bluuebunny/crossref_metadata_2025 huggingface dataset:
The effect of phonological variation on adult learner comprehension
Authors: Eisenstein Miriam , Berkowitz Diana
Published: November 2008

This research report focuses on the relationship of English phonological variation to intelligibility for adult second language learners of English. Fifty-eight adult ESL learners (29 advanced beginners and 29 high intermediates) were tested' on their ability to understand working-class (New Yorkese), educated (Standard English) and foreign-accented speakers of English. Subjects were also asked to rank these speakers according to expected ease of comprehension. Findings from this study indicate Standard English to be more intelligible than either New Yorkese or foreign accented English for these learners.
The Social Context of “Special” Second Language Acquisition
Authors: Rickford John , Hancock Ian
Published: November 2008

Although this is not standard SSLA practice, weare pleased to publish these two complementary reviews of a collective volume that, while it focuses on creolization and pidginization, addresses issues relevant to the social context of second language acquisition. Seldom dowe have the opportunity of having two leading specialists in a particular field apply to the same work different theoretical perspectives and varying familiarity with areas of the field.
On Communication Strategies: Focus on Interaction
Authors: Labarca Angela , Khanji Rajai
Published: November 2008

Although previous work has analyzed the communication strategies (CS) of L2 learners (e.g., Faerch & Kasper, 1983; Tarone, 1980, 1977), this research is an attempt at studying CS used by both 12 speakers and their interviewers during a videotaped interview. Fifty-three beginning French learners were interviewed and the videotape was analyzed for CS used by both interactants; learners' performance was also rated (ACTFL, 1982). Subjects were instructed either by Total Physical Response (Asher, 1977, 1969) or Strategic Interaction (SI) (Di Pietro, 1982, 1979).

Data were subjected to correlational and MANOVA analyses. Significant differences were detected between groups. In general, the higher the proficiency rating, the lower was the use of CS. The method of instruction appeared to have an important bearing on amount and kind of CS used and on proficiency ratings received. SI students and their interviewers used fewer CS; SI learners also received higher ratings.

It appears that SI facilitated students' development of at least two underlying competencies, linguistic and interactional (Palmer, 1979; Canale & Swain, 1980). It is also posited that SI students' diminished use of CS indicates a higher degree of control over the activity, as proposed by the Vygotskian paradigm (Vygotsky, 1962).
Universal unfoldings in Banach spaces: reduction and stability
Authors: Magnus R. J.
Published: November 2008

This article is concerned with the bifurcation of critical points of a smooth real-valued function on a Banach space. There are two somewhat different aspects considered. The first, referred to as reduction, is a refinement of the Liapunov-Schmidt procedure by incorporating techniques from the theory of determinate germs and their universal unfoldings. A definite reduction procedure is given, whereby a germ or unfolding given in a Banach space may be replaced by a germ or unfolding in a finite-dimensional space in such a way that the geometry of the bifurcation set and the overlying critical point set is preserved. The aim is to provide a practicable tool, not an exhaustive theoretical discussion. How practicable it is may be seen from another paper (Magnus and Poston(7)) in which it is applied to a problem in elasticity; indeed, the present paper in part was originally the ‘machinery’ section of that paper. The second aspect, stability, is of more theoretical interest. The well known structural stability of universal unfoldings (see (5)) is extended to infinite-dimensions. Only a local theory is given.
Order-continuous functions and order-connected spaces
Authors: Burgess D. C. J. , McCartan S. D.
Published: November 2008

We introduce and compare four procedures for defining the order-continuity of a function from one topological ordered space into another, where each reduces to the usual conception when the orderings of the two spaces are trivial. Chiefly for the purposes of this comparison, we use the idea of an ‘order-connected’ space, and in the course of investigating under which types of order-continuous functions this property is preserved, we are helped in assessing their relative importance.
Sorting vectors
Authors: Davidson Rollo
Published: November 2008

1. Introduction: In what follows, X will be a real inner-product linear space (inner product (x, y), norm |x| = (x, x)½); En will be n-dimensional real Euclidean space. We shall be dealing with sets {as: 1 ≤ s ≤ rk; r ≥ 1, k ≥ 2} of elements of X; we write a for the vector (a1; …, ark), and put . We suppose that |as| ≤ ∈ (< 0) for every s.
On the capacity of the circular disc condenser at small separation
Authors: Leppington Frank , Levine Harold
Published: November 2008

