bluuebunny/crossref_metadata_2025 huggingface dataset:
An application of local uniformization to the theory of divisors
Authors: Northcott D. G.
Published: November 2008

If V is an irreducible variety and W is an irreducible simple subvariety of V, then one of the properties of the quotient ring of W in V is that it is a unique factorization domain. A proof of this theorem has been given by Zariski ((2), Theorem 5, p. 22), based on the structure theorems for complete local rings, and the fact that the local rings which arise geometrically are always analytically unramified. Here the theorem is deduced from certain properties of functions and their divisors which will be established by entirely different considerations. The terminology which will be employed is that proposed by A. Weil in his book(1), and we shall use, for instance, F-viii, Th. 3, Cor. 1, when referring to Corollary 1 of the third theorem in Chapter 8. Before proceeding to details it should be noted that Weil and Zariski differ in then-definitions, and that in particular the terms ‘variety’ and ‘simple point’ do not mean quite the same in the two theories. The effect of this is to make Zariski's result somewhat stronger than Theorem 3 of this paper.
A note on Riesz potential
Authors: Auluck F. C. , Kothari L. S.
Published: November 2008

The object of the present paper is to discuss the Fourier expansion of the Riesz potential. For this purpose a new definition of the electromagnetic potentials, depending upon an arbitrary parameter α is given. It is shown that this definition is a generalization of the Wentzel potentials in the α-plane, whereas that given by Fremberg (3) is a generalization of the Maxwell potentials. The analysis is applied to the problem of eliminating, in a straightforward way, the longitudinal part of the potential describing the electromagnetic field. The problem of the quantization of the field, based on its Fourier expansion, will be considered in another paper. The recent work of Tomonaga, Schwinger and Dyson, and the regularization process of Pauli has lifted the theory of quantum electrodynamics to a much higher level of rigour and fruitful applicability. All the same, a further study of Riesz potential seems to us of some interest in this field.
Non-measurable interpolation sets
Authors: Noble M. E.
Published: November 2008

In two classical papers (1, 2) J. M. Whittaker introduced the study of integral functions bounded at the lattice points m + in(m, n = 0, ± 1, …,). He succeeded in showing (cf. also G. Polya(3)) that an integral function of at most the minimum type of order 2 uniformly bounded at the lattice points was necessarily constant. This result was improved almost simultaneously by A. Pflüger(5) and V. Ganapathy Iyer(11), who showed that the result was true also for functions of type K<½12π of order 2. The example of Weierstrass's σ(z) function shows that theirs is a best possible result in this direction.
Some geometrical properties of plane flows
Authors: Hansen A. G. , Martin M. H.
Published: November 2008

The geometrical configuration consisting of five straight lines tangent to the stream line (ψ = const.), isobar (p = const.), isocline (θ = const.), isovel (q = const.) and isopycnic (ρ = const.) at a point P of the physical plane in the plane flow of a fluid has a number of interesting properties. The study of this geometric configuration for a wide class of ideal fluids (which includes polytropic gases and incompressible fluids), in particular its properties at sonic points, is the purpose of this paper.
Tables for numerical integration at non-equidistant argument steps
Authors: Hartley H. O.
Published: November 2008

1. Published tables facilitating the numerical evaluation of the definite integral

from given numerical valuesf(xi) of the integrand in the form of a weighted sum

essentially deal with the following situations:

(a) The argumentsxiare equidistant, i.e.xi=x0+iw;i=integer and the weight coefficientsciare tabulated for all argumentsxioccurring in the summation. Among these tables the most important are:

(a1) panels in which the end-points of the integration a andbare two of thexi, and in particular

(a11) panels confined to the range of the integration in which

(a12) panels extending beyond the range of integration which usually have the same number of arguments,k, below a and beyondb, so that

(a2) panels in which a and b are at distance ½wfrom one of thexi, and in particular

(a21) panels confined to the range of integration in which

(a22) panels extending beyond the range of integration which usually have the same number of (k) points belowaand beyondb, so that

(b) Thenargumentsxiare not equidistant but are specially chosen so that for a givennthe sumSapproximates toIwith highest precision. Gauss's formula (see, for example, Milne-Thomson (4)) and various formulae of approximate product integration (see, for example, Beard (l)) are examples of these. In Chebyshev's integration formula (see, for example, Runge and König (5)) thexirequired for equal weights ctare used, resulting in an important formula when the ordinatesf(xi) are affeected by observational errors of equal variance.
A note on Janossy's mathematical model of a nucleon cascade
Authors: Ramakrishnan Alladi
Published: November 2008

In a contribution (3) to these Proceedings the author considered stochastic problems in physics where one had to deal with a stochastic variable representing the number of particles distributed in a continuous infinity of states characterized by a parameter E, and where the distribution varied with another parameter t which might be continuous or discrete (if t represents time or thickness, it is of course continuous). The author introduced the concept of product densities and derived some general results relating to the functions representing these densities. Recently, Janossy(2), using a certain mathematical model for a nuclear cascade, introduced certain functions which bear a close relation to the product-density functions. The object of this note is to establish a complete correspondence between these two sets of functions and apply them to the specific problem of the development of a nucleon cascade. The diffusion equations involving product densities can be derived from the diffusion equations involving Janossy's functions.
Overlapping energy bands and the theory of collective electron ferromagnetism
Authors: Lidiard A. B.
Published: November 2008

