bluuebunny/crossref_metadata_2025 huggingface dataset:
Electronic states at the surfaces of crystals
Authors: Goodwin E. T.
Published: November 2008

The wave functions and energies of both the normal and surface electronic states of a finite linear chain are determined in terms of the overlap integrals by the approximation of tight binding, it being assumed that the interaction of atomic s-states with p-states can be neglected. It is shown that the existence of surface states depends on the ratio of two overlap integrals being greater than unity, and reasons are given for expecting this to be so.
The number of configurations in an assembly and cooperative phenomena
Authors: Chang T. S.
Published: November 2008

We consider an ideal problem of adsorption of single and double particles upon a solid surface which has its sites of accommodation regularly arranged, and by comparing the equilibrium properties obtained by Bethe's method with the ordinary statistical formulae, we obtain approximate expressions for:

(1) g(N, n, X), the number of ways of arranging n particles upon N sites of a lattice so that the number of neighbouring sites occupied by the particles is X.

(2) g2(N, n, X), the number of ways of arranging n double particles upon N sites so that each of the double particles takes up two adjacent sites and the number of neighbouring sites occupied by two different particles is X.

Both these expressions are found to agree with the exact values when the N sites lie on a straight line. When we use the first expression to construct the configurational partition functions of certain physical assemblies and expand them in powers of 1/kT, they are found to agree with the corresponding rigorous expressions as far as (1/kT)3, which is the highest power which we can find rigorously at present. With the help of the first expression, formal equations for superlattice formation in an alloy with the composition 1: 1 and equations for the separation into phases of regular liquids are given. Lastly we show that atoms and molecules in a regular liquid may dissociate or recombine suddenly accompanied by a latent heat. This is a new cooperative phenomenon, which may bear some resemblance to the melting process between the solid and liquid states.
Contributions to the theory of Ramanujan's function τ(n) and similar arithmetical functions
Authors: Rankin R. A.
Published: November 2008

Suppose that

is an integral modular form of dimensions −κ, where κ > 0, and Stufe N, which vanishes at all the rational cusps of the fundamental region, and which is absolutely convergent for Then

where a, b, c, d are integers such that ad − bc = 1.
Critical conditions in neutron multiplication
Authors: Peierls R.
Published: November 2008

It is well known that a single neutron may cause a nuclear reaction chain of considerable magnitude, if it moves in a medium in which the number of secondary neutrons which are produced by neutron impact is, on the average, greater than the number of absorbed neutrons. From recent experiments it would appear that this condition might be satisfied in the case of uranium.
The reflexion of sound pulses by convex parabolic reflectors
Authors: Friedlander F. G.
Published: November 2008

The reflexion of a train of simple harmonic waves by a convex paraboloid of revolution, and by a parabolic cylinder, has been discussed by Lamb. In the present paper these results are extended to the reflexion of plane waves of arbitrary form. It is found that on the introduction of suitable variables the equation of sound propagation transforms (in each case) into a simpler equation whose general integral can be obtained by quadratures. Two unknown functions are introduced during the integration, which have to be determined from the boundary conditions. This involves in both cases the solution of a Volterra integral equation, which is effected numerically by calculation of the first terms in the series development of the resolving kernel. An interesting feature of the solutions obtained is that when a suitable time scale is introduced (for a sharp-fronted pulse the time must be counted from the onset of the wave), the reflected wave experienced is the same at all points on any paraboloid (or parabolic cylinder) confocal with the reflector.
On Lorentz invariance in the quantum theory. II
Authors: Eddington Arthur
Published: November 2008

1. I appreciate the careful reply which Dirac, Peierls and Pryce (hereafter referred to as DPP) have made to my criticism of the usual theory of the eigen-functions of a hydrogen atom. Their paper will be generally welcomed because there has not, I think, previously been available an authoritative and clear statement of what is assumed and the grounds for assuming it. Their defence, which is partly on lines I had not expected, has required very serious consideration. But the result of making the arguments more explicit is two-fold. The theory is perhaps more self-consistent than it had appeared to be; but on the other hand, the pressing need for amendment becomes too plain to be overlooked. I have endeavoured to show in the last twelve years that this amendment opens out very fertile extensions.
A method of calculating the integral breadths of Debye-Scherrer lines
Authors: Stokes A. R. , Wilson A. J. C.
Published: November 2008

It is shown that, for crystals of the cubic system, the reciprocal of the integral breadth of a Debye-Scherrer line is cos θ/λ times the volume average of the thickness of the crystal measured at right angles to the reflecting plane. The result is applied to calculate the integral breadths of reflexions from crystals having the external forms of rectangular parallelepipeds, tetrahedra, octahedra and spheres. Except for spheres, the integral breadths are a function of the indices of reflexion as well as of the size of the crystal and the angle of reflexion. For cubes the variation with indices of reflexion is about 15%.
Some generalizations of the ergodic theorem
Authors: Pitt H. R.
Published: November 2008

Throughout this paper we shall suppose that denotes a set of elements x in which a Lebesgue measure is defined and that itself is measurable and has finite measure. A (1, 1) transformation T of into itself is called an equimeasure transformation if the transform T E of any measurable subset E of is measurable and has measure equal to that of E. Then, if f(x) is integrable in , it is plain that f(Tx) is also integrable and that
The number of configurations of a cooperative assembly
Authors: Miller A. R.
Published: November 2008

Formulae are found for the number of configurations of particles on two-and three-dimensional lattices when each particle (a) occupies two closest neighbour sites, and (b) consists of three groups which occupy three sites on the lattice in such a way that adjacent groups in the molecule occupy closest neighbour sites on the lattice. Bethe's method is used to determine the equilibrium conditions of the corresponding order-disorder problem, and the number of configurations is determined from these equilibrium conditions. For the case in which the molecules occupy two closest neighbour sites on the surface the determination of the number of configurations from geometrical considerations is discussed.

It is found that for molecules which occupy two closest neighbour sites the number of configurations of particles for a square lattice, a simple cubic lattice and a body-centred cubic lattice respectively are and For molecules which occupy three sites on the lattice the corresponding results are and when the molecules are perfectly flexible and and when the molecules are completely inflexible, Ns being the total number of sites in the lattice.

The author wishes to thank Dr J. K. Roberts for suggesting this work; the geometrical treatment given in § 3 was developed from manuscript notes prepared by him. The problem arose during an investigation of the vapour pressure equations of solutions in which the solute molecules are chain molecules consisting of a large number of groups, undertaken as part of the programme of fundamental research of the British Rubber Producers' Research Association, whom the author wishes to thank for the grant of a Research Scholarship.
Conduction of heat in a slab in contact with well-stirred fluid
Authors: Jaeger J. C.
Published: November 2008

The problem of conduction of heat in a solid whose surface is in contact with fluid, which is so well stirred that its temperature is constant throughout its mass, often arises in practice, for example in calorimetry, or in the warming of a closed room. Despite their importance, problems of this type have not been much studied since the presence of the mass of fluid at the surface gives rise to a boundary condition for which the classical methods of solution do not apply without modification; the Laplace transformation method, however, has the advantage that it can be applied in the same way to all cases.
Note on the equation
Authors: ErdéLyi A. , Kermack W. O.
Published: November 2008

In the solution of certain boundary-value problems by means of complex integrals or by means of the Laplace transformation, transcendental equations of the form

arise, and it is then usually necessary to show that this equation has no roots in the right half of the complex z-plane.
Totally heterogeneous continua
Authors: Besicovitch A. S.
Published: November 2008

A possible degree of heterogeneity in the structure of continua is investigated in this paper.
On Mathisson's variational equation of relativistic dynamics
Authors: Shanmugadhasan S.
Published: November 2008

The paper considers the solution of the general variational equation of relativistic dynamics using the methods of Mathisson with a generalization at one stage. The solution gives us two equations describing the translational motion and rotational motion. The rotational equation is also investigated by setting up a variational equation using the conservation of angular momentum, and this is found to be of the same form as that obtained from Mathisson's variational equation.
A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type
Authors: Crank J. , Nicolson P.
Published: November 2008

This paper is concerned with methods of evaluating numerical solutions of the non-linear partial differential equation

where

subject to the boundary conditions

A, k, q are known constants.

Equation (1) is of the type which arises in problems of heat flow when there is an internal generation of heat within the medium; if the heat is due to a chemical reaction proceeding at each point at a rate depending upon the local temperature, the rate of heat generation is often defined by an equation such as (2).
A rational formulation of the equations of plastic flow for a Bingham solid
Authors: Oldroyd J. G.
Published: November 2008

In this paper we are concerned with a material which can support a finite stress elastically without flow and which flows with constant mobility(1) (or plastic fluidity) when the stresses are sufficiently great. Following Bingham(1) and Houwink(2), such a material is called a Bingham solid and the type of flow (purely) plastic. The transition from elastic to plastic behaviour takes place at the yield point.
The lattice properties of asymmetric hyperbolic regions
Authors: Cassels J. W. S.
Published: November 2008

Let θ > 0 and α ≠ 0 be real numbers, and let θ be irrational. Khintchine has shown, by the use of continued fractions, that there is an infinite number of pairs of positive integers (p, q) which satisfy the inequality

for any given K > 5−½; and, more recently, Jogin has shown the same is still true with K = 5−½. The condition that p and q shall be positive is, of course, essential, as otherwise there is the classical result K = ¼ due to Minkowski.
Rectilinear plastic flow of a Bingham solid
Authors: Oldroyd J. G.
Published: November 2008

It is apparent from the two previous papers of the same main title (1, 2) that velocity distributions in steady rectilinear plastic flow of a Bingham solid between moving boundaries are not easy to determine, even when the boundaries are of simple shape. In the familiar case of a Newtonian liquid (which can be regarded as a special case of a Bingham solid with zero yield value) the velocity ω is a harmonic function and can be obtained by a conformal transformation of the region of flow of the type

In order to generalize and include materials of finite yield value, in which ω is not harmonic, one must first regard the Newtonian case from a slightly different point of view. The transformation

must be regarded as defining a change of coordinates, from Cartesian coordinates x, y to the natural curvilinear coordinates for a particular problem ω, W, one of which is the velocity. For a Bingham solid in general, it is shown in the present paper that natural coordinates ω, W exist, but are not obtainable by a conformal transformation from Cartesian coordinates, except in the limit when the yield value tends to zero. In practice it may be difficult to determine the natural coordinate system in a given problem, but when it is found the velocity distribution is automatically known.
The condition for the detachment of the shock wave from a wedge in a supersonic stream
Authors: Pack D. C.
Published: November 2008

The flow past a wedge which is moving with supersonic velocity into a gas at rest has been treated by a number of investigators. In the analysis of the problem given in the present note, the condition for the detachment of the shock wave is examined, and an error in a previous treatment by Epstein is corrected.
A statistical theory of cascade multiplication
Authors: Woodward P. M.
Published: November 2008

The secondary emission electron multiplier is chosen to illustrate the phenomenon of ‘cascade multiplication’. A method is given for deriving the semi-invariants of the probability distribution for the number of output electrons after any number of identical stages of multiplication, in terms of the corresponding semi-invariants for a single stage. The output distribution is not, in general, either of the Poisson or Gaussian types, though it tends to a limiting shape as the number of stages becomes very large. The special case in which each stage replaces a single primary electron by a Poisson distribution of secondaries is considered. The overall output distribution after many stages is still not Gaussian unless the mean amplification per stage is large compared with unity.
The lattice properties of asymmetric hyperbolic regions
Authors: Cassels J. W. S.
Published: November 2008

1. Let

where a > 0, be an indefinite quadratic form, so that d = b2 − 4ac > 0. A classical theorem of Minkowski states that, if (x0, y0) is any pair of real numbers, there are numbers (x, y) congruent (mod 1) to (x0, y0), such that

and, more recently, Davenport has shown that this theorem can be sharpened for certain special f, for instance that it is always possible to satisfy
The dissection of equilateral triangles into equilateral triangles
Authors: Tutte W. T.
Published: November 2008

In a previous joint paper (‘The dissection of rectangles into squares’, by R. L. Brooks, C. A. B. Smith, A. H. Stone and W. T. Tutte, Duke Math. J. 7 (1940), 312–40), hereafter referred to as (A) for brevity, it was shown that it is possible to dissect a square into smaller unequal squares in an infinite number of ways. The basis of the theory was the association with any rectangle or square dissected into squares of an electrical network obeying Kirchhoff's laws. The present paper is concerned with the similar problem of dissecting a figure into equilateral triangles. We make use of an analogue of the electrical network in which the ‘currents’ obey laws similar to but not identical with those of Kirchhoff. As a generalization of topological duality in the sphere we find that these networks occur in triplets of ‘trial networks’ N1, N2, N3. We find that it is impossible to dissect a triangle into unequal equilateral triangles but that a dissection is possible into triangles and rhombuses so that no two of these figures have equal sides. Most of the theorems of paper (A) are special cases of those proved below.
Experimental evidence for the theory of local isotropy
Authors: Townsend A. A. , Taylor Geoffrey
Published: November 2008

Some new measurements of isotropic turbulence produced behind a biplane grid have been made at high Reynolds numbers, and these results are compared with the predictions of the theory of local isotropy developed by A. N. Kolmogoroff. The transverse double-velocity correlation has been measured at mesh Reynolds numbers up to 3·2 × 105, and the observed form agrees well with the predicted form. Measurements of the skewness factor of velocity differences over finite intervals have also been made, and the factor is nearly constant and equal to −0·38, if the interval is small compared with the integral scale. The invariance of dimensionless functions of the velocity derivatives has been confirmed for the flattening factor of ∂u/∂x, namely,

which is nearly constant over a wide range of conditions. It is concluded that the theory of local isotropy is substantially correct for isotropic turbulence of high Reynolds number.
The definition of measure in function space
Authors: Pitt H. R.
Published: November 2008

A fundamental result in the theory of measure in the space Ω of real functions x(t) of a real variable t is the following theorem of Kolmogoroff:

Theorem 1. Suppose that functions F(t1, …, tn; b1 …, bn) = F(t; b) are defined for positive integers n and real numbers t1, …, tn, b1, …, bn, and have the following properties:

(1·1) For every fixedt1, …, tn, F(t; b) has non-negative differences

with respect to the variables bl, b2,…, bn, and is continuous on the right with respect to each of them;

if (i1, …, in) is any permutation of (1, 2, …, n). Then a measure P(X) can be defined in a Borel system of subsets of Ω in such a way that the set of functions satisfying

is measurable for any realbi, tiand has measure F(t; b).
On the existence of metric differential geometries based on the notion of area
Authors: Brickell F.
Published: November 2008

The problem of constructing an n-dimensional metric differential geometry based on the idea of a two-dimensional area has given rise to several publications, notably by A. Kawaguchi and S. Hokari (1), E. T. Davies (2), and R. Debever (3). In this geometry the area of a two-dimensional plane element is defined by a fundamental function L(xi, uhk), where the xi are point coordinates and the uhk are the coordinates of the simple bivector representing the plane element. L is supposed to be a positive homogeneous function of the first degree with respect to the variables uij, and to possess continuous partial derivatives up to and including those of the fourth order. With these assumptions the problem of the construction of the metric differential geometry splits into two problems; the first of these is the problem of constructing a metric tensor gij(xr, uhk), and the second is the problem of constructing an affine connexion. We deal with the first problem only in this paper.
Rayleigh's problem for a semi-infinite plate
Authors: Howarth L.
Published: November 2008

The problem considered in this paper is the fluid motion arising from a thin semi-infinite plate started to move impulsively from rest (in viscous incompressible fluid at rest) with a velocity, subsequently maintained uniform, parallel to the edge. Two solutions are given, one obtained in polar coordinates is in the form of an infinite series, whilst the other, derived operationally in parabolic coordinates, leads to a single integral for the velocity distribution. The former is convenient for computation in the vicinity of the edge, the latter being more convenient elsewhere.

A hypothesis originally introduced by Rayleigh after he had discussed the corresponding flow for an infinite plane has been used here to draw deductions about the steady flow past a quarter-plane whose leading edge is normal to the direction of flow and also to obtain approximate expressions for the effect of the edges on the skin friction of a sufficiently broad rectangle whose length is parallel to the incident stream.
The expected number of aggregates in a random distribution of n points
Authors: Mack C.
Published: November 2008

When n points are distributed at random (equally likely anywhere) over a large area or volume, they may form small clusters or aggregates. In this paper a k-aggregate is defined more precisely as k points which can be covered by a small figure of given size, shape and orientation, but whose position is not otherwise specified. Exact formulae for the expected number of such k-aggregates are given in some cases, while a good approximate formula (accurate in most practical cases) and limits are stated, covering all cases.
Extensions and applications of a tauberian theorem due to valiron
Authors: Noble M. E.
Published: November 2008

1. Introduction. In the course of an important memoir on integral functions of finite order†, G. Valiron discusses ‘fonctions orientées’, that is, functions with zeros an such that arg an tends to a limit as n → ∞. He obtains results that include the following two theorems, in which Vρ(x) denotes a function of the form , where α1, …, αν are positive integers, and n(r) has its usual significance in the theory of integral functions:.
The stopping power of liquid water
Authors: Appleyard R. K.
Published: November 2008

Experiments are described in which the stopping power of liquid water relative to air has been re-determined for 5 MeV. α-particles, using only a limited portion of the range. The results confirm those of earlier experimenters, who concluded that liquid water did not obey the Bragg law.
On a theorem of Zygmund
Authors: Kac M.
Published: November 2008

Zygmund (1) proved in 1936 that if f(x) ∈Lp(−∞, ∞) (1 ≤ p ≤ 2), the Fourier transform

exists for almost every λ, in the sense that

exists for almost every λ. The purpose of this note is to provide a short proof of this theorem.
On semi-groups in which xr = x
Authors: Green J. A. , Rees D.
Published: November 2008

In this note we are concerned with the semi-group Snr, generated by n of its elements, in which each element x satisfies the equation xr = x, the semi-group being otherwise free. The main result proved is that the proposition

(Ar) Snris finite for all finite n, is equivalent to the proposition

(Br) the group Bn, r−1, generated by n of its elements, in which each element x satisfies the equation xr−1 = 1, the group being otherwise free, is finite for all finite n.
The Hahn-Banach theorem for non-Archimedean-valued fields
Authors: Ingleton A. W.
Published: November 2008

1. The Hahn-Banach theorem on the extension of linear functionals holds in real and complex Banach spaces, but it is well known that it is not in general true in a normed linear space over a field with a non-Archimedean valuation. Sufficient conditions for its truth in such a space have been given, however, by Monna and by Cohen‡. In the present paper, we show that a necessary condition for the property is that the space be totally non-Archimedean in the sense of Monna, and establish a necessary and sufficient condition on the field for the theorem to hold in every totally non-Archimedean space over the field. This result is obtained as a special case of a more general theorem concerning linear operators, which is analogous to a theorem of Nachbin ((6), Theorem 1) concerning operators in real Banach spaces.
Slow motion of viscous liquid near a half-pitot tube
Authors: Dean W. R.
Published: November 2008

1. The problem of this paper was first attempted some years ago at the suggestion of Sir G. I. Taylor, to whom the writer makes grateful acknowledgement, in connexion with experimental work in which he and others (5,6) had used half-pitot tubes in order to measure fluid velocities as near as possible to a fixed boundary.
On the slow motion of a sphere along the axis of a rotating fluid
Authors: Stewartson K.
Published: November 2008

We consider the perturbation of the velocity of an inviscid incompressible fluid rotating about an axis with uniform angular velocity Ω, due to the slow uniform motion, after an impulsive start, of a sphere of radius a along the axis with velocity − V. This problem was first considered by Taylor (4), who obtained a family of solutions for the case of steady motion of a sphere along the axis of rotation of the fluid. All the solutions satisfied the boundary condition at the surface of the sphere and also the condition that the relative velocity should vanish at infinity. It was thought possible that the indeterminacy lay in the manner in which the motion was started. However, an experiment carried out in connexion with the problem showed that if V/aΩ was less than about 0·16 a column of liquid of the same diameter as the sphere was apparently pushed along in front of it. This agreed with Proudman's observation (3) that if the perturbation was small and steady it would be two-dimensional, while in Taylor's solutions the relative motion of the fluid was not small. Taylor (5) suggested that three possibilities presented themselves. Either the motion never becomes steady, or it becomes steady but not small, or it becomes steady and two-dimensional. Of these he preferred the latter, which seems to occur in practice.
On accessibility of plane sets and differentiation of functions of two real variables
Authors: Davies R. O.
Published: November 2008

1. The paper is in three parts, of which the first is devoted to the proof of certain lemmas, which form the basis for the results proved in Parts II and III, and which are summed up in Lemma 6, §3. In Part II we consider questions relating to linear accessibility: a member of a set of points in the plane is said to be (linearly) accessible if through it there exists a straight line (infinite in both directions) containing no other point of the set. The main results are Theorems 3 and 5. Theorem 5 extends the result of Nikodym (2) that a set can be constructed which is of full measure in a square, and each point of which is accessible. Theorem 3 was conjectured by Besicovitch.
The Hausdorff α-dimensional measure of Brownian paths in n-space
Authors: Taylor S. J.
Published: November 2008

Dvoretsky, Erdös and Kakutani (3), showed that Brownian paths in 4 dimensions have zero 2-dimensional capacity, and their method gives the same result for Brownian paths in n-space whenever n ≥ 3. However, if we apply the method indicated by Hausdorff (4) for constructing a linear set having measure 1 with respect to a given measure function h(x), the Cantor-type set we obtain when h(x) = xα log log 1/x, where 0 < α < 1, is easily seen to have zero α-capacity but infinite α-measure; and similar methods apply for other values of α. Thus the result mentioned above does not imply that Brownian paths in n-space (n ≥ 3) have zero 2-measure. The other relevant result is due to Lévy (6), who showed that Brownian paths in the plane have zero Lebesgue measure (and therefore zero Hausdorff 2-measure) with probability 1. However, his method of proof cannot be extended to deal with Brownian paths in n-space.
The use of a ‘floating address’ system for orders in an automatic digital computer
Authors: Wilkes M. V.
Published: November 2008

This note is concerned with automatic digital computing machines in which orders are expressed in coded numerical form and held in the same store as numbers. It is possible in such machines to modify orders by performing arithmetical operations on them. The storage locations are numbered serially, and their contents can be referred to when necessary by these numbers.
On p-quotients and star diagrams of the symmetric group
Authors: Farahat H.
Published: November 2008

The star diagram of a diagram [λ] was first obtained by Robinson (5) who denned and studied it, thus giving a unique characterization of the hook structure of [λ]. Staal (6) continued Robinson's work and obtained some important results. Recently, Littlewood (2) published a paper on the modular representations of the symmetric group in which he defined ‘congruent partitions’, ‘p-quotients’, etc., relating to the diagram [λ]. Littlewood's ideas, though superficially different from those of Robinson and Staal, are in essence equivalent to them.
A short proof of the Minkowski-Hlawka theorem
Authors: Cassels J. W. S.
Published: November 2008

Theorem. Let K be a bounded star-bodyin (n + 1)-dimensional space symmetric about the origin and of volume V < 2ζ(n +1). Then there is a lattice of unit determinant with a point at the origin and having no other points in K. (See (1, 2,3,4).)
Algebraic integers with two conjugates outside the unit circle
Authors: Samet P. A.
Published: November 2008

In this paper we consider algebraic integers, θ, such that θ and θ¯ (θ ╪ θ¯) are the only algebraic conjugates of θ that lie outside the unit circle.

Let S1 be the set of algebraic integers, ρ, necessarily real, all of whose conjugates, except ρ, satisfy | z < 1.
On the flow downstream of separation in an incompressible fluid
Authors: Stewartson K.
Published: November 2008

When a stream is flowing along a wall against an adverse pressure gradient, the skin friction at the wall decreases to zero, at which point or at least very near it the flow seems to leave the surface altogether. Between the main stream and the wall there may be a slow back-flow. Upstream of separation the influence of the wall on the flow is satisfactorily explained by the boundary-layer theory, which among other things presupposes that this influence is confined to a narrow region near the wall. Such a theory is clearly inadequate downstream, however, because this layer rapidly increases in width and, further, the main stream is seriously affected, and consideration of the Navier-Stokes equations would seem to be necessary before a full theoretical explanation of separation could be given. Apart from a note by Dean (2) on the relations between the pressure and velocity at separation, there has, however, been no relevant attack on these equations.
Triangular representations of linear algebras
Authors: Drazin M. P.
Published: November 2008

It is well known that the elements of any given commutative algebra (and hence of any commutative set) of n × n matrices, over an algebraically closed field K, have a common eigenvector over K; indeed, the elements of such an algebra can be simultaneously reduced to triangular form (by a suitable similarity transformation). McCoy (5) has shown that a triangular reduction is always possible even for matrix algebras satisfying a condition substantially weaker than commutativity. Our aim in this note is to extend these results to more general systems (our arguments being, incidentally, simpler than some used for the matrix case even by writers subsequent to McCoy).
Experiments with a rotating cylinder viscometer in liquid helium II
Authors: Hollis-Hallett A. C.
Published: November 2008

Liquid helium II was contained in the annular space between two co-axial cylinders, the inner of which was suspended by a torsion fibre while the outer was rotated at constant speeds. The torque upon the inner cylinder produced by the rotating fluid was measured for various steady velocities between 0·1 and 3 cm.sec.1, and was not found to be directly proportional to the velocity of rotation at any temperature between the lambda-point and 1·15° K. This result suggests that there must be some new type of non-linear frictional force acting in the liquid, possibly in addition to the Gorter-Mellink force of mutual friction.

Extrapolation of the experimental results to zero velocity gives values of the coefficient of viscosity of the normal component which agree with the oscillating disk values between the lambda-point and about 1·6° K. At lower temperatures, the present results are significantly lower, suggesting, perhaps, that the values of the normal component density used in the analysis of the oscillating disk results were too low.
The calculation of transients in dynamical systems
Authors: Ward E. E.
Published: November 2008

The paper shows that the calculation of transients by Tricomi's method using Laguerre functions is a practical alternative to the use of partial fractions. By taking numerical examples of Laplace transforms ranging from quadratic to sixth-power polynomials it shows that the approximation offered by this method would be acceptable for many practical uses. It analyses the composition of the coefficients of the Laguerre series, thus disclosing the conditions for quick convergence; for some transforms these conditions are not attainable.
The dimension of Cartesian product sets
Authors: Marstrand J. M.
Published: November 2008

Given a plane set E, we denote by Ex the set of its points whose abscissae are equal to x.

Throughout this paper we use the letters A and B to denote subsets of the x-axis and y-axis respectively, and we denote by A × B their Cartesian product set. We use the letters s and t to denote positive numbers; we denote by ΛsE the outer Hausdorff s-dimensional measure of the set E.
Separation of angle variables for helium
Authors: Jackson T. A. S.
Published: November 2008

The Schroedinger equation of the helium atom is separable between the radial and angular variables. The separated equations in the radial variables are found by using a new technique of calculation with symmetric traceless tensors. Account is taken of reflexion and permutation characters.
A note on reductions of ideals with an application to the generalized Hilbert function
Authors: Northcott D. G. , Rees D.
Published: November 2008

1. Throughout this note Q will denote a local ring, m will denote its maximal ideal, q will denote a primary ideal belonging to m and k will denote the residue field Q/m. It will not be assumed that k is infinite, but we shall suppose that Q and k both have the same characteristic. Now let υ1, υ2 …,υd be a system of parameters contained in q, so that d = dim Q; then according to the definition given in (2) the ideal (υl υ2,…, υd) is a reduction of q if (υ1 υ2, …, υd) qm = qm+1 for at least one value of m. The use of the concept lies in the fact that such a reduction is, in a certain sense, a very good approximation to q itself; but the notion does, however, suffer from a minor disadvantage in that, if k is finite, q need not have any reductions. In §3 we shall generalize the notion of a reduction in such a way that we overcome this difficulty, and in such a way that the results concerning reductions obtained in (2) acquire some useful extensions.
A note on commutators in free products. II
Authors: Griffiths H. B.
Published: November 2008

The purpose of this note is to improve Theorem 1·1 of (1) to the following result:

Theorem 1. Let H be the ordinary free product of a sequence of groups Gn, and let m be any integer > 0. If

where each gi є Gi and is not the identity, then h is not equal to any product of fewer than m commutators from H.
The solution of linear differential equations
Authors: Dennis S. C. R. , Poots G.
Published: November 2008

A number of methods are at present available for the numerical solution of linear ordinary differential equations over a restricted range of the independent variable. Of these perhaps the most commonly used are methods based on finite differences which have as their aim the construction of a table of values of the dependent variable at stated intervals of the independent variable. The appropriate method to be used depends upon the boundary conditions. Thus for second-order equations, for example, if boundary conditions occur at each end of the range of integration the relaxation method of Southwell (6) is suitable, whereas if this is not the case resort may be made to step-by-step methods or the recently proposed method of Allen and Severn(1). This method, applicable to equations of any order, brings the problem within the scope of normal relaxation methods at the expense of an increase in the order. In all such methods the accuracy attainable for a given (small) amount of labour is a worth-while consideration, and methods have been suggested by Fox (3) and Fox and Goodwin (4) for improving this by the use of comparatively large intervals of the independent variable.
Atomic wave functions for gold and thallium
Authors: Douglas A. S. , Hartree D. R. , Runciman W. A.
Published: November 2008