A pair of identical circular discs, held at equal and opposite potentials, forms a condenser whose capacityCdepends on the ratio ε of separation against diameter. The determination of an asymptotic expansion forCwhen ε is small poses an axisymmetric boundary-value problem for harmonic functions that has engaged the attention of numerous investigators over a long span of time. It is a simple matter to construct a Fredholm integral equation of the first kind for the charge density ± σ on the discs, in terms of which the potential field and the capacity are implicitly determined, but the equation is unsuitable if ε ≪ 1. Integral equations of the second kind and of the dual variety have also been proposed as a means of securing a more manageable formulation of the boundary-value problem. An elementary approximation follows from the hypothesis that the charge density is almost the same as though the discs were of infinite extent, except for a region close to the edges, and leads to the resultC∼ l/8ε as ε → 0. Kirchhoff considerably improved on this crude estimate by suggesting a plausible edge correction which yields two further terms forC, of orders log ∈ and a constant, respectively, and his results have been rigorously established by the more refined analysis of Hutson. In the present work an integral equation of the first kind for the distribution of potential off the discs is derived and utilized to obtain an approximation forCwhen ε is small, reproducing the result of Kirchhoff and Hutson. Furthermore, an estimate of the error provides explicit details regarding the next term in the asymptotic expansion ofC, which is of the order ε(log ε)2.
Spectral decay for definite kernels and functions
Authors: Oehring Charles
Published: November 2008

A classical theorem of Weyl [7] guarantees that the eigenvalues, ordered according to decreasing absolute values, of a symmetric kernel of class Cm (m ≥ 0) satisfy λn = o(n−m−½). Reade [5, 6] recently proved that if K is, in addition, positive definite, then λn = o(n−m−1;). He has also in [4] made similar improvements of classical spectral estimates for kernels of class Lip α. James Cochran pointed out to me that allied theorems for trigonometric Fourier coefficients seem to have been neglected in the literature. The trigonometric versions turn out to be elementary; nevertheless, in their conclusions concerning the decreasing rearrangement {f^*(n)} they generalize known results about the behaviour of monotone trigonometric transforms. Furthermore they suggest that the Cm hypothesis of Reade's theorem could be relaxed.
On the Problem of Four Particles
Authors: Brill J.
Published: November 2008

The object of the present communication is to give an account of an investigation of the problem of four particles on the lines of my investigation of the problem of three particles, an account of which has been recently published by the Society.
Elliptic inclusions in a stressed matrix
Authors: Bhargava R. D. , Radhakrishna H. C.
Published: November 2008

This paper treats an extension of the problem considered by the authors in a recent paper (1). The minimum energy principle of the classical theory of elasticity was used in the above paper for evaluating the elastic field when an elliptic region (the inclusion, which could be of a material different from the rest) undergoes spontaneous dimensional change in an otherwise unstrained infinite medium (the matrix). By modification of this method, it has been possible to deal with the case when the inclusion is spherical or circular and the matrix is under uniform tension at infinity (2). The present paper deals with the much more general case when the matrix is under tension, at infinity, inclined at any angle to the major axis of the elliptic inclusion. The solution has been possible by the combination of the complex variable method coupled with minimum energy principle and superposition methods of linear elasticity theory. As a consequence we immediately derive almost without further calculation many particular cases, viz. (i) the inclusion problem in a matrix under axial tension parallel to either of the axes, (ii) under all round uniform tension (or pressure) etc. It is obvious that the results for the respective cases of a circular inclusion can be deduced from these results.

It also solves the problem of composite sections under external forces at infinity because of the complete freedom in choosing the elastic constant of the inclusion which can be different from that of the matrix. As a corollary, it solves the problem of a cavity under stress at infinity.
Equicontinuity and -recursive transformation groups
Authors: Allsbrook Janet , Bagley R. W.
Published: November 2008

In this paper we obtain results on equicontinuity and apply them to certain recursive properties of topological transformation groups (X, T, π) with uniform phase space X. For example, in the special case that each transition πt is uniformly continuous we consider the transformation group (X,Ψ,ρ), where Ψ is the closure of {πt|t∈T} in the space of all unimorphisms of X onto X with the topology of uniform convergence (space index topology) and p(x, φ) = φ(x) for (x, φ)∈ X × Ψ. (See (1), page 94, 11·18.) If π is a mapping on X × T we usually write ‘xt’ for ‘π(x, t)’ and ‘AT’ for ‘π(A × T)’ where A ⊂ X. In this case we obtain the following results:

I. If (X, T, π) is almost periodic [regularly almost periodic] and πxis equicontinuous, then (X,Φ,ρ) is almost periodic [regularly almost periodic]

II. Let A be a compact subset of X such that. If the left and right uniformities of T are equal and (X, T, π) is almost periodic [regularly almost periodic], then (X, T, π) is almost periodic [regularly almost periodic].
On the solution of an equation occurring in quantum mechanics
Authors: Grubb M. Anthea
Published: November 2008

The following equation arose in a study of the evolution of a one-dimensional quantum system:

where α and κ are positive constants, and

It will be shown that this system provides an example of exactly exponential decay. This is of interest in view of recent papers discussing the validity of the exponential decay law (see, for example, Khalfin (1)and Martin (2)).
The Photoelectric Absorption of Gamma Rays
Authors: Gray L. H.
Published: November 2008

No satisfactory formula has so far been derived theoretically for the photoelectric absorption of X-rays and γ-rays. The empirical law

has hitherto been generally accepted as giving approximately the variation of the photoelectric absorption coefficient per electron, with atomic numberZand wave length λ for X-rays of wave length greater than 100 X.U., and the validity of this law has often been assumed for γ-rays also.
On groups of linear substitutions which contain irreducible metacyclical subgroups
Authors: Burnside W.
Published: November 2008

If G is a group of finite order which contains an operation P of prime order p, permutable only with its own powers, the order of G must, by Sylow's theorem, be of the form (1 + kp) pś, where s is a factor of p – 1. The greatest subgroup of G, which contains self-conjugately {P};, the subgroup generated by P, must be a metacyclical subgroup {S, P}, where

while g is a primitive root of the congruence gs = 1 (mod. p).
On Abelian continuous groups
Authors: Newman M. H. A.
Published: November 2008

An abstract continuous group, G, is an abstract space whose points have a continuous associative multiplication law, with division. The object of this note is to sketch a proof, which will appear in full elsewhere, that if the space G is locally Cartesian and compact, and the group G is Abelian (commutative), then G is the ordinary closed translation group in n dimensions: i.e. the space is that obtained from the “cube”
The densities of the regular polytopes, Part 2
Authors: Coxeter H. S. M.
Published: November 2008

Here the notion of density is made more precise, by definition in terms of the fundamental orthoscheme. (The fact that reciprocal polytopes have the same density is a direct consequence of the precise definition.) A new criterion for regular polyhedra is given, superior to Petrie's in that it applies to degenerate, as well as to finite, polyhedra. The cases are examined in which the actual vertex figures of one polytope bound another; and this idea is developed into the process called alternation, which provides an alternative method for calculating density in four dimensions.
Contribution to the Theory of the Diffusion Pump
Authors: Wertenstein L.
Published: November 2008

There are two effects involved in the sucking action of the diffusion pump. One is the penetration of gas molecules into the mercury vapour jet, and the second is the existence of a gradient of the partial pressure of the gas in the direction of the jet, so that the pressure at the origin of the jet is much lower than at the end of it.
Notes on the electrical counter
Authors: Emeléus K. G.
Published: November 2008

1. The brush discharges from a counting point have been examined with the microscope.

2. Magnetic fields have been shown not to affect counting under certain conditions.

3. The function of the series resistance has been further studied.

4. An approximate measurement has been made of the mechanical reaction on the electrodes.

5. An attempt has been made to detect hysteresis in the discharge, with negative results.

6. An effect of intense ionization has been noted.
The β-ray spectrum of Ra E
Authors: Madgwick E.
Published: November 2008

In the course of an investigation of the β-ray spectrum of Ra (B + C) by the ionisation method, Chadwick(1) concluded that the line spectrum is superimposed on a strong continuous background. The ionisation method, however, is not one which admits of high resolving power, and doubts have been entertained whether the continuous spectrum has in fact any real existence. In this connection the case of Ra E is one of special importance, since no line spectrum from this body has been detected by the more sensitive photographic method. Experiments have therefore been undertaken with the object of determining the distribution with velocity of the numbers of particles in the β-ray spectrum of Ra E.
Complexes of Conics and the Weddle Surface
Authors: James C. G. F.
Published: November 2008

A complex or system ∞3 of conics in space of four dimensions is such that a finite number of conics pass through an arbitrary point. Linear complexes are those for which this number is unity, and are such that their curves are defined by conditions of incidence with fixed surfaces, curves and points. In this paper are discussed briefly the linear complexes defined by the condition that their curves meet an irreducible curve in four points. Denoting by a curve of order m and genus p it is found that the curves in question are The complex associated with is considered in greater detail, since it is found to have an interesting connection with the well-known Weddle quartic surface of ordinary space. In fact the conics of the system touching a space (of three dimensions) do so in the points of such a surface. The main properties of this surface can be thence deduced. In addition we discuss certain results in connection with this curve . The paper closes with certain enumerative results which were obtained in the course of the researches giving the results recorded and which we believe are worth record.
On the Equation of State of an Ionised Gas
Authors: McCrea W. H.
Published: November 2008

The equation of state of an ionised gas has been discussed from time to time by various writers*. It is of importance chiefly in astrophysical applications. The simplest form of the problem, and the only form considered in this note, is to find the pressure in a gas of given temperature in which the numbers of atoms per unit volume at the different possible stages of ionisation are supposed known. In describing the ionisation an electron is counted as “free” when its total energy is positive. If this problem can be solved we should have a good estimate of the relative importance of the various physical factors at work, without tackling the more difficult, though more natural, problem of finding the ionisation for given temperature and pressure.
The Added Mass of Prisms Floating in Water
Authors: Browne A. D. , Moullin E. B. , Perkins A. J.
Published: November 2008

When a floating body performs free simple harmonic vibrations in a vertical plane it will create a field of pulsating current flow in the liquid, and the total kinetic energy of the whole moving system will correspond to that of the floating body with an enhanced mass. The total apparent mass of the body is often called the “virtual mass” and the apparent increment the “added mass,” and these terms will be used subsequently. We are concerned in this paper with finding the “added mass” of rectangular and triangular prisms floating in water. The authors' concern with this problem is related to their investigation of the natural frequency of lateral vibration of a ship's hull, and they have contributed already two papers to these Proceedings which should be associated with the present one. In their ultimate problem the body is not moving as a whole in a vertical plane but is flexing about two or more nodal points, but in the present problem the floating body is supposed rigid and all parts of it are moving vertically with the same velocity. It is believed the present investigations are applicable to the lateral vibrations of ships' hulls, but whether or not this is so the investigation is complete in itself: for example it may be used to calculate the natural frequency of a punt or pontoon which has, say, been relieved suddenly of a dead load.
Homogeneous variables in classical dynamics
Authors: Dirac P. A. M.
Published: November 2008

The well-known methods of classical mechanics, based on the use of a Lagrangian or Hamiltonian function, are adequate for the treatment of nearly all dynamical systems met with in practice. There are, however, a few exceptional cases to which the ordinary methods are not immediately applicable. For example, the ordinary Hamiltonian method cannot be used when the momenta pr, defined in terms of the Lagrangian function L by the usual formulae pr = ∂L/∂qr, are not independent functions of the velocities. A practical case of this kind is provided by the electromagnetic field, considered as a dynamical system with an infinite number of degrees of freedom, since the momentum conjugate to the scalar potential at any point vanishes identically. Again, for the very simple example of the relativistic motion of a particle of zero rest-mass in field-free space, the Lagrangian function vanishes and the usual Lagrangian method is not applicable.
Rational normal octavic surfaces with a double line, in space of five dimensions: Addition
Authors: Babbage D. W.
Published: November 2008

In a recent paper in these Proceedings on the rational normal octavic surfaces with a double line in [5] I found four such surfaces, , and representable on a plane respectively by the systems of curves C5 (22, 19), C6(26, 14), and C7 (3, 28), with the base points in each case lying on an elliptic cubic. Inadvertently I overlooked a solution of certain indeterminate equations which leads to a fifth type represented by the plane system C9(38, 1).
Born's electrodynamics in complex form
Authors: Weiss P.
Published: November 2008

Born's electrodynamics, together with its quantization, has been developed in several papers by Born and Infeld†. It is a general concept, containing many different systems of electrodynamics, of which Maxwell's theory is one particular case. Each system is characterized by a Lagrangian (or Hamiltonian) function of two invariants F and G (or P and Q). These invariants are built up by means of two antisymmetrical tensors, which are the “field quantities”.
Momentum distribution in molecular systems
Authors: Duncanson W. E.
Published: November 2008

The momentum distribution for the electron in the hydrogen molecular ion has been calculated for various wave functions, including the one used by James with which he obtained such a good value for the binding energy. The method adopted for this particular wave function is outlined and the results show appreciable change with improvement in the wave function. In conclusion there are discussed the implications of the present calculations on similar work on the H2 molecule.
On Young's criteria for the convergence of Fourier series and their conjugates
Authors: Sargent W. L. C.
Published: November 2008

1. Young's criterion for the convergence of a Fourier series may be stated in the following form:

The integral

tends to zero as ω → ∞ whenever

(A) ø(t) is absolutely integrable in (0, a),

(B) ø(t) → 0 in mean as t → 0,

(C) tø(t) is of bounded variation to the immediate right of t = 0 and, as t → 0,
On Certain Loci of Three Dimensions Representable on Ordinary Space by means of Cubic Surfaces, and the Cremona Transformations for Ordinary Space obtained by Projection of such Loci
Authors: Semple J. G.
Published: November 2008

If we have one birational representation of a threefold locus V on an ordinary space S3, and if we can project V birationally on to another space S3′, then clearly there will be a Cremona correspondence between the two spaces S3 and S3′. This paper deals, in the first place, with threefold loci in higher space which can be represented birationally on ordinary space by means of cubic surfaces; and Cremona transformations for ordinary space are then obtained, as indicated above, by projecting such loci. In particular it will be shown that most of the more important cubic transformations and their reverses can be obtained very simply by this method.
An analogy for beams of particles of a reciprocal optical theorem due to Helmholtz
Authors: Fowler R. H.
Published: November 2008

There is a theorem in geometrical optics due to Helmholtz which states that if a ray of light (1) after any number of refractions and reflections at plane or nearly plane surfaces gives rise (among others) to a ray (2) whose intensity is a certain fraction f12 of the intensity of the ray (1), then on reversing the path of the light an incident ray (2)′ will give rise, among others, to a ray (1)′ whose intensity is a fraction f21 of the ray (2)′, such that

The surfaces if not plane must be such that their radius of curvature is large compared with the lateral extent of the ray and this lateral extent must itself be large compared with the wave-lengths in the ray. In view of the many recent studies of reflection and transmission (refraction) of beams of electrons at potential walls, it seems worth while to ask whether there exists an analogy of the optical theorem for beams of electrons (or other particles). It is easy to see that the analogy must exist by using the principle of detailed balancing required by an assembly in statistical equilibrium. But a direct “dynamical” proof is also easy and may prove of sufficient practical interest to put on record here.
The mean value of the fluctuations in pressure and pressure gradient in a turbulent fluid
Authors: Taylor G. I.
Published: November 2008

In the general systems of vortices represented by (1) the mean variation in pressureis

whereKis a number which varies between 1 and √2. When the vortices are confined to cubical partitions, the case most nearly analogous to that of free turbulence,K= 1·06, so that the conjecture which I made some years ago, thatwould be equal tois probably nearly correct.
The magnetic properties of antimony
Authors: Shoenberg D. , Uddin M.Zaki
Published: November 2008

The magnetic susceptibility of antimony both parallel and perpendicular to the trigonal axis is independent of field down to 4° K. The numerical value of the susceptibility parallel to the trigonal axis decreases with increasing temperature, similarly to that of bismuth, but perpendicular to the trigonal axis there is no temperature dependence. The results at higher temperatures are compared with earlier measurements and the comparison suggests that the susceptibility of antimony, like that of bismuth, is very sensitive to addition of foreign elements.
A repetition of the Bothe-Geiger experiment
Authors: Burcham W. E. , Lewis W. B.
Published: November 2008

A repetition has been made, with the introduction of electrical counting methods, of the experiment of Bothe and Geiger on the simultaneous production of a scattered quantum and of a recoil electron in the scattering of X-rays. Evidence has been obtained that such coincidences do actually occur, in agreement with the results of the original experiment, and with the Compton theory of scattering.
Superlattice formation of the type AB in alloys with applications to the theory of adsorption
Authors: Chang T. S.
Published: November 2008

Calculations are made to investigate the behaviour of ordering near the critical temperature for alloys with a superlattice of the type AB for various compositions. No fluctuation among the sites of the same kind, which may lead to the possible existence of some other types of superlattice, say, AB3, is considered. Results showing the change of ordering with T near Tc are plotted.

The theory is also applied to the adsorption of gases upon a solid surface, where the sites of accommodation are regularly arranged. It is shown that under certain conditions, a superlattice formation in the adsorbed layer is possible.
A reversible head for the automatic curve following device
Authors: Williams F. C.
Published: November 2008

In the preceding paper, Blackett and Williams described an automatic curve follower for use with the differential analyser. The device would only operate if the x-coordinate was either continuously increasing or continuously decreasing; it could not handle input functions where the x value pulsated, as may often happen when the x-coordinate is not the independent variable.
The exchange force between three heavy particles due to the meson exchange field
Authors: Jánossy L.
Published: November 2008

The exchange forces between two heavy particles (neutrons, protons) have been treated on the basis of the meson theory by Yukawa, Sakata and Taketani (5), and by Fröhilch, Heitler and Kemmer (2), assuming the existence of positive and negative mesons only. The calculations have been extended by Kemmer (3) for a meson field which includes in addition neutral particles (neutretto's). The first order approximation of the exchange potential, i.e. the potential due to the emission and subsequent absorption of one meson only, gives qualitatively the correct interaction between the unclear particles.
Momentum Distribution in Molecular Systems
Authors: Coulson C. A. , Duncanson W. E.
Published: November 2008

The momentum distribution has been calculated for an electron in one of the hybridized orbitals of a carbon atom, and is shown as a function of the degree of hybridization. The mean momentum shows a steady increase from pure s to pure p binding. Similar calculations have been made for the momentum distribution in the C—H bond, and it is shown that this also depends upon the degree of s-p mixing. The mean momentum for the bond is less than for an isolated hybridized carbon orbital, but behaves similarly to such an orbital when the degree of hybridization is varied. The differences are about within the limit of experimental determination.
The motion of a solid through an infinite liquid under no forces
Authors: Ursell H. D.
Published: November 2008

1. Introduction. It is a classical result in hydrodynamics that a solid moving in an infinite liquid under no forces is capable of steady translational motion in any one of three mutually perpendicular directions. In the general case such a motion is only possible in three directions, though of course particular solids are capable of steady motion without rotation in an infinity of directions.
A generalization of additive ideal theory
Authors: Grundy P. M.
Published: November 2008

The division of ideal theory into the additive and multiplicative branches of the subject is a marked one of long standing. It is the object of this paper to carry the distinction a stage further, by showing how the group-theoretic methods of the additive branch can be extended to modules, that is, to Abelian groups, provided the ring of operators is commutative.
The complete system of the binary (4, 1) form
Authors: Todd J. A.
Published: November 2008

1. In this paper we shall show that the complete system of concomitants of a binary (4, 1) form in two sets of binary variables (x0, x1), (y0, y1) consists of 63 forms. The methods employed are those of a previous paper.
Note to the preceding paper by C. V. H. Rao
Authors: Baker H. F.
Published: November 2008

If (a1, b1, c1), (a2, b2, c2), (a3, b3, c3) be three sets of numbers subject to the six relations such as and with a positive determinant, then the four tetrahedra, M, M1, M2, M3, whose angular points have coordinates given by the rows of the four matrices

are such that every two are mutually inscribed. The sixteen points so arising are the nodes of a Kummer surface having for one form of its equation
A note on white light fringes
Authors: Shoenberg D.
Published: November 2008

It is shown that the white light fringe obtained with Brewster's fringes or with ‘scatter’ fringes has a breadth which decreases with the reflexion coefficient R of the film surfaces. The decrease of intensity from the centre of the fringe is given by the usual Fabry-Perot formula except that R is replaced by R2.
Rectilinear flow of non-Bingham plastic solids and non-Newtonian viscous liquids. II
Authors: Oldroyd J. G.
Published: November 2008

The equation of steady rectilinear motion of a material of variable viscosity is transformed, when the pressure gradient vanishes, into a second-order differential equation which is linear in the dependent variable. A general solution of the linearized equation can be obtained in the case of a wide class of plastic solids and viscous liquids. By making use of the idealized materials which lend themselves to this method of solution, a theoretical comparison can be made of the behaviour of plastic solids of different types under the same conditions. As an illustration, a plastic solid which yields very sharply when the yield point is first exceeded is compared with one which yields only gradually at the same yield value; the flow caused by the longitudinal motion of an (approximately) elliptic cylinder in an infinite mass of material is investigated in each case.
On a theorem of S. Bernstein
Authors: Chang Shih-Hsun
Published: November 2008

S. Bernstein (see(l), pp. 198–204) has proved that if

is an entire function of genus zero, then the series

is convergent. By considering the most unfavourable case, when all the (βn) are positive real numbers, and writing

which is convergent by hypothesis, he proved that

.
The transients arising from the addition of heat to a gas flow
Authors: Stocker P. M.
Published: November 2008

Problems involving the addition of heat to a flowing gas have received considerable attention in recent years. The practical interest lies in the application to ram jets and similar engines, and the presence of entropy gradients makes the problem of theoretical interest.
The long-wave paradox in the theory of gravity waves
Authors: Ursell F.
Published: November 2008

The theory of long waves in shallow water under gravity employs two different approaches, which have given rise to a well-known paradox remarked by Stokes, but hitherto not fully resolved. On the one hand it was shown by Airy (1) that, if the pressure at any point in the fluid is equal to the hydrostatic head due to the column of water above it, then no wave form can be propagated without change hi shallow water of uniform depth; on the other hand, it was shown by Rayleigh's theory (8) of the solitary wave that this conclusion may be incorrect. In the present paper an attempt is made to elucidate the paradox. Waves of small amplitude η0 and large horizontal wave-length λ (compared with the depth h of the water) are studied, and it is shown that Airy's conclusion is valid if η0λ2/h3 ia large, whereas the solitary wave has η0λ2/h3 of order unity. Equations of motion are derived corresponding to large, moderate and small values of η0λ2/h3; these can be summarized in a single equation for the profile η(x, t):

due to Boussinesq (3).

It is also shown that the linearized theory of surface waves is valid only if η0/λ and η0λ2/h3 are both small. Some remarks are made on the generation of the solitary wave, and on the breaking of waves on a shelving beach.
A geometrical factor in the variability of sensations of warmth evoked by radiation
Authors: Weight G. H. , Rankin R. A.
Published: November 2008

It is a commonplace observation in experiments on cutaneous sensation that a constant stimulus, applied under supposedly constant conditions, evokes a varial response. In this paper it will be shown that the variability is in part a geometri consequence of the fact that the sense organ is not a continuum but an aggregati of discrete, sensitive ‘spots’. The total extent of the ‘spots’ for one modality, acti at a given time, is only a small fraction of the area of the site tested; this is illustrat in all of the classical maps, and is particularly true of the warmth-spots (Rein (8)). I purposes of theoretical discussion, the sensory spots may therefore be treated points that form an array or pattern. This pattern is unlikely to be regular and mt indeed, be highly irregular. No assumptions about its regularity are necessary the present argument.
On the definition of tangents to sets of infinite linear measure
Authors: Besicovitch A. S.
Published: November 2008

Given a plane set E of positive and finite linear measure, the tangent at a point x at which the upper density of E is positive (which is so at almost all points of E and is not so at almost all points outside E) is defined in the following way. The line l through x is said to be the tangent to the set at the point x if in any angle vertex x that leaves the line outside the density of E is zero. This definition when applied to sets of infinite linear measure leads often to the conclusion that no tangent exists in cases when the structure of the set singles out some lines that have strong claim to be tangents to the set.
Classical electrodynamics as a distribution theory
Authors: Taylor J. G.
Published: November 2008

In this paper a precise distribution-theoretic formalism for the description of point charges interacting through a classical electromagnetic field is given. The distribution solutions of the Maxwell equations are shown to reduce to the Lienard–Wiechert fields, which are summable over the whole of space-time. It is not possible to obtain directly a unique field intensity, on the world lines, from these distribution solutions. A brief analysis shows that analytic continuation methods also do not give a unique field intensity on the world lines. The energy tensor for the field is a distribution which can only be rigorously defined in terms of the motions of the charges. Thus only the value of the field energy or momentum residing in any finite region of space-time can be given any meaning. Equations of motion for the charges may be derived from the field energy tensor, using the principle of conservation of total energy and momentum for the system. These equations agree with those obtained previously by Dirac (2). Also any canonical formalism in which the field potentials and the particle variables enter in an equivalent manner is not possible. The difficulty of the definition of products of distributions in the stress tensor does not occur.
A theory of the propagation of bores in channels and rivers
Authors: Abbott M. R. , Lighthill M. J.
Published: November 2008

A theory is presented of the non-linear propagation of waves and bores in channels of varying cross-section with a basic steady flow governed by frictional resistance; this corresponds to the flow in tidal rivers. The theory provides a condition on the tidal range required to produce a bore, in terms of the geometry and friction parameters of the river, and the propagation of such a bore is then described. The theory is applied to the River Severn and the results agree satisfactorily with observation. The results for the special case of waves moving into still water in a channel of varying section are also noted in detail.
A consistency condition for electron wave functions
Authors: Eliezer C. Jayaratnam
Published: November 2008

Dirac's wave equation of an electron enables one to solve for the wave function of an electron moving in an electromagnetic field. The wave function ψ has 4 components ψ1, ψ2, ψ3, ψ4, and the electromagnetic field is described by a four-vector Aμ, consisting of a scalar potential φ and a vector potential A.

In the usual problem, we solve for the wave function ψ when the potential Aμ is given. By doing so, we obtain the wave function of an electron which moves in a given electromagnetic field. It is of some interest to consider the reverse question: given the wave function ψ, what can we say about the electromagnetic potential Aμ, which is connected with ψ by Dirac's equation? Is Aμ uniquely determined, and if not, what is the extent to which it is arbitrary ?
The concept of order
Authors: Bastin E. W. , Kilmister C. W.
Published: November 2008

Quantum mechanics is formulated in terms of the sequences introduced in earlier papers of the series. This reformulation resolves some difficulties in the foundations of quantum theory and gives a new picture of an elementary particle.
The volume of a certain set of matrices
Authors: Jack Henry , Macbeath A. M.
Published: November 2008

The object of the present paper is to evaluate m(k), the invariant measure (defined in §2) of the set of real matrices with determinant 1 and norm bounded by a positive real number k. The norm is that induced on the linear transformations of Euclidean n-space by the usual Euclidean norm in that space, and is also defined in § 2. We were led to this problem as a result of work arising from the Geometry of Numbers. A knowledge of the behaviour of m(k) was necessary in order to generalize a certain mean value formula (Macbeath and Rogers (9)) from Riemann to Lebesgue integrable functions.
Roots and Wreath Products
Authors: Baumslag Gilbert
Published: November 2008

B. H. Neumann (5) has shown, by using free products with an amalgamated subgroup, that every group can be embedded in a divisible group. This theorem can also be proved, in an entirely different way, by making use of the wreath product (Baumslag(1)). In view of this connexion between roots in groups and the wreath product, a study is made here of the conditions which ensure that the wreath product (and also the unrestricted wreath product) shares certain properties regarding the existence and uniqueness of roots with the groups from which it is composed.
A remark on the proof of dispersion relations in quantum field theory
Authors: Taylor J. G.
Published: November 2008

In order to complete fully the proof of the fixed momentum-transfer dispersion relations for elastic scattering in quantum field theory given by Bremermann, Oehme and Taylor (1) it is necessary (2) to discuss more fully the analyticity of the absorptive part A(t, γ, Δ) in the mass variable γ. We are using the notation of (1), with 2Δ the momentum transfer and 4t2 the total energy of the initial system. For simplicity of discussion we will only treat the case of the elastic scattering of equal mass particles of mass m, the other cases discussed in (1) can be treated similarly. The crucial point of the proof, as first pointed out in (3), is to prove that

is regular in γ in a strip . It was only proved explicitly in (1) that A(t, γ, Δ) is regular for each t, in a strip , where δ may depend on t. In order to prove the analyticity in S of the integral in equation (1) it is necessary to show that δ(t) may be chosen to be independent of t, at least above some finite value of t (possibly large).
On the solution of certain integral equations by generalized functions
Authors: Miller J. B.
Published: November 2008

This note is concerned with a method by which generalized solutions can be shown to exist for certain types of integral equations

where f(x) ∈ L2(0, ∞). The method is briefly this. An extended meaning is given to such equations by using generalized functions of a particular type. Then, if (1) denotes a transformation which has no everywhere-defined inverse in the usual sense, it may be possible to define in the extended sense an inverse transformation

so that if f is given, a generalized function g = can be determined as a solution of (1).
A Uniqueness Theorem for the Exponential Series of Herglotz (II)
Authors: Verblunsky S.
Published: November 2008

The notation of the introduction of the first paper (1) will be used without further explanation. All page references, and also references to Theorems 1, 2, 3, 4 or to numbered equations, will be understood to refer to the pages, theorems or equations* so numbered in (1). The theorems of the present paper are denoted by primed numbers.
On the thickness of the complete bipartite graph
Authors: Beineke Lowell W. , Harary Frank , Moon John W.
Published: November 2008

A graph consists of a finite set of points and a set of lines joining some pairs of these points. At most one line is permitted to join any two points and no point is joined to itself by a line. A graph G′ is a subgraph of the graph G if the points and lines of G′ are also points and lines of G. The union of several graphs having the same set of points is the graph formed by joining two points in this set if they are joined in at least one of the original graphs. A graph is planar if it can be drawn in the plane (or equivalently, on a sphere) so that no lines intersect. The thickness of a graph G is defined as the smallest integer t such that G is the union of t planar subgraphs.
Solutions of unsteady boundary-layer equations
Authors: Sarma G. N.
Published: November 2008

The unsteady two-dimensional boundary-layer equations, linearized as by Lighthill are studied. A unified method is developed, from which the results for the stagnation flow, the flow along a flat plate, the flow in a converging canal, etc., can be derived as special cases. Solutions are obtained in two systems, one when the main stream is in unsteady motion and the wall is at rest and the other when the main stream is in steady motion and the wall is in an arbitrary motion. The stagnation flow has been done by Glauert and generalized by Watson. The flow along a flat plate and the flow in a converging canal are considered in detail.
Polynomial modules and general polynomials
Authors: Reufel M.
Published: November 2008

Throughout k denotes a field and k[X] = k[X1, …,Xn] the polynomial ring in n indeterminates over k.
On topological games
Authors: Pears A. R.
Published: November 2008

1. A topological game is a game in which the positions are the points of a topological space; the set of possible moves from any such point varies continuously with the point. Play can start from any point of the space and at each point it is specified which of the players has the initiative. Play ends when a position is encountered at which the set of positions from which the player with the initiative can choose is empty. The payoff to each player depends on the set of positions met in the play. The notion of topological game is due to Berge (1), (2).
On the Hausdorff dimension of general Cantor sets
Authors: Beardon A. F.
Published: November 2008

Introduction and notation. In this paper a generalization of the Cantor set is discussed. Upper and lower estimates of the Hausdorff dimension of such a set are obtained and, in particular, it is shown that the Hausdorff dimension is always positive and less than that of the underlying space. The concept of local dimension at a point is introduced and studied as a function of that point.
Note on the relations between series equations and integral equations
Authors: Ashour A. A.
Published: November 2008

Williams ((3)) has transformed certain series equations in Legendre functions to Bessel integral equations; his main result being that if

then

and vice versa. (Here, n is written for Williams's n + m. The coefficients an are related to Williams's An by a factor. There is an extra factor 2 in equation (7) in Williams's paper which appears to be an error.)
On Dieudonné's paper: Sur les espaces de Köthe
Authors: Phuong-Các Nguyen
Published: November 2008

An error in Dieudonné's paper, Sur les espaces de Köthe, published in Journal d' Analyse Mathématique, 1 (1951), 81–115, is pointed out and a correction put forward.
On the Mathieu group of degree twelve
Authors: Whitelaw T. A.
Published: November 2008

Introductory. In the following pages a miscellany of information about M12, the Mathieu group of degree 12, is presented, mainly concerning its subgroups. The development is inspired by the work of Coxeter and Todd on the representation of M12 as a group of collineations in PG (5,3)—i.e. five-dimensional projective space over the field of 3 elements.
An asymptotic bound for the residual area of a packing of discs
Authors: Larman D. G.
Published: November 2008

Suppose that a sequence of discs

arranged in decreasing order of diameters, forms a packing within the unit plane square I2. It has been shown, by Florian(1), that the area of

is at least O(a), where a is the radius of θn. However, Gilbert (2) has produced some empirical results for the Apollonius packing 71 of discs which seem to suggest that for such a packing, the area of the set

is at least O(as) for some positive real number s, less than one. As Gilbert remarks, it is difficult to imagine that the Apollonius packing is not the extremal case, and so, that it would seem likely that there exists a positive real number s, less than one, such that for a general packing, the area of

is at least O(as). The purpose of this paper is to establish this result by showing that 0·97 is an allowable value for s.
On ‘translated quasi-Cesàro’ summability
Authors: Kuttner B.
Published: November 2008

Corresponding to a fixed sequence {μn}, the Hausdorff method of summability (H, μn) is defined by the sequence-to-sequence transformation†

where we write

The quasi-Hausdorff method (H*, μn) is defined by the transformation

thus the matrix of the (H*, μn) transformation is the transpose of that of the (H*, μn) transformation. A method introduced by Ramanujan (9), which we will call‡ (S,μn) is given by the transformation

Thus the elements of row n of the matrix of the (S, μn) transformation are those of the corresponding row of the (H*, μn) transformation moved n places to the left.
The integration of generalized hypergeometric functions
Authors: Srivastava H. M.
Published: November 2008

In the usual notation for generalized hypergeometric functions we let

where

and (a) denotes the sequence of parameters

Throughout the present paper we shall suppose that there are A of the a parameters, Bof the b parameters, and so on. Thus ((a))m is to be interpreted as

and similar interpretations hold for ((b))m, etc.