The theory of collective electron ferromagnetism given by Stoner applies to a system of electrons in a single energy band; for iron, cobalt and nickel this is identified with the band of states derived from atomic 3d functions. To bring in the generally assumed overlapping of the 3d band by the wide 4s band in these metals, the theory has been extended to take account of the transfer of electrons from 3d to 4s states with change of temperature. A previous calculation of this transfer effect must be regarded as inadequate, for the part played by the exchange energy in determining the distribution of electrons between the two sets of states was omitted. The general equations are derived in § 2 and used as a basis for discussion of the properties of nickel-copper alloys at absolute zero in § 3. In §§4 and 5 numerical results are presented which show the effect of the overlapping 4s band on the magnetic properties of a system such as nickel both above and below its Curie point. Comparison with the measured paramagnetic susceptibility of pure nickel above the Curie point strongly suggests that for this metal the overlapping 4s band has only a minor influence, although in principle the effect could be very large (cf. Fig. 4). This result is not unambiguous, however, because values thus inferred for the two unknown parameters lead to inaccurate predictions below the Curie point. First, the predicted values for the spontaneous magnetization are too small. Secondly, the theory demands that the nickel-copper alloys should only be ferromagnetic below a copper content of about 20 %, whereas experimentally the limit is known to be about 60 %. In conclusion, it is suggested that the implicit assumption of Stoner's theory that the exchange integrals between all pairs of 3d states are equal to one another is a more serious restriction on the theory than the consideration of only a single energy band.
The isotherms of an imperfect gas
Authors: ter Haar D.
Published: November 2008

The liquid drop model of an imperfect gas in the form introduced by Wergeland is discussed by using the method of the grand ensembles and the equation of state of the system is derived. This equation of state is of the same general type as the one derived by Mayer for a more general model. It is shown that in both cases the isotherms consist of two analytically different parts in the limit where the number of particles in the system, N, goes to infinity.
Conduction of heat in a solid with periodic boundary conditions, with an application to the surface temperature of the moon
Authors: Jaeger J. C.
Published: November 2008

The object of this note is to indicate a numerical method for finding periodic solutions of a number of important problems in conduction of heat in which the boundary conditions are periodic in the time and may be linear or non-linear. One example is that of a circular cylinder which is heated by friction along the generators through a rotating arc of its circumference, the remainder of the surface being kept at constant temperature; here the boundary conditions are linear but mixed. Another example, which will be discussed in detail below, is that of the variation of the surface temperature of the moon during a lunation; in this case the boundary condition is non-linear. In all cases the thermal properties of the solid will be assumed to be independent of temperature. Only the semi-infinite solid will be considered here, though the method applies equally well to other cases.
A Frequency-function form of the central limit theorem
Authors: Smith Walter L.
Published: November 2008

The central limit theorem in the calculus of probability has been extensively studied in recent years. In its simplest form the theorem states that if X1, X2,… is a sequence of independent, identically distributed random variables of mean zero, then under general conditions the distribution function of Zm = (X1 + … + Xn)/√ n converges as n → ∞ to the normal or Gaussian distribution function. This form of the theorem in terms of distribution functions is the one required in statistical work, since it enables statements to be made about the limiting behaviour of prob {a ≤ Zn ≤ b}.
A case of gaseous accretion
Authors: Dodd K. N.
Published: November 2008

The isothermal flow of gas past a gravitating point-sink has been studied for four values of the undisturbed gas velocity, all of which are supersonic. From these studies, estimates have been made of the rates of accretion of an isothermal gas by a star moving with a supersonic velocity. These estimates are of the same order as those obtained by earlier workers who used simpler mathematical models.
Modified propagation functions in perturbation theory
Authors: Salam Abdus
Published: November 2008

Modifications to the Feynman propagation functions Sp and Dp are suggested in order to deal with field theoretic problems involving the presence in the initial and the final states, of free fermions or bosons, having all values of momentum (and energy) within a certain specified range.
A free-electron theory of conjugated molecules
Authors: Griffith J. Stanley
Published: November 2008

The values of a free-electron eigenfunotion at the carbon nuclei of a conjugated hydrocarbon are found to satisfy a system of algebraic equations. These equations are similar in form to those obtained in the method known as the linear combination of atomic orbitale but only coincide with them for linear polyenes and benzene. The symmetry, degeneracy and energy of the eigenvectors of these free-electron equations correspond exactly to those of the free-electron wave functions found by the usual methods. From this correspondence, a theorem is deduced about the free-electron charge density in alternant hydrocarbons.
Self-consistent electrodynamics
Authors: Buneman O.
Published: November 2008

