, xn and use the usual vector notation.
On totally complex quartic fields with small discriminants
Authors: Godwin H. J.
Published: November 2008

A table of totally complex quartic fields, with small discriminants, which possess quadratic subfields was given by Delone and Faddeev(1); the purpose of the present paper is to give a table of such fields which have no subfield. The method used is based on an idea used recently (2) in the case of totally real quartic fields and depends on a special case of the following theorem, which is stated in full generality.
Commutative products on spheres
Authors: James I. M.
Published: November 2008

Consider a topological n–sphere Sn, where n ≥ 1. By a commutative product I mean a continuous function f: Sn × Sn → Sn such that
Green's functions for thin isotropic plates containing holes
Authors: Bassali W. A. , Dawoud R. H.
Published: November 2008

This paper is concerned with the small transverse displacement of an infinite thin plane isotropic plate due to the application of a transverse force applied at an arbitrary point of the plate. The plate has its outer edge free, and is clamped along and bounded internally by a closed curve that can be mapped onto the unit circle by means of a polynomial. Three polynomials are considered and in each of these cases the deflexion is obtained in finite terms. Circular and elliptic holes as well as curvilinear polygonal holes are included as special cases.
On a theorem of van der Corput on uniform distribution
Authors: Coles W. J.
Published: November 2008

Van der Corput has shown (2), using a general criterion of Weyl (1), that a necessary and sufficient condition, that a sequence of points Pn = (αn, βn) (n = 1, 2,…) in two-dimensional space be uniformly distributed modulo 1, is that for all pairs of integers (u, v) other than u = v = 0 the one-dimensional sequence (uαn + vβn) (n = 1, 2,…) is uniformly distributed modulo 1. The object of this paper is to give a quantitative form to the sufficiency part of this qualitative criterion.
On whether elastic wave surfaces possess cuspidal edges
Authors: Musgrave M. J. P.
Published: November 2008

Consideration of the direction of the normal at any point on a continuous sheet of a surface yields a sufficient condition for the existence of parabolic points on that sheet. This condition has been used to derive some simple inequalities between elastic constants, whose fulfilment determines the existence of parabolic points upon the inverse surfaces of media of orthorhombic, tetragonal, cubic or hexagonal symmetry; in virtue of the polar reciprocal relation between inverse and wave surfaces, the existence of cusp points on the latter is thereby simultaneously established. It is also pointed out that conditions for external conical refraction may prevail in hexagonal media.
On the numerical solution of a class of partial differential equations
Authors: Douglas A. S.
Published: November 2008

For a suitable choice of E*, the solution as t becomes large of the equation

where Y is given independent of t over the space boundaries, tends to the ground state solution of the wave equation

with the same boundary conditions on P as on Y. As a preliminary to using this relation to solve wave equations in more than one variable, the solution of the equation

has been studied. Methods of numerical solution are discussed, and the convergence of these is examined. Some practical experiments using an electronic computing machine are described.
Problems concerning the bending of isotropic thin elastic plates subject to various distributions of normal pressures
Authors: Bassali W. A. , Swinnerton-Dyer H. P. F.
Published: November 2008

Within the limitations of the small-deflexion plate theory, complex variable methods are used in this paper to obtain an exact solution for the problem of a thin circular plate supported at several interior or boundary points, and subjected to a certain normal loading spread over the area of an eccentric circle, the boundary of the plate being free. The load considered includes as a special case a linearly varying load over the circle and, as the radius of the loaded circle tends to zero, this load can be made to tend to a couple nucleus at its centre. As limiting cases the procedure adopted provides us with solutions appropriate to a circular plate, an infinitely large plate and a half-plane having free boundaries and acted upon by any normal system of concentrated forces and concentrated couples in equilibrium. Formulae for the moments, shears and deflexions relating to special examples are worked out in detail.
Transverse bending of infinite and semi-infinite thin elastic plates. III
Authors: Bassali W. A. , Nassif M. , Swinnerton-Dyer H. P. F.
Published: November 2008