Before the war, self-consistent field calculations for the Au+ ion had been carried out by W. Hartree but were left still unpublished at his death (see prefatory note in (5)). These results have been used by Brenner and Brown (1) in a relativistic calculation of the K-absorption edge for gold, and they were also used in obtaining initial estimates for the partial self-consistent field calculations for thallium of which results are given in §§3–5 of the present paper. In the meantime an independent calculation for Au+ has been carried out by Henry (6), and his results agree closely with those of W. Hartree. However, it still seems desirable to publish the latter, since they give directly the radial wave function P(nl; r) at exact values of r, whereas Henry used log r as independent variable, as had been done for similar calculations for Hg(4), and has tabulated r½P(nl; r) which is the natural dependent variable to use with log r as independent variable (2); in some applications it is more convenient to have the radial wave functions themselves.
Pairs of matrices with a non-zero commutator
Authors: Goddard L. S. , Schneider H.
Published: November 2008

1. This note takes its origin in a remark by Brauer (1) and Perfect (5): Let A be a square complex matrix of order n whose characteristic roots are α1,…, αn. If X1 is a characteristic column vector with associated root α and k is any row vector, then the characteristic roots of A + X1 k are α1 + KX1, α2, …, αn. Recently, Goddard (2) extended this result as follows: If x1; …, xr are linearly independent characteristic column vectors associated with the characteristic roots α1, …, αr of the matrix A, whose elements lie in any algebraically closed field, then any characteristic root of Λ + KX is also a characteristic root of A + XK, where K is an arbitrary r × n matrix, X = (x1, …, xr) and Λ = diag (α1, …, αr). We shall prove some theorems of which these and other well-known results are special cases.
On the Taylor coefficients of rational functions
Authors: Mahler K. , Cassels J. W. S.
Published: November 2008

Let F(z) be a rational function of z which is regular at z = 0 and so possesses a convergent power series

The problem arises of characterizing those rational functions F(z) that have infinitely many vanishing Taylor coefficientsfh. After earlier and more special results by Siegel(2) and Ward(4) I applied in 1934(1) a p-adic method due to Skolem(3) to the problem and obtained the following partial solution.
The thermodynamics of bodies in static electromagnetic fields
Authors: Heine V.
Published: November 2008

The thermodynamic behaviour of a body in a static electromagnetic field depends only on the field over the region of the body itself, whence its own thermodynamic behaviour is determined even in the presence of other polarizable media. Its behaviour depends only on the entropy density, for in fact no real energy density need exist. Also new formulae are developed for the energy and for the entropy of a body in electric and magnetic fields, including those due to permanent magnets.
On families of domains
Authors: Besicovitch A. S.
Published: November 2008

The boundary of a simply-connected plane domain in the simplest case is a Jordan curve, but in the general case it is of a very complicated nature. An interesting problem arises about the nature of the boundary, when we consider not a single domain but some extensive family of domains. The question is whether only exceptional domains of the family have a complicated boundary, the rest having simple ones.
The self-consistent field with exchange for some 10 and 12 electron systems
Authors: Froese Charlotte
Published: November 2008

The results of self-consistent field calculations with exchange for the ground-state configurations of F−, Fe+16, +14, Al+3, +1 are presented.
Another transference theorem of the geometry of numbers
Authors: Birch B. J.
Published: November 2008

In a recent note (1) I sharpened existing theorems, which state, roughly speaking, that unless a symmetric convex body K has an excessively small homogeneous minimum, it must have a reasonably small inhomogeneous minimum; in other words, if the translates of K centred at points of the integer lattice form an efficient packing, then by expanding them we may derive an efficient covering. In this note I will prove the converse, that a bad packing leads to a bad covering; the result is a much less useful one, as the worst case occurs when we foolishly try to pack rather spiky bodies point to point. A result of this kind has previously been proved by Mahler (4), but his result is less precise from our present point of view; a full account is given by Cassels(3).
The rheology of some two-dimensional disperse systems
Authors: Oldroyd J. G.
Published: November 2008

An investigation is made of the deformation and flow properties of two-dimensional disperse systems consisting of small circular patches of one component widely dispersed in a continuous component with different rheological properties. Attention is restricted to small deformations (or small rates of deformation in the case of a fluid system) so that the equations of state are linear and the properties of each component can be characterized by two elastic moduli, or by two operators involving d/dt which take the place of moduli. Surface tension in each component and boundary tension between the components are taken into account, so that the theory can be applied to interfacial films at liquid-liquid or liquid-gas interfaces as well as to thicker sheets and films. General formulae are derived, expressing the two modulus operators of a disperse system in terms of those of the components, and their use is illustrated by means of examples.
Polar modules
Authors: Rees D.
Published: November 2008

This paper is concerned with what, for lack of a better name, the author proposes to call geometric modules over a fieldk. These modules are defined as follows. A geometric moduleMconsists of a vector space, also denoted byM, overk, together with a ring of linear transformationsAofMinto itself subject to the following conditions:

(i)Ais a homomorphic image of the ringSn=k[X1, …,Xn] of polynomials innindeterminatesX1, …,Xnfor some value ofn;

(ii)M, considered as anA-module, is finitely generated.
Conjugate Banach spaces
Authors: Ruston A. F.
Published: November 2008

The purpose of this note is to present two characterizations of conjugate Banach spaces. More precisely, we present two conditions, each necessary and sufficient for a (real or complex) Banach space to be isomorphic to the conjugate space of a Banach space, and two corresponding conditions for to be equivalent to the conjugate space of a Banach space. Other characterizations, in terms of weak topologies, have been given by Alaoglu ((1), Theorem 2:1, p. 256, and Corollary 2:1, p. 257) and Bourbaki ((4), Chap, IV, §5, exerc. 15c, p. 122). Here, by the conjugate space* of a Banach space we mean ((2), p. 188) the space of continuous linear functionals over . Two Banach spaces and are said to be isomorphic if there is a one-one continuous linear mapping of onto (its inverse is necessarily continuous by the inversion theorem ((2), Théorème 5, p. 41; (6), Theorem 2·13·7, Corollary, p. 29)); they are said to be equivalent if there is a norm-preserving linear mapping of onto .((2), p. 180).
Precompact linear operators in locally convex spaces
Authors: Ringrose J. R.
Published: November 2008

This paper is concerned with the properties of precompact and compact linear operators from a locally convex Hausdorff space into itself, the field of scalars being the complex number field.

The Riesz–Schauder theory for the equations

where T is a compact linear operator from the Bacach space into itself, is well known (see, for example, Banach(1), Chapter 10, §2) and has been extended to the more general setting of locally convx Hausdorff spaces by Leray (4). In particular, the ‘Fredholm alternative theorem’ remains valid.
An integral for distributions
Authors: Burkill J. C.
Published: November 2008

The theory of distributions, systematized by L. Schwartz (3), has many applications in pure mathematics and physics. A number of different approaches to the theory are possible. Schwartz's own account requires for its comprehension a familiarity with the topology of general spaces. I give in this note another approach to the theory based on a process of integration of the Stieltjes type.
On random walks with restricted reversals
Authors: Domb C. , Fisher M. E.
Published: November 2008

1. Introduction. Problems of unrestricted random walks on lattices have been considered by many authors and methods have been discovered for the exact enumeration of the number of walks between two points, for obtaining asymptotic estimates of the distribution functions and for evaluating special parameters such as the probability of ultimate retum to the starting point. By an unrestricted walk we mean one in which the probability of the next step at any stage is not influenced by the previous choice of steps. The corresponding problems for restricted (or correlated) random walks are more difficult and fewer results have been obtained. Restricted walks are, however, of considerable physical interest in connexion with the statistical behaviour of long chain molecules, the theory of cooperative phenomena in crystals and in other applications.
Double refraction phenomena in quantum field theory
Authors: Stephen M. J.
Published: November 2008

It is shown that the phenomena of double refraction—optical activity, Faraday rotation, Kerr effect, etc.—may be treated as scattering problems in quantum field theory. An expression for the scattering cross-section of an atom or molecule for photons may be easily obtained and from this expression using the idea of Stokes operators, equating their expectation values to the classical Stokes parameters, the usual quantum mechanical expressions for optical activity, etc. may be obtained. This approach does not require the ideas of refractive index or polarization of the medium, and emphasizes what are the actual observables in these experiments.
The analysis of singularly loaded and rigidly clamped thin elastic slabs with curvilinear boundaries. I
Authors: Bassali W. A.
Published: November 2008

In this paper complex variable methods are used to derive exact solutions in closed forms for the small deflexions of certain thin elastic plates due to transverse concentrated forces or couples applied at arbitrary or specified points. The isotropic plates considered are bounded by curvilinear edges of certain types along which the plates are rigidly clamped. Plates bounded by quartic curves having the forms of the inverses of an ellipse with respect to its centre or its focus are included as special cases.
On 3N points in a plane
Authors: Birch B. J.
Published: November 2008

In this note I prove the following theorem:

Theorem 1. Given 3N points in a plane, we can divide them into N triads such that, when we form a triangle with the points of each triad, the N triangles will all have a common point.

The proof depends on three lemmas; the first two are well known, and are quoted without proof; the third I believe to be new, and leads directly to the proof of the theorem.
The Radial Charge Densities for the Ti+2 Argon Core
Authors: Hartree D. R.
Published: November 2008

The radial charge densities tabulated opposite were communicated some years ago by the late D. R. Hartree to R. J. Weiss, of the Ordnance Materials Research Office, Watertown Arsenal, Watertown 72, Massachusetts, U.S.A. At his suggestion they are now published, no self-consistent field calculations for Ti being available. It is not known how Hartree obtained the wave functions which he describes as ‘estimated’, but it is presumed that they are interpolated. Radial charge densities for the 3d electrons for Ti+1 have already been published (l).