The idea of direct action between streams is applied to a continuous charged fluid and combined with the new formulation of the electrodynamical laws of motion in terms of conservation of circulation. A simple and rigorous integrated formulation is thus obtained from the Maxwell-Lorentz differential equations, applicable to co-existing positive and negative fluids, as well as vacuum. Exact solutions are obtained, among them one which represents self-consistent, self-maintained flow in a hollow tubular region of infinite axial extent. It is hoped this tube might be bent into a torus and that an electron model will result from merely quantizing the one or two vortices around which this flow-pattern circulates.
Lattice coverings with four-dimensional spheres
Authors: Bambah R. P.
Published: November 2008

1. Let Jn be a sphere with volume V(Jn) in the n-dimensional Euclidean space Rn. Let Λ be a lattice of determinant d(Λ) such that every point in Rn lies in one at least of the bodies obtained from Jn by applying to it all possible lattice translations. Then Λ is called a covering lattice for Jn and V(Jn)/d(Λ) is called the density of the lattice covering by Jn provided by Λ. The lower bound θn of V(Jn)/d(Λ) taken over all covering lattices Λ for Jn is called the density of the thinnest lattice covering by Jn.
Tauberian theorems in a Banach lattice, with applications to the LP spaces
Authors: Burgess D. C. J.
Published: November 2008

In a previous paper (2) of the author, there was given a treatment of Tauberian theorems for Laplace transforms with values in an arbitrary Banach space. Now, in § 2 of the present paper, this kind of technique is applied to the more special case of Laplace transforms with values in a Banach lattice, and investigations are made on what additional results can be obtained by taking into account the existence of an ordering relation in the space. The general argument is again based on Widder (5) to which frequent references are made.
The concept of order I. The space-time structure
Authors: Bastin E. W. , Kilmister C. W.
Published: November 2008

Even if ideas of measurement did not exist, a great deal of the simple mechanical and electromagnetic structure of the physical world could be understood. To see how this can be so, the orthodox ideas of the relationship of mathematics to physics are inadequate. In this paper a development of this fundamental non-metrical physics is made to depend on a different view of this relationship.
A note on Wild's solution of the Boltzmann equation
Authors: Klein Martin J.
Published: November 2008

It is shown that Wild's formal solution of the Boltzmann integro-differential equation can be used to obtain Maxwell's classical relaxation behaviour of the second velocity moments of the molecular distribution function.
The evaluation of the basic confluent hypergeometric functions
Authors: Slater L. J.
Published: November 2008

This paper contains all those formulae which are necessary for the numerical evaluation of the basic confluent hypergeometric functions, including asymptotic formulae, and three skeleton tables which can be used as a basis for the study of the numerical behaviour of these functions over a wide field.
Bounds for the integral of a non-negative function in terms of its Fourier coefficients
Authors: Longuet-Higgins M. S.
Published: November 2008

The first 2N + 1 Fourier coefficients of an unknown, non-negative function f(θ) are given, and it is required to find bounds for ∫Ef(θ) dθ, where E is some given region of integration. We also wish to find the interval E for which the bounds are most strict, when the width of E is specified. f(θ) may represent a distribution of energy in the interval 0 ≤ θ ≤ 2π; the object is to determine where the energy is chiefly located.

In the present paper we show that if the energy is located mainly in the neighbourhood of not more than M distinct points, significant lower bounds for ∫Ef(θ) dθ can be found in terms of the first 2M + 1 Fourier coefficients. The effectiveness of the method is illustrated by applying the inequalities to some known functions.

The results have application in determining the direction of propagation of ocean waves and other forms of energy.
Functions regular in the unit circle
Authors: Linden C. N. , Cartwright M. L.
Published: November 2008

Let

be a function regular for | z | < 1. With the hypotheses f(0) = 0 and

for some positive constant α, Cartwright(1) has deduced upper bounds for |f(z) | in the unit circle. Three cases have arisen and according as (1) holds with α < 1, α = 1 or α > 1, the bounds on each circle | z | = r are given respectively by

K(α) being a constant which depends only on the corresponding value of α which occurs in (1). We shall always use the symbols K and A to represent constants dependent on certain parameters such as α, not necessarily having the same value at each occurrence.
Two classical theorems of ideal theory
Authors: Rees D.
Published: November 2008

The purpose of this note is to present new proofs of two classical theorems in ideal theory, both due to W. Krull: the intersection theorem (see Krull(1) for the original proof, and Northcott (2), Chapter III, where the theorem is stated in the form proved below), and the principal ideal theorem ((1), or (2), Chapter III, Theorem 6). Both proofs depend on a device introduced by the author in (3) in another connexion.
Abstract dilatations and infinitely near points
Authors: Northcott D. G.
Published: November 2008

In the following pages there is developed an abstract theory of infinitely near points for two-dimensional regular local rings which have infinite residue fields. The scope of the theory may be indicated by the fact that it contains generalizations of Noether's formula for intersection multiplicities (Theorem 8), the theory of proximate points (§ 8) and the conditions for a set of integers to be curve multiplicities at a given finite sequence of consecutive points (§ 10). To this extent our account resembles that given by van der Waerden ((6), Chap. 9), but the starting point and (with the exception of the concluding section) the type of reasoning employed are very different. There is also a marked contrast with the methods used by Zariski (7) in an account of related topics.
Note on a result of Krull
Authors: Gravett K. A. H. , De Wet J. S.
Published: November 2008

The following theorem was stated and proved by Krull(l). We give a new and simple proof.

Theorem.Let K be afield with valuation υ, value group Γ and residue class field R.
A generalization of isometries to uniform spaces
Authors: Rhodes F.
Published: November 2008

Isomorphisms are, in many ways, the generalizations of isometrics to uniform spaces. Yet some theorems on isometries of metric spaces only generalize to uniform spaces in terms of more restricted transformations of the uniform space. In § 1, in the course of a discussion of a theorem on transitive groups of automorphisms, we define such a transformation and call it anisobasism. It appears that in many respects isobasisms, rather than isomorphisms, are the generalizations of isometries to uniform spaces. The results of Freudenthal and Hurewicz (7) on contractions, expansions and isometries of totally bounded metric spaces are generalized, in § 2, to contractions, expansions and isobasisms of totally bounded uniform spaces. These results, together with generalizations of some theorems of Eilenberg (6) on compact groups of homeomorphisms of metric spaces which are obtained in §3, give a characterization of isobasisms. The language of Bourbaki (2,3,4) is used throughout this note.
The real zeros of the confluent hypergeometric function
Authors: Slater L. J.
Published: November 2008

This paper contains a discussion of various points which arise in the numerical evaluation of the small real zeros of the confluent hypergeometric function

where

There are two distinct problems, first the determination of those values of x for which M(a, b; x) = 0, given a and b, and secondly the study of the curves represented by M (a, b; x) = 0, for fixed values of x. These curves all lie on the surface M(a, b; x) = 0, of course.
Transverse bending of a thin circular plate loaded normally over an eccentric circle
Authors: Bassali W. A.
Published: November 2008

Complex variable methods have been applied to isotropic and aelotropic plate problems by several authors. The notation used here is that of Stevenson(14). Dawoud(5) has expressed the continuity conditions between two differently loaded regions in terms of the complex potentials and the particular integrals for the two regions.

The problem of a transverse load at any point of a clamped circular plate was solved by Clebsch(4), Michell(11), Melan(10) and Flügge(6). A series solution for the simply supported circular plate under the same load was given by Foeppel(8). Using Stevenson's tentative method Dawoud(5) applied complex potentials to solve the problem of an eccentric isolated load under certain boundary conditions. Applying Muskhelishvili's method, Washizu(15) obtained the same results for clamped and simply supported boundaries.

It is easy to get solutions for a circular plate concentrically and uniformly loaded. For non-uniform loadings there are the solutions found by Sen (13) for certain distributions of normal thrust over the complete plate or over a concentric circle and the solution of Flügge (7) for a linearly varying load over the simply supported circular plate. The present author and Dawoud(3) obtained the solutions for a circular plate with the load over the complete plate or over a concentric circle, under a general boundary constraint including as special cases the usual clamped and hinged boundaries. Ghose (9) worked out the problem of a clamped circular plate when the load is uniformly distributed between two concentric circles and two radii. Schmidt (12) found the solution for a clamped circular plate uniformly loaded over an eccentric circle. The complex variable method was applied by the author and Dawoud(2) to obtain the solutions for a circular plate having an eccentric circular patch symmetrically loaded with respect to its centre under the general boundary condition mentioned before. The author (1) also found the solution for a linearly varying load over an eccentric circle under the same boundary condition. In this paper the power of the complex variable method is exhibited by rinding the appropriate complex potentials corresponding to the load

over an eccentric circular patch, where R, θ are measured from the centre of the patch and the common diameter of the plate and the patch. Since the two cases n = 0, 1 require special consideration and were dealt with separately (in (2) and (1) respectively), we see that this paper completes the solution of the problem of a circular plate with an eccentric circular patch symmetrically loaded with respect to the common diameter of the plate and patch, the load being in this case expressible in the form .

For a clamped boundary the solution is obtained in finite terms.
On the transformation of a continuous spectrum by refraction
Authors: Longuet-Higgins M. S.
Published: November 2008

When waves are propagated through a medium whose velocity of propagation varies gradually from place to place, the wave direction and intensity vary according to the laws of refraction. Although the geometry of ray-paths has been well explored, and so also the laws governing the intensity of a coherent train of waves, little attention has apparently been given to the variation in intensity of an incoherent beam having a broad spectrum. The transformation of the energy spectrum is of practical importance in branches of geophysics, for example, in the study of sea waves entering shallow water, or of microseismic waves propagated through inhomogeneous regions of the earth's crust. Accordingly, it seems worth while to state and prove the rule governing the transformation of the two-dimensional spectrum function of a wave disturbance undergoing refraction.
On a scale of Abel-type summability methods
Authors: Borwein D.
Published: November 2008

1. Introduction. In this note Abel-type summability methods (Aλ) are defined and some of their properties investigated.

Let

and let {sn} be any sequence of numbers. If

is convergent for all x in the open interval (0,1) and tends to a finite limit s as x → 1 in (0,1), we shall say that the sequence is Aλ-convergent to s and write sn → s (Aλ;). The A0 method is the ordinary Abel method.
The static spherically symmetric solutions in a unified field theory
Authors: Moffat John
Published: November 2008

A brief account is given of the fundamental properties of a new generalization ((1), (2)) of Einstein's gravitational theory. The field equations are then solved exactly for the case of a static spherically symmetric gravitational and electric field due to a charged particle at rest at the origin of the space-time coordinates. This solution provides information about the gravitational field produced by the electric energy surrounding a charged particle and yields the Coulomb potential field. The solution satisfies the required boundary conditions at infinity, and it reduces to the Schwarzschild solution of general relativity when the charge is zero.
A note on the global dimension of polynomial rings
Authors: Northcott D. G.
Published: November 2008

It is well known that if K is a commutative field then the polynomial ring K[x1, x2, …, xn] has global dimension (in the sense of homology theory) equal to the number of variables. This, of course, is not an isolated result but, in view of the special interest which attaches to such a familiar object as a polynomial ring, a short and comparatively elementary proof of this particular result may be of interest. The present paper is devoted to such a proof and therefore it is proper that this first section should indicate, in some detail, what knowledge of homological algebra is presupposed.
Legendre polynomials and trinomial random walks
Authors: Good I. J.
Published: November 2008

The main purpose of this note is to show some connexions between Legendre polynomials and random walks. More specifically we shall be concerned with what may be called ‘trinomial’ random walks i.e. walks on the points 0, ± 1, ± 2, …, such that the probabilities of ‘currently’ taking steps of – 1, 0, + 1 are p–1, p0, p1.
Homology and ringoids. I
Authors: Hilton P. J. , Ledermann W.
Published: November 2008

In this paper we bring together two aspects of homological algebra, one modern and the other classical. MacLane(6) has suggested the desirability of avoiding superfluous references to the objects of the category in proving fundamental results in homology theory. Accordingly, we consider categories whose maps form a ringoid in the sense of Barratt(1) and impose on the ringoid axioms, expressed in the language of ring theory, but having a close affinity to the axioms of Buchsbaum(2) for an exact category. (See also Cartan and Eilenberg(3); Appendix.) The purpose of these axioms is to ensure that the standard concepts of homology theory are applicable to the ringoid. Consequently, we call such a ringoid a homological ringoid.
An application of the Weber-Orr transform to the problem of transonic flow past a finite wedge in a channel
Authors: Helliwell J. B.
Published: November 2008

In an earlier paper (Helliwell and Mackie(3)) it was shown that steady two-dimensional flow patterns of a compressible inviscid fluid at high subsonic speed past a finite wedge could be determined quite simply when sonic velocity is attained at the shoulder of the wedge and thereafter the flow breaks away from the shoulder with a free streamline. In a subsequent paper (Helliwell (4)) a similar method of analysis has been applied to determine a flow pattern of the same general type past a finite wedge symmetrically placed in a channel, from which the case of the wedge in the free stream may be deduced as a special case. However, in a general investigation into transonic flow past wedges (Mackie and Pack (5)) it was argued that when the wedge angle or the free stream (subsonic) velocity is too small no supersonic region would develop on the wedge side, and the flow would break away from the wedge shoulder with some higher subsonic velocity, giving a free stream line. The present note examines the flow pattern which develops under these conditions for a wedge symmetrically placed in a channel with parallel walls.
Some self-reciprocal functions and kernels
Authors: Rao V. V. L. N.
Published: November 2008

1. The object of this note is to deduce some self-reciprocal functions and kernels for Hankel transforms. We denote by Rμ the class of functions which are self-reciprocal for Hankel transforms of order μ; that is, those that satisfy

where Jμ(x) is a Bessel function of order μ. In particular we write R8 for R½ and Rc for R−½.
Two bodies at rest in general relativity
Authors: Marder L.
Published: November 2008

It is well known that there is no static axisymmetric two-body solution of Einstein's gravitational field equations, if it is assumed that the bodies are separated in a certain definite sense. In this paper it is shown, by the construction of a complete physically sensible model, that static two-body solutions do exist for systems in which one body is hollow and contains the other. The stability of the particular system described is briefly discussed.
The distribution of the sizes of images reflected in a random surface
Authors: Longuet-Higgins M. S.
Published: November 2008

It has been shown recently that a fruitful method of studying the formation of waves on the sea surface is through the average intensity of reflected sunlight, as seen from different angles ((3), (4)). This gives, in effect, the statistical distribution of the components of surface slope.

Following the same line of thought, we may inquire what information could be derived from the distribution of the sizes of the reflected images of the sun. The size of an image depends essentially on the total curvature of the surface at the point of reflexion (see § 2).
Cubic forms representing arithmetic progressions
Authors: Watson G. L.
Published: November 2008

The following result has recently been proved by Lewis (3), Davenport (2) and Birch (1).

There exists an integer n0 such that every cubic form with rational coefficients and at least n0 integral variables represents zero non-trivially.

The arguments of Lewis and Birch are simple, and yield also various generalizations of this result. Davenport's proof is complicated, but it shows that the minimal n0 satisfies
The Equilibrium Stability of a System of Disk Dynamos
Authors: Lebovitz Norman R.
Published: November 2008

The stability of the equilibrium solutions of the equations describing the behaviour of a system of coupled disk dynamos is investigated. In the absence of viscous damping, it is found that systems consisting of more than two dynamos are unstable. That a single disk dynamo is stable is known. The stability of the undamped two-dynamo system has not been ascertained. When viscous damping is present, there are two equilibrium solutions, one in which all the currents are zero, and one in which they are finite. In the zero-current case, a stability criterion is found. Stability criteria are also found in the finite-current case. Further, the existence of the finite-current equilibrium state excludes the stability of the zero-current equilibrium state.
The Fredholm Solution as the Limit for a Sum of Separable Potentials
Authors: Fairlie D. B.
Published: November 2008

The scattering amplitude for a potential expressible as a sum of n separable potentials is obtained as the solution of a set of n simultaneous linear equations. As n→∞ the Fredholm solution for the integral form of the Schrödinger equation is recovered.
Linking spheres
Authors: Epstein D. B. A.
Published: November 2008

Andrews and Curtis have shown (1) that one can embed two Sn's in En+2 for n = 2, in such a way that one sphere cannot be shrunk to a point in the residue space of the other. In this paper the result is shown to be true for any n ≥ 1. (The result is obvious for n = 1.) The method is to calculate the appropriate homotopy group of the residue space of one sphere, and to show that the embedding of the other sphere represents a non-zero element of the group. The two spheres can both be embedded analytically.
Energy transfer by gravitation in Newtonian theory
Authors: Bondi H. , McCrea W. H.
Published: November 2008

The problem is considered as to whether, in accordance with Newtonian theory, energy can be transferred from one system to another across empty space by gravitational interaction alone. Familiar examples of apparent energy transfer by this means do not give an unambiguous answer since they involve some net change of gravitational potential energy and this is not localized in the theory. Two examples are given here of systems in which the potential energy is the same at the beginning and end of an operation that does produce a resultant energy transfer. The establishment of this result is significant as a preliminary to the discussion of energy transfer according to general relativity theory. The appendix gives a particular illustration of one of the examples that admits exact mathematical treatment.
On a Generalization of the Euclidean Algorithm
Authors: Cohn P. M.
Published: November 2008

In any valuated ringR, more specifically, a ring withadegree-functiond(a) as defined in §2, the notion of rightR-dependence may be denned as follows: (i) The elementsa1, …,arofRareright R-dependentif there existb1, …,br∈Rsuch that. (ii) An elementaofRis said to beright R-dependenton the elementsa1, …,arofRifa= 0 or if there existc1, …,cr∈Rsuch that.
A Variant on the Proofs of Some Inequalities
Authors: Åkerberg Bengt
Published: November 2008

THEOREM 1. If av > 0 and is a convergent series, then
The Kinematics of inertial frames
Authors: Capildeo R.
Published: November 2008

The recurring controversy about clocks and relativity reveals the need for an examination of the meaning and methods of time measurement.
A central limit theorem for systems of regressions
Authors: Hannan E. J.
Published: November 2008

A central limit theorem is proved for the estimates of the regression coefficients in a multiple system of regressions when the vector process generating the residuals is a linear process with coefficient matrices the sum of whose norms converges. The regressor variables are assumed to satisfy conditions, due to Grenander, which make their generalized harmonic analysis possible.
The transmission of surface waves under surface obstacles
Authors: Ursell F.
Published: November 2008

A train of surface waves (water waves under gravity) is normally incident on a cylinder with horizontal generators fixed near the free surface, and is partially transmitted and partially reflected. At a great distance behind the cylinder the wave motion tends to a regular wave train travelling towards infinity; the ratio of its amplitude to the amplitude of the incident wave is the transmission coefficient . The transmission coefficient is studied when the wavelength is short compared to the dimensions of the body; physically (though not for engineering applications) this is the most interesting range of wavelengths, which corresponds to the range of shadow formation and ray propagation in optics and acoustics. The waves are then confined to a thin layer near the free surface, and the transmission under a partially immersed obstacle is then small. In the calculation the boundary condition at the free surface is linearized, viscosity is neglected, and the motion is assumed to be irrotational.

At present the transmission coefficient is known only for a few configurations, all of them relating to infinitely thin plane barriers. A method is now given which is applicable to cylinders of finite cross-section and which is worked out in detail for a half-immersed cylinder of circular cross-section. The solution of the problem is made to depend on the solution of an integral equation which is solved by iteration. Only the first two terms can be obtained with any accuracy, and it appears at first that this is not sufficient to give the leading term in the transmission coefficient at short wavelengths; this difficulty is characteristic of transmission problems. By various mathematical devices which throw light on the mechanism of wave transmission, it is, nevertheless, found possible to prove that the transmission coefficient for waves of short wavelength λ and period 2π/ω incident on a half-immersed circular cylinder of radius a is asymptotically given by

when N = 2πα/λ = ω2α/g is large. Earlier evidence had pointed towards an exponential law. It is suggested that transmission coefficients of order N−4 are typical for obstacles having vertical tangents and finite non-zero radius of curvature at the points where they meet the horizontal mean free surface. For obstacles having both front and rear face plane vertical to a depth a, is probably of order e−2N approximately; if only one of the two faces is plane vertical, is probably of order e−N approximately. Thus is seen to depend critically on the details of the cross-section.
On non-negative forms in real variables some or all of which are non-negative
Authors: Diananda P. H.
Published: November 2008

In a recent paper (3) Mordell proposed for solution

PROBLEM 1. To find the constants a1, …, an so that

where xn+i = xi ≥ 0 for all i.
Stationary Cylindrically Symmetric Clusters of Particles in General Relativity
Authors: Raychaudhuri A. K. , Som M. M.
Published: November 2008

The case of a cylindrically symmetric cluster of particles moving in circles perpendicular to the axis of symmetry is discussed on the basis of the equations of general relativity. A number of special solutions are obtained, and two interesting results on the relation between the gravitational mass and the sum of the free particle masses for these clusters are deduced.
Auerbach's theorem and tensor products of Banach spaces
Authors: Ruston A. F.
Published: November 2008

In his well-known treatise, Banach mentions that there exist complete normed bi-orthogonal systems in any finite-dimensional (real) Banach space—a result he attributes to H. Auerbach ((1), p. 238). The purpose of this note is to present a proof of this result (valid for complex as well as real Banach spaces), and to give some applications to the theory of tensor products of Banach spaces.
A Punch Problem for a Transversely Isotropic Layer
Authors: England A. H.
Published: November 2008

In this note we consider the axially symmetric indentation of a transversely isotropic layer resting on a rigid foundation. The corresponding half-space problems both for isotropic and transversely isotropic materials have been solved previously (see (l), (2), (3), p. 455) and, more recently, the present problem has been solved for isotropic materials in (4) and (5) using methods different from that given in this paper. In § 2 the problem is reduced to the solution of a Fredholm integral equation of the second kind. In § 3 and subsequently the case of incomplete penetration by the punch, when the radius of the circle of contact is less than that of the end face of the punch, is considered. Finally, in §4 the results are illustrated by considering a hemispherically ended punch.
Differential equations in Banach spaces and the extension of Lyapunov's method
Authors: Lakshmikantham V.
Published: November 2008

The concept of Lyapunov's function is an important tool in studying various problems of ordinary differential equations. In the present paper we shall extend the Lyapunov's method to study some problems of differential equations in Banach spaces. Continuing the theory of one parameter semi-groups of linear and bounded operators founded by Hille and Yoshida, Kato(4) presented some uniqueness and existence theorems for the solutions of linear differential equations of the type

where A(t) is a given function whose values are linear operators in Banach space. Krasnoselskii, Krein and Soboleveskii (5,6) also considered such equations including non-linear differential equations of the type

Mlak (9) obtained some results concerning the limitations of solutions of the latter equation.
On well-quasi-ordering lower sets of finite trees
Authors: Nash-Williams C. St J. A.
Published: November 2008

A set Q is quasi-ordered if a reflexive and transitive relation ≤ is defined on Q. It is well-quasi-ordered if it is quasi-ordered and, for every infinite sequence u1, u2,… of elements of Q, there exist i, j such that i < j and ui ≤ uj. A lower set of Q is a subset P of Q such that, if x ≤ y ∈ P, then x ∈ P. The class of lower sets of Q, quasi-ordered by ⊂, is denoted by LQ, and L2Q = L(LQ), etc. The set (, say) of all finite trees is quasi-ordered by writing T1 ≤ T2 if T1 is homeomorphic to a subtree of T2. It is proved that is well-quasi-ordered for all n.
The direction of time and the equivalence of ‘expanding’ and ‘contracting’ world-models
Authors: Schumacher D. L.
Published: November 2008

It is assumed initially that only the increase of entropy defines locally the sense of advance of time. This assumption, together with the feature of global coherence of statistical processes, which is provided by the cosmic line-element, imply that there is a unique way of associating the standards with the cosmic statistical processes. There is hence a connexion between the sense of divergence of geodesics of fundamental observers and the cosmic time-sense. This is in keeping with the unmodified conservation laws of gravitation and the analogous adiabatic conservation law quite generally without dependence on the explicit form of the line element. These remarks correspond exactly to consequences of the ‘elliptic’ model in the case of de Sitter space, obtained by entirely separate geometrical considerations. A remark is made with regard to the irreversibility associated with quantum measurement.
Elastico-viscous boundary-layer flows I. Two-dimensional flow near a stagnation point
Authors: Beard D. W. , Walters K.
Published: November 2008

The Prandtl boundary-layer theory is extended for an idealized elastico-viscous liquid. The boundary-layer equations are solved numerically for the case of two-dimensional flow near a stagnation point. It is shown that the main effect of elasticity is to increase the velocity in the boundary layer and also to increase the stress on the solid boundary.
On a problem about the Epstein zeta-function
Authors: Ennola Veikko
Published: November 2008

1. Let

be a positive definite binary quadratic form with determinant αβ − δ2 = 1. A special form of this kind is

We consider the Epstein zeta-function

the series converging for . The function Zh(s) can be analytically continued over the whole s-plane and it is regular except for a simple pole with residue π at s = 1.
The transport equation with plane symmetry and isotropic scattering
Authors: Smith M. G.
Published: November 2008

The double integral equation, which takes the place of the Milne equation in the one-dimensional problem, is derived from the governing partial differentio-integral equations. An analytical solution of the problem of a distribution of point sources on a plane, when the other boundaries are at infinity, is then found. The possibility of more complicated boundary conditions is discussed.
Betweenness and order in semigroups
Authors: Gilder J.
Published: November 2008

A betweenness semigroup is a semigroup possessing a ternary relation, ‘b lies between a and c’, which is invariant under the semigroup operation. We take as our axioms of betweenness those suggested by Shepperd in (2) and given below in 1·01–1·04. It was shown in (2) that these axioms lead to just two types of betweenness; namely a linear betweenness corresponding to the order of points on an undirected straight line or a cyclic betweenness of just four points on a circle. In (3), Shepperd characterized betweenness groups and we here generalize his results to betweenness semigroups.
Stokes flow between two intersecting circular arcs
Authors: Ranger K. B
Published: November 2008

This paper considers a family of viscous flows closely related to the exact Jeffery-Hamel solution ((l), (2)) of the two-dimensional Navier-Stokes equations, for diverging or converging flow in a channel. It is known that if the walls of the channel intersect at an angle less than π then there is a unique solution of the Navier-Stokes equations in which the streamlines are straight lines issuing from the point of intersection of the walls and the flow is everywhere diverging or everywhere converging. The flow parameters depend on the total fluid mass M emitted at the point of intersection and the angle 2α between the walls. By taking the Reynolds number R = M/ν, where v is the kinematic viscosity, the stream function can be expanded in a power series in R in which the leading term is a Stokes flow. Alternatively the solution can be developed by perturbing the Stokes flow and is one of very few examples known in which a Stokes flow can be regarded as a uniformly valid first approximation everywhere in an infinite fluid region. The class of flows to be considered is a generalization of the Jeffery–Hamel flow by taking the flow region to be finite and bounded by two circular arcs which intersect at an angle less than π At one point of intersection fluid is forced into the region and an equal amount is absorbed out at the other point. It is found to the first order that the flow at the two points of intersection corresponds to the zero Reynolds number limit for diverging and converging flow, respectively. Now since the flow at these points can be developed by perturbing the Stokes flow solution it is reasonable to assume that the zero Reynolds number flow in the entire finite region bounded by the arcs is a Stokes flow since the most likely region in which this approximation becomes invalid is locally at the points of intersection but here the validity of the approximation is ensured. A comparison of the convection terms with the viscous terms verifies that this conclusion is borne out.
On embedding 4-manifolds in R7
Authors: Hirsch Morris W.
Published: November 2008

Let M denote a connected, compact, unbounded, piecewise linear (= P L) manifold of dimension 4. Denote by the normal Stiefel-Whitney class in dimension 3, with integer coefficients (twisted if M is not orientable).
Unknotting tori in codimension one and spheres in codimension two
Authors: Wall C. T. C.
Published: November 2008

We shall present this paper in the framework and terminology of differential topology though all our arguments are valid in the piecewise linear ease also, under local un-knottedness hypotheses. In particular we use Rp for Euclidean space of dimension p, Sp−1 for the standard unit sphere in it, and Dp for the disc which it bounds.
Inequalities for permanents and permanental minors
Authors: Brualdi R. A. , Newman M.
Published: November 2008

Let Ωndenote the convex set of alln×ndoubly stochastic matrices: chat is, the set of alln×nmatrices with non-negative entries and row and column sums 1. IfA= (aij) is an arbitraryn×nmatrix, then thepermanentofAis the scalar valued function ofAdefined by

where the subscriptsi1,i2, …,inrun over all permutations of 1, 2, …,n. The permanent function has been studied extensively of late (see, for example, (1), (2), (3), (4), (6)) and it is known that ifA∈ Ωnthen 0 <cn≤ per (A) ≤ 1, where the constantcndepends only onn. It is natural to inquire if per (A) is a convex function ofAforA∈ Ωn. That this is not the case was shown by a counter-example given by Marcus and quoted by Perfect in her paper ((5)). In this paper, however, she shows that per (½I+ ½A) ≤ ½ + ½ per (A) for allA∈ Ωn. HereI=Inis the identity matrix of ordern.