Within the restrictions of the classical plate theory, complex variable methods are used in this paper to develop an exact expression for the transverse displacement of an infinitely large isotropic plate having a free outer boundary and elastically restrained at an inner circular boundary, the plate being subjected to a general type of loading distributed over the area of a circle. The limiting case of a half-plane clamped along the straight edge and acted upon normally by the same loading is also considered.
A theorem of continuation for functions of several complex variables
Authors: Taylor J. G.
Published: November 2008

In the last few years it has been found useful to apply known theorems in the theory of functions of several complex variables to solve problems arising in the quantum theory of fields (11). In particular, in order to derive the dispersion relations of quantum field theory from the general postulates of that theory it appears useful to apply known theorems on holomorphic continuation for functions of several complex variables ((2), (10)). The most important theorems are those which enable a determination to be made of the largest domain to which every function which is holomorphic in a domain D may be continued. This domain is called the envelope of holomorphy of D, and denoted by E(D). If D = E(D) then D is termed a domain of holomorphy. E(D) may be defined as the smallest domain of holomorphy containing D. Only in the special cases that D is a tube, semi-tube, Hartogs, or circular domain has it been possible to determine the envelope of holomorphy E(D) ((3), (7)). An iterative method for the computation of envelopes of holomorphy has recently been given by Bremmerman(4). It is also possible to use the continuity theorem (1) in a direct manner, though in most cases this is exceedingly difficult.
Which distributive lattices are lattices of closed sets?
Authors: Papert S.
Published: November 2008

1. An elegant theorem due to Tarski states that a completely distributive complete Boolean algebra is isomorphic with a lattice of sets, and in fact the lattice of all the subsets of some aggregate. The obvious generalization of the question underlying this theorem is to ask whether one can pick out by means of a distributivity condition those lattices (not necessarily Boolean algebras) which are isomorphs of lattices of sets. The answer is no. The real numbers with their natural order form a complete lattice which satisfies the strongest possible distributivity conditions and yet is not iso-morphic with any lattice of sets.
Sound pulses in a conducting medium
Authors: Friedlander F. G.
Published: November 2008

1. Introduction. The subject of this paper is the propagation of small disturbances in a compressible fluid which is also a conductor of electricity, in the presence of a magnetic field. The characteristic feature of the problem is that it combines sound waves and magneto-hydrodynamic waves. The theory is based on linearized equations which generalize the equations of motion of the theory of sound. The disturbances with which it deals might therefore be called magneto-acoustic waves. Energy dissipation by viscosity, heat conduction and Joule heat is neglected, and so is the displacement current. Even so, a hyperbolic system of seven partial differential equations of the first order is obtained, whose solution requires analysis of considerable complexity.
On a Result in Polynomial Rings
Authors: Wall C. T. C.
Published: November 2008

Let denote the polynomial algebra over the integers in countably many variables ui (i ≥ 1). Let ∂ be the derivation of defined on the generators by. Thus if is graded by dim ui=i, then∂ is homogeneous, of degree − 1. The result is that ∂ is onto, and that its kernel is a polynomial algebra in , where is homogeneous of degree i and the coefficient of ui in is p if i is a power of a prime p, and 1 if i is not a prime power.
Blowing up Chern Classes
Authors: Porteous I. R.
Published: November 2008

The behaviour of the Chern classes or of the canonical classes of an algebraic variety under a dilatation has been studied by several authors (Todd (8)–(11), Segre (5), van de Ven (12)). This problem is of interest since a dilatation is the simplest form of birational transformation which does not preserve the underlying topological structure of the algebraic variety. A relation between the Chern classes of the variety obtained by dilatation of a subvariety and the Chern classes of the original variety has been conjectured by the authors cited above but a complete proof of this relation is not in the literature.
A uniqueness theorem for the exponential series of Herglotz
Authors: Verblunsky S.
Published: November 2008

If {λ;n}, {bn} are sequences of complex numbers, and we consider the series ∑bn exp (−λnx), given as convergent in (0, 1) (i.e. the open invertal (0,1)) to f(x)∈L, then, writing

(if λn = 0 the corresponding term is ½bnx2) where the series is supposed is to be uniformly convergent in (0, 1), we have

for 0<h<h(x).If we know that the second member of (2) tends to f(x) as h → +0, it will follow that F(x) is a repeated integral of f(x) ((1), 671). If there is a sequence {φv(x)} of integrable functions with the property that

then, on multiplying (1) by φv(x) and integrating over (0,1), we obtain a formula for bv in terms of F(x). On integrating by parts twice, bv will be expressed in terms of f(x), and this will constitute a uniqueness theorem for the series ∑bn exp (−λnx).
An extension of the Newtonian law of gravitation
Authors: Bastin J. A.
Published: November 2008

An attempt is made to relate such diverse phenomena as kinetic energy, rest mass energy, the size of the universe, the velocity of propagation both of light and gravitation, and the recession of the galaxies. The correlation is made by considering an extension of the Newtonian gravitational law which covers a particular simple case when the two attracting bodies are in relative motion. The extension is treated as postulatory, although in the last section using the idea of gravitational flux, the assumed gravitational law will be shown to be the simplest of a number of possible extensions to the Newtonian law. The paper implies a new approach to special relativity, and it is therefore hoped to treat aspects of the work in greater detail subsequently.
Rational Euler Characteristics
Authors: Wall C. T. C.
Published: November 2008

Let ℒ be the class of discrete groups G which admit a finite complex X as classifying space. On ℒ we may define the Euler characteristic χ(G) = χ(X); that this is independent of X follows from the homological interpretation of characteristics.
On Chern Characters and the Structure of the unitary group
Authors: Adams J. F.
Published: November 2008

The purpose of this paper is twofold. In order to state our first aim, let U denote the ‘infinite’ unitary group, and let BU be a classifying space for U. Then Bott (2),(3) has shown that we propose to investigate the ‘Postnikov system’ of BU.
Bordism and Cobordism
Authors: Atiyah M. F.
Published: November 2008

In (10), (11) Wall determined the structure of the cobordism ring introduced by Thom in (9). Among Wall's results is a certain exact sequence relating the oriented and unoriented cobordism groups. There is also another exact sequence, due to Rohlin(5), (6) and Dold(3) which is closely connected with that of Wall. These exact sequences are established by ad hoc methods. The purpose of this paper is to show that both these sequences are ‘cohomology-type’ exact sequences arising in the well-known way from mappings into a universal space. The appropriate ‘cohomology’ theory is constructed by taking as universal space the Thom complex MSO(n), for n large. This gives rise to (oriented) cobordism groups MSO*(X) of a space X.
Note on the Preceding Paper
Authors: Burgess D. C. J.
Published: November 2008

It is worth remarking that if e, f are two primitive idempotents in a semigroup S which commute, then: (i) if S has a zero 0, either e = f or ef = 0; (ii) if S has no zero, e = f. Clearly the product ef is under each of the primitive idempotents e and f.
An infinite Legendre integral transform and its inverse
Authors: Clemmow P. C.
Published: November 2008

A consideration of Legendre's equation, of order zero, expressed in terms of the angular variable θ, leads heuristically, but methodically, to an infinite integral representation of the delta function in terms of two independent non-periodic solutions Ev−½(θ), E−v−½(θ). This result yields an integral transform and inverse formula, where in the former the kernel is E−v−½(θ) sin θ and the integration over θ runs from − ∞ to ∞. By rigorous contour integral methods, the inverse is verified when the original function is a rectangular pulse, and the standard series expansion in terms of Legendre polynomials is recovered when the original function is even and of period 2π in θ. Analogies with the theory of Fourier transforms are emphasized.
The idea of size in large-scale physics
Authors: Bastin E. W.
Published: November 2008

The current controversy over the interpretation of radio-stellar counts shows, in my view, that in large-scale physics as well as on the very small scale the idea of size, as applied to distributions of matter, has broken down. I shall expand this view.
Landau damping for non-Maxwellian distributions
Authors: Hudson John F. P.
Published: November 2008

It has been shown by Landau(8) that in a uniform plasma with a Maxwelliau velocity distribution longitudinal waves are damped. Penrose(9) has considered more general distributions and developed stability criteria. In particular, if the velocity distribution function has only one maximum, then the plasma is stable. In this paper the values of the Landau damping for some stable single maximum distributions are compared in order to assess the sensitivity of Landau damping to the form of the distribution function, and to investigate the usefulness of approximating to the Landau damping in a Maxwellian plasma by using an algebraically simpler velocity distribution function.

It is shown also that, for at least some velocity distributions having a maximum particle velocity, the behaviour of the plasma can no longer be described in terms of exponential damping, since there will be longer lasting perturbations with phase velocity equal to the maximum particle velocity.
On Certain Classes of Locally Soluble Groups
Authors: Roseblade J. E.
Published: November 2008

A group G is called locally soluble if every finitely generated subgroup of G is soluble. Terms like ‘locally nilpotent’ and ‘locally finite’ are defined similarly.
Absorption Probabilities for Sums of Random Variables Defined on a Finite Markov Chain
Authors: Miller H. D.
Published: November 2008

This paper is essentially a continuation of the previous one (5) and the notation established therein will be freely repeated. The sequence {ξr} of random variables is defined on a positively regular finite Markov chain {kr} as in (5) and the partial sums and are considered. Let ζn be the first positive ζr and let πjk(y), the ‘ruin’ function or absorption probability, be defined by The main result (Theorem 1) is an asymptotic expression for πjk(y) for large y in the case when , the expectation of ξ1 being computed under the unique stationary distribution for k0, the initial state of the chain, and unconditional on k1.
A Class of Conserved Tensors in an Arbitrary Gravitational Field
Authors: Collinson C. D.
Published: November 2008

Four-index tensors with all possible terms either (a) quadratic in the Riemann curvature tensor, or (b) linear in its second derivatives, with coefficients constructed from products of metric tensors, are enumerated for the case of Riemannian space-time. All linear combinations of these which have zero divergence, and which might therefore be associated with the super-energy in gravitational theory, are found by elementary methods. The only independent such combination reduces in empty space-time to the tensor previously found by Bel and Robinson.
Diffraction by a Screen Causing Large Random Phase Fluctuations
Authors: Mercier R. P.
Published: November 2008

The propagation of a scalar wave is studied, after it has suffered changes of phase in its passage through an irregular refracting medium, such as the ionosphere. The rms phase fluctuations may be large; i.e. greater than one radian. The paper is in three parts. In the first, the receiver is at a great distance from the screen (Fraunhofer diffraction), and a formula is found for all even order moments of the amplitude, and for the correlation function of the even powers of the amplitude. The probability distribution of the amplitude is proved to be a Rice distribution. The multiple integrals expressing the moments of the amplitude reduce to sums of correlations of the original phase fluctuations amongst points over the screen, which are evaluated using combinatorial arguments.

In the second part the variance of the square of the amplitude is calculated numerically for points near the screen (Fresnel diffraction). When the rms phase fluctuation is greater than one the variance is maximum near the point where the irregularities in the screen would bring the rays to a focus.

In the third part radio star scintillation and scatter propagation are discussed in the light of the theory.
Equally Inclined Spheres
Authors: Mauldon J. G.
Published: November 2008

If five spheres in 3-space are such that each pair is inclined at the same non-zero angle θ, then where b1, …, b5 (the ‘bends’ (1) of the spheres) are the reciprocals of their radii. To prove this result, establish a system of rectangular cartesian coordinates (x, y, z) and let the spheres have centres (xi, yi, zi) and radii , where i = 1,…, 5. Then for x5, y5, z5, r5 we have the equations which, on subtraction, yield three linear equations and one quadratic equation. Solving the three linear equations for x5, y5, z5 and substituting, we see that the required relation is algebraic (indeed quadratic) in r5 and hence in b5. Since it is also symmetric in b1,…, b5, it follows that it can be expressed as a polynomial relation in the elementary symmetric functions p1, …, p5 in b5, …, b5.
On the Stability and Boundedness of Differential Systems
Authors: Lakshmikantham V.
Published: November 2008

Consider the differential systems

where A(t), g(t, y) and g(t, y) are operators acting in the real Banach space E, A(t) is an unbounded, closed, linear operator for each t in 0 ≤ t < ∞ and x0, y0 belong to the domain of definition of the operator A (t0). Let ‖x‖ denote the norm of an element x ε: E and R(λ, t) the resolvent of A(t). Here and in the following the prime denotes the right-hand derivative.
A class of integral equations
Authors: Williams W. E.
Published: November 2008

It was pointed out by Copson(1) in 1947 that the solution of the integral equation for the electrostatic problem for a circular disc could be reduced to the solution of Abel integral equations and hence a solution obtained in a fairly elementary manner. This result was obtained independently by Lebedev(2), who also obtained a similar result for the electrostatic problem for the spherical cap. The solution for the spherical cap was also obtained independently by Collins (3). In view of the relative simplicity of the approach it seems to be of interest to examine whether there exist other integral equations which can be treated in a similar fashion. In the present paper one such class of integral equations will be considered.
On axially symmetric waves
Authors: Craggs J. W.
Published: November 2008

The wave equation with axial symmetry is reduced, by the assumption of dynamic similarity, to an equation in two variables, r/t and θ A basic solution, having some of the attributes of a source, is introduced, and it is shown that, by use of this solution and a suitable contour integral representation, the solution of any appropriate boundary-value problem may be reduced to the determination of an analytic function, with boundary conditions given by an integral equation.

The method of solution is illustrated by reference to some basic problems in the theory of linearized compressible flow.
General transition probabilities for finite Markov chains
Authors: Neuts Marcel F.
Published: November 2008

We consider a stationary discrete-time Markov chain with a finite number m of possible states which we designate by 1,…,m. We assume that at time t = 0 the process is in an initial state i with probability (i = 1,…, m) and such that and .
The partition of energy in cosmical physics
Authors: Tupper B. O. J.
Published: November 2008

Eddington's energy partition in cosmical physics is discussed and the partition equations are found to have a very simple form. It is shown that the rigid-field convention leads to a triviality.
A continuum theory of anisotropic fluids
Authors: Green A. E.
Published: November 2008

In 1955 Noll (7) gave a definition of an anisotropic fluid. Noll's idea is developed in more detail and the results of applying the theory to some simple flows are compared with those obtained by Ericksen (1) from his theory of transversely isotropic fluids.
Small disturbances in a conducting fluid in the presence of a current-carrying conductor
Authors: Davis A. M. J.
Published: November 2008

1. Introduction. The problem considered here derives its motivation from a paper by Friedlander (8) on the propagation of small disturbances in a compressible, conducting fluid in the presence of a uniform magnetic field (see also Courant and Hilbert (3), VI, §3a). In this the displacement current and energy dissipation by viscosity, heat conduction and Joule heat are neglected and a system of linear partial differential equations is obtained, which generalizes the equations of motion of the theory of sound. Their solution is in general the superposition of an arbitrary incompressible Alfven wave and a magneto-acoustic disturbance. This latter was considered by constructing a Green's function by means of suitable combinations of plane wave solutions and it was found that there are fast and slow wave fronts diverging from a point disturbance. The latter are conoidal in shape and have a singularity at their vertices which propagate along the field line in either direction from the source.
Power-free groups
Authors: Geddes A.
Published: November 2008

1. Introduction. Consider a set P with elements a, b, c,… and a certain distinguished element e. Assume that P satisfies the following conditions.

(I) With each ordered pair (a, b) of distinct elements a and b of P there is associated a unique element of P (to be denoted by ab and called the product of a and b). Further, with the pair (e, e) there is associated a unique element ee of P.

(II)(ab)c = a(bc) for all elements a, b, c in P for which both sides are defined.

(III) ea = a = ae for all elements a ∈ P.

(IV) To each element a ∈ P, there corresponds an element a′ ∈ P such that
The Tchebysheffian approximation of one rational function by another
Authors: Talbot A.
Published: November 2008

In a previous paper we discussed a uniform algebraic method of solution of problems in which a prescribed real rational function (or polynomial) g(x) was to be approximated in a given finite interval by a real rational function (or polynomial)f(x) with prescribed numerator and denominator degrees, the approximation being Tchebysheffian, i.e. such as to make the ‘deviation’ of f, max |f − g| in the interval, as small as possible.
A differential equation for undamped forced non-linear oscillations. III
Authors: Morris G. R.
Published: November 2008

The most general differential equation to which the dynamical description of the title applies is

where dots denote differentiation with respect to t. The essential problem for this equation is to determine the behaviour of solutions as t → ∞. When we attack this problem, the most obvious question is whether, under reasonable conditions on p(t), every solution is bounded as t → ∞ this question is open except when g(x) is linear. In the special case when p(t) is periodic, (1·1) may have periodic solutions; it is clear that any such solution is bounded, and it is worth mentioning that finding periodic solutions is the easiest way of finding particular bounded ones. So long as the bounded-ness problem is unsolved, there is a special interest in finding a large class of particular bounded solutions: if we know such a class then, although we cannot say whether the general solution is bounded or not, we can make the imprecise comment that either the general solution is in fact bounded or the structure of the whole set of solutions is quite complicated.
Ignorable coordinates and steady motion in classical mechanics
Authors: Kilmister C. W. , Pirani F. A. E.
Published: November 2008

It is shown, for a classical dynamical system with a Lagrangian, that the existence of an ignorable coordinate is equivalent to the vanishing of a certain Lie derivative. On this covariant description is based a new definition of steady motion. A definition given earlier by Synge is criticized.
A general theory of unsteady compressible boundary layers with and without suction or injection
Authors: Sarma G. N.
Published: November 2008

The theory of unsteady two-dimensional compressible boundary layers is studied in two parts. Part I deals with the solutions when the main stream is steady and the body is in an arbitrary motion while Part II deals with the solutions when the main stream is steady and the body is at rest with an arbitrary unsteady suction or injection perturbing about a zero mean. General methods of obtaining solutions are given when the coefficient of viscosity is proportional to temperature and the Prandtl number is unity. The main feature of the work in this paper lies in subjecting the equations to a transformation similar to that of Howarth and in linearization similar to that of Lighthill. Following the work of Sarma in incompressible flows two types of solutions are developed, one for large times and the other for small times. General expressions for the skin friction, the pressure ratio, the heat transfer and the difference between the normal coordinates of compressible and incompressible flows are obtained.
Some expansions of generalized Whittaker functions
Authors: Srivastava H. M.
Published: November 2008

In the usual notation let

where (a)n = a(a + 1)(a + 2)…(a + n − 1), (a)0 = 1.

Dr L. J. Slater ((3), page 628) gave an expansion of the form

for all values of n and t, real or complex.
On multilattice groups. II
Authors: Mcalister D. B.
Published: November 2008

Conrad ((2)), has shown that any lattice group which obeys (C.F.)each strictly positive element exceeds at most a finite number of pairwise orthogonal elementsmay be constructed, from a family of simply ordered groups, by carrying out, alternately, the operations of forming finite direct sums and lexico extensions, at most a countable number of times. The main result of this paper, Theorem 3.1, gives necessary and sufficient conditions for a multilattice group, which obeys (ℋ*), to be isomorphic to a multilattice group which is constructed from a family of almost ordered groups, by carrying out, alternately, the operations of forming arbitrary direct sums and lexico extensions, any number of times; we call such a group a lexico sum of the almost ordered groups.
A note on the temperature distribution in a channel bounded by two co-axial cylinders
Authors: Bhatnagar P. L. , Tikekar V. G.
Published: November 2008

If we neglect the viscous dissipation, the temperature distribution in an incompressible fluid flowing in the annulus between two coaxial cylinders with radii r1 and r2 (> r1) is determined by the equation

Initial conditions

Boundary conditions

where ∂Q/∂t is a function of time alone representing the rate of heat generation per unit volume per unit time in the fluid.
Inclusion sets of regular summability matrices. III
Authors: Baker J. W. , Petersen G. M.
Published: November 2008

Let A = (am, n) be a (regular summability) matrix. Then will denote the set of bounded sequences which are summed by A. If {Ai} (i = 1, 2, …, N) is a finite set of such matrices, and if consists of every bounded sequence then we shall say that the matrices span the bounded sequences. Ifx = {xn} belongs to then we denote the value to which A sums x by A-lim x. If y = {yn} is any sequence, then the A-transform of y (if it exists) is the sequence {Aμ(y)}, where
On integral operators with kernels involving Bessel functions
Authors: Okikiolu G. O.
Published: November 2008

By representing integral operators defined by kernels involving Bessel functions as compositions of two operators, we determine their mapping properties and, in one case, derive an inversion process. The operators considered are of the form

where τ(t) is given respectively by and and in the last three cases the integrals converge in norm.