bluuebunny/crossref_metadata_2025_split huggingface dataset:
Some solutions of the equations of flow of a viscous compressible fluid
Authors: Illingworth C. R. 
Published: November 2008

AbstractThe exact (Navier-Stokes) equations for the flow of a viscous compressible fluid are examined in a search for simple solutions, in which the equations reduce to ordinary differential equations. Such solutions are found for the uniform shearing motion in the Couette flow between both parallel flat plates and coaxial circular cylinders in relative motion. For a compressible fluid there are no simple solutions corresponding to certain other flows (such as the Poiseuille flow between fixed parallel flat plates) described by simple solutions for an incompressible fluid. A solution is found for the flow of a gas over an infinite porous plate with uniform suction through the plate (gravity neglected), and the relation of the solution to asymptotic solutions of the boundary-layer equations is discussed. Ordinary differential equations are obtained for the problems of (i) the flow due to the rotation of a horizontal disk in a gas (dissipation neglected), and (ii) the steady circulatory flow round a porous circular cylinder with uniform suction (gravity neglected). Discussion of the gas-dynamical problems is preceded by a description of the state of a horizontally stratified gas in static and thermal equilibrium.
The lateral diffusion of a cloud of falling droplets
Authors: Davies D. R. 
Published: November 2008

AbstractThis paper is an application of O. G. Sutton's theory of eddy diffusion to a case for which it was not originally developed; the theory is used to predict the dispersion of a cloud of falling droplets released from a point source at great height above the surface of the earth. Some experimental and theoretical values are given for the widths of ground area covered by liquid drops released from sources situated at heights between 1000 and 5000 ft., and agreement between theory and experiment is seen to be quite reasonable. This paper therefore provides further support for the use of O. G. Sutton's theory in problems of this type. The theory has practical applications, e.g. in the spraying of crops from aeroplanes.
A process for the step-by-step integration of differential equations in an automatic digital computing machine
Authors: Gill S. 
Published: November 2008

AbstractIt is advantageous in automatic computers to employ methods of integration which do not require preceding function values to be known. From a general theory given by Kutta, one such process is chosen giving fourth-order accuracy and requiring the minimum number of storage registers. It is developed into a form which gives the highest attainable accuracy and can be carried out by comparatively few instructions. The errors are studied and a simple example is given.
Multiplicative norms on Banach algebras
Authors: Edwards R. E. 
Published: November 2008

1. Mazur(1) has shown that any normed algebra A over the real field in which the norm is multiplicative in the sense thatis equivalent (i.e. algebraically isomorphic and isometric under one and the same mapping) to one of the following algebras: (i) the real numbers, (ii) the complex numbers, (iii) the real quaternions, each of these sets being regarded as normed algebras over the real field. Completeness of A is not assumed by Mazur. A relevant discussion is given also in Lorch (2).
The solution of algebraic equations on the EDSAC
Authors: Brooker R. A. 
Published: November 2008

AbstractThis paper is an account of the methods that have been used with the EDSAC for the solution of algebraic equations. Three repetitive or iterative methods are examined: Bernoulli's method, the root-squaring method, and the Newton-Raphson method. Experience with the EDSAC has shown that, as in hand computing, quadratically convergent methods are to be preferred to those less rapidly convergent. In particular, the Newton-Raphson method has proved the most useful. Several examples are given in the appendix.
Generalized magneto-hydrodynamic formulae
Authors: Hines C. O. , Bondi H. 
Published: November 2008

AbstractThe formulae for current and force which were employed in some of the initial work on magneto-hydrodynamic theory are not valid over the whole range of phenomena which the subject itself embraces. Generalized formulae, suitable for wider application, are derived by an extension of the magneto-ionic approach, and are applied to the simple case of plane-wave propagation in the direction of a primary magnetic field. The dispersion equation which results, as well as the basic current and force formulae, are reduced by approximations in a variety of limiting circumstances. Some earlier results are thereby regained and, in particular, the region of known validity of Alfvén's propagation relation is extended.
On the flow between two rotating coaxial disks
Authors: Stewartson K. 
Published: November 2008

AbstractThe steady motion of a viscous fluid confined between two coaxial rotating disks is discussed both experimentally and theoretically. It is found experimentally that when the disks rotate in the same sense the main body of the fluid rotates as well, but if they rotate in opposite senses the main body of the fluid is almost at rest. An adequate theory is found to explain this.
On the energy scattered from the interaction of turbulence with sound or shock waves
Authors: Lighthill M. J. 
Published: November 2008

ABSTRACTThe energy scattered when a sound wave passes through turbulent fluid flow is studied by means of the author's general theory of sound generated aerodynamically. The energy scattered per unit time from unit volume of turbulence is estimated (§3) aswhere I is the intensity and ∧ the wave-length of the incident sound, and is the mean square velocity and L1 the macro-scale of the turbulence in the direction of the incident sound. This formula does not assume any particular kind of turbulence, but does assume that ∧/L1 is less than about 1. For turbulence which is isotropic and homogeneous, the energy scattered, and its directional distribution, are obtained for arbitrary values of ∧/L1. It is predicted that components of the turbulence with wave-number κ will scatter sound of wave-number K at an angle 2 sin−1 (k/2k). The statistics of multiple successive scatterings is considered (§4), and it is predicted that sound of wave-length less than the micro-scale λ of the turbulence will become uniform (i.e. quite random) in its directional distribution in a distance approximately .The theory is extended (§5) to the case of an incident acoustic pulse. However, this extended theory cannot be applied directly to the case of a shock wave, for which it would predict infinite scattered energy. This is due to the perfect resonance between successive rays emitted forwards which would occur if the shock wave were propagated at the speed of sound. By taking into account (§6) the true speed of the shock wave (subsonic relative to the fluid behind it), the theory is improved to give a finite value, 0·8s tunes the kinetic energy of the turbulence traversed by a weak shock of strength s, for the total energy scattered. However, the greater part of this energy catches up with the shock wave, and probably is mostly reabsorbed by it, and only the remainder (tabulated as a function of s in Table 1) is freely scattered, behind the shock wave, as sound. The energy thus freely scattered when turbulence is convected through the stationary shock-wave pattern in a supersonic jet may form an important part of the sound field of the jet.
Fixed points on rotating continua
Authors: Reifenberg E. R. 
Published: November 2008

Suppose I is a bounded plane continuum whose complement is a single domain (I) and that is a (1–1) bicontinuous transformation of the plane onto itself which leaves I invariant. Cartwright and Littlewood(1) have proved the following theorem:Theorem A. Suppose that(I) has no prime end fixed underand the frontier of I contains a fixed point P. Ifis any prime end of(I) containing P, and C is any curve in(I) converging tosuch that PeC¯, the closure of C, then for some‡ integer N the continuum I(FN) consisting oftogether with the sum B(FN) of its interior complementary domains contains all fixed points in I.
Real inequalities with applications to function theory
Authors: Hayman W. K. , Stewart F. M. 
Published: November 2008

Suppose that f(x) is non-negative for x ≥ 0 and letIn a recent paper (4) inequalities were proved relating fn(x) with the nth derivative f(n)(x) of f(x). However, the two main results, Theorems 3 and 4, were proved only for n = 1 and 2. In the first part of this paper we shall prove the corresponding results for all positive integral n under a slightly less restrictive hypothesis (Theorem 3). We shall then give an application showing that from the hypothesis f(x) ≤ μ(x) for all x it is sometimes possible to deduce f(n)(x) = O{μ(n)(x)} for a set of values x depending only on μ(x).
On the use of asymptotic approximations of Green's type when the coefficient has zeros
Authors: Jeffreys Harold 
Published: November 2008

ABSTRACTAllowance is made for the error terms in the cases where χ has a simple zero at x = 0, or two simple zeros at a; = ± 1 (with χ either positive or negative for − 1 < x < 1). The errors of the approximations given by neglecting the error terms are found to be small in all cases, sometimes much smaller than there was any previous reason to expect.
The final period of decay of non-homogeneous turbulence
Authors: Phillips O. M. 
Published: November 2008

ABSTRACTIn this paper, a study is made of the properties of the motion due to a general localized disturbance in an infinite fluid initially at rest. Two invariants of the motion are found, which are identified with the net linear momentum, or effective dipole strength, and the net angular momentum imparted to the fluid, and an equation is derived expressing the rate of change of the effective quadrupole strength of the motion. Solutions are obtained for the final period of decay of such a motion, and it is shown that, when the net linear momentum of the fluid is non-zero, the motion corresponds to a type of viscous vortex ring. The length scale increases with time t as [ν(t – t0)]½, and the total energy decays as [v(t–t0)]-1, where ν is the kinematic viscosity and t0 is a virtual time origin. If the net momentum is zero, the total energy decays as , and expressions are given for the velocity field in this case also.It is shown that, if it is attained, the final period of any turbulent motion can be found from the vortex ring type of solution described above. The final period of homogeneous turbulence is clearly one particular example of such a motion. The final period of an axially symmetrical turbulent wake is treated in this way and is shown to consist generally of the superposition of two types of motion: (i) a mean streaming motion whose energy per unit length of the wake decreases as [ν(t – t0)]-1, and (ii) a superimposed fluctuating motion whose energy per unit length decays as [v(t–t0)]-1. Expressions are given for the distribution of mean-square vorticity and the energy spectrum tensor in the wake.
Bending of a circular plate with an eccentric circular patch symmetrically loaded with respect to its centre
Authors: Bassali W. A. , Dawoud R. H. 
Published: November 2008

ABSTRACTThe complex variable method is applied to obtain solutions for the deflexion of a supported circular plate with uniform line loading along an eccentric circle under a general boundary condition including the clamped boundary , a boundary with zero peripheral couple , a boundary with equal boundary cross-couples , a hinged boundary and a boundary for which , η being Poisson's ratio. These solutions are used to obtain the deflexion at any point of a circular plate having an eccentric circular patch symmetrically loaded with respect to its centre. Expressions for the slope and cross-couples over the boundary and the deflexions at the centres of the plate and the loaded patch are obtained.
On the Sturm-Liouville equation with two-point boundary conditions
Authors: Douglas A. S. 
Published: November 2008

ABSTRACTIn numerical solution of a Sturm-Liouville system, it is necessary to determine an eigenvalue by a method of successive approximation. A relation is derived between the estimated accuracy of an approximate eigenvalue and the accuracy at every point of its corresponding eigenfunction. A method is also described whereby the correction to a trial eigenvalue, required for convergence to its true value, can be automatically determined. This method has been successfully used in solving radial wave-function equations, both with and without ‘exchange’, arising from the Hartree-Slater-Fock analysis of Schrödinger's equation.
Angular terms in the electron density for the phosphine molecule
Authors: Ballinger R. A. , March N. H. 
Published: November 2008

ABSTRACTAn attempt is made to calculate the first few angular terms in an expansion of the electron density for the phosphine molecule in Legendre polynomials. Such an expansion is appropriate for a model in which the three hydrogen nuclei are smeared to form a circular line charge. The Thomas–Fermi approximation has been used in conjunction with the variational method. The variational density employed includes p and f angular terms. An approximate charge density map is constructed for a plane containing the molecular axis in order to demonstrate the effect of the angular terms.
A suggested experiment on the clock paradox
Authors: Cochran W. 
Published: November 2008

ABSTRACTAn experiment is suggested to settle the question whether two clocks which are synchronized and subsequently separated must necessarily show identical readings on being brought together again. Charged π-mesons play the role of a clock travelling with a velocity comparable with c, this clock being read by passing the mesons through a scintillation counter or counters. It is pointed out that the experiment can be designed in a way which makes it technically feasible for a laboratory possessing an accelerating machine capable of producing mesons in the 100 Me V. energy range.Some considerations suggested by the proposed experiment are examined, and it is concluded that only the orthodox result—that in general the two clocks will record the passage of different time intervals between successive coincidences—is consistent with the result of an experiment which has already been made, or with the special theory of relativity.
The real solutions of a certain non-linear system of equations
Authors: Lebovitz N. R. 
Published: November 2008

In a recent paper on the behaviour of a system of disk dynamos(1), a problem of a purely algebraic character arose. The problem is to find all the real solutions of the non-linear, inhomogeneous system of 2n equationswhere x0 = xn and the parameter ρ is real and not zero, but otherwise arbitrary. The real solutions were correctly given in (1), but the proof that they are the only real solutions was incomplete. A different, and complete, proof is given here.
Scattering of plane elastic waves at rough surfaces. I
Authors: Abubakar Iya 
Published: November 2008

ABSTRACTAn approximate solution of the two-dimensional problem of reflexion of plane harmonic P and S V waves by an irregular boundary is obtained using a modification of Rice's perturbation method of approximation on the assumption that the curvature of the surface is everywhere small. The case of a periodic surface is treated in more detail. It is found that the reflected waves are composed of specularly reflected waves and various diffracted waves, propagating in both horizontal directions if the wavelength of the incident waves is long compared with that of the surface. If the wavelength of incident distortional waves is long compared with that of the surface, the amplitudes of some of the scattered waves decrease exponentially with depth. In general the phases of the waves change on reflexion and the phase angles of the reflected waves are functions of the wavelength of the corrugation and the angle of incidence. It is verified, in the case of zero angle of incidence, that the energy going into the scattered radiation is obtained at the expense of the energy of the specularly reflected waves.
Partial orderings on real linear topological spaces
Authors: Simons S. 
Published: November 2008

The following properties of the real numbers are well known:If (xn) is a sequence of real numbers, xn → x and xn ≥ 0 for each n then x > 0. (1)If (xn) is a sequence of real numbers, xn → x and x > 0 then, for all sufficiently large n,xn > 0.
Some explicit results for an asymmetric two-dimensional random walk
Authors: Barnett V. D. 
Published: November 2008

AbstractThree distinct methods are used to obtain exact expressions for various characteristics of a particular asymmetric two-dimensional random walk. The results obtained include, for the transient unrestricted walk, the probability of return to the starting-point and the average number of arrivals at the general lattice point; and, for a walk restricted within a rectangular absorbing barrier, the average number of arrivals at any accessible point and the absorption probabilities on the boundary. Whilst there is some duplication of results by using the three different methods of analysis, this is not extensive and provides a useful check on the results. Also the methods are of some general interest in themselves.
The elastic stresses produced in the mid-plane of a slab by pressures applied symmetrically at its surface
Authors: Greenwood J. A. 
Published: November 2008

AbstractFormulae for the elastic stress distribution in an infinite slab due to symmetrical pressure distributions on the upper and lower faces have been given by Filon (two dimensions) and by Sneddon (axial symmetry). Few numerical values have been published except for the limiting cases of line and point loads. The stresses in the mid-plane are given here for a uniform pressure acting over circles and bands of different sizes, and the published values for a point load are shown to be wrong.
A central limit theorem for processes defined on a finite Markov chain
Authors: Keilson J. , Wishart D. M. G. 
Published: November 2008

We shall be concerned in this paper with a class of temporally homogeneous Markov processes, {R(t), X(t)}, in discrete or continuous time taking values in the spaceThe marginal process {X(t)} in discrete time is, in the terminology of Miller (10), a sequence of random variables defined on a finite Markov chain. Probability measures associated with these processes are vectors of the formwhereWe shall call a vector of the form of (0·2) a vector distribution.
Note on the abstract group (2,3,7;9)
Authors: Leech John 
Published: November 2008

The abstract groupis finite for n = 4,6,7,8, and the relations are incompatible for n = 1,2,3,5. A criterion of Coxeter ((1)) suggests that (2,3,7; n) should be infinite for all n ≥ 9, but its applicability to these groups is unproved, and it is not known whether there are any further examples of finite groups (2,3,7; n). However, (2,3,7; 9) has been proved infinite by Sims ((3)), and it follows at once that (2,3,7; n) is infinite whenever n is a multiple of 9 as it then has an infinite factor group.
On convex sets and measures
Authors: Larman D. G. , Ward D. J. 
Published: November 2008

If α is a positive real number then, for each set E in R3, we definewhere U(ρ, E) is any countable collection of convex sets, each of diameter at most ρ, whose union covers E, and Aα is a positive real number. Then the convex α-dimensional measure Λα(E), of E, is . In this paper we shall only be considering the cases α = 1,2 where, as is usual, we take A1 = 1 and A2 = ¼π. The symbols ,Λs(E) will denote the spherical (circular) measure of E, i.e. when the coverings are restricted to being collections of spheres (circles), and E(z) is the intersection of a set E (in R3) with the plane (x, y) at z.
On generalized inverses of matrices
Authors: Pearl M. H. 
Published: November 2008

The notion of the inverse of a matrix with entries from the real or complex fields was generalized by Moore (6, 7) in 1920 to include all rectangular (finite dimensional) matrices. In 1951, Bjerhammar (2, 3) rediscovered the generalized inverse for rectangular matrices of maximal rank. In 1955, Penrose (8, 9) independently rediscovered the generalized inverse for arbitrary real or complex rectangular matrices. Recently, Arghiriade (1) has given a set of necessary and sufficient conditions that a matrix commute with its generalized inverse. These conditions involve the existence of certain submatrices and can be expressed using the notion of EPr matrices introduced in 1950 by Schwerdtfeger (10). The main purpose of this paper is to prove the following theorem:Theorem 2. A necessary and sufficient condition that the generalized inverse of the matrix A (denoted by A+) commute with A is that A+ can be expressed as a polynomial in A with scalar coefficients.
Addenda to processes defined on a finite Markov chain
Authors: Keilson Julian , Wishart David M. G. 
Published: November 2008

Introduction. Two previous papers (1,2) have dealt with additive processes defined on finite Markov chains. Such a process in discrete time may be treated as a bivariate Markov process {R(k), X(k)}. The process R(k) is an irreducible Markov chain on states r = 1, 2, …, R governed by a stochastic transition matrix B0 with components brs. The marginal process X(k) ‘defined’ on the chain R(k) is a sum of random increments ξ(i) dependent on the chain, i.e. if the ith transition takes the chain from state r to state s, ξ,(i) is chosen from a distribution function Drs(x) indexed by r and s. The distribution of the bivariate process may be represented by a vector F(x, k) with componentsThese are generated recursively by the relationwhere the increment matrix distribution B(x) has components brsDrs(x). We denote the nth moment of B(x) by Bn = ∫xndB(x), so that B0 = B(∞).
Abstract Köthe Spaces. I
Authors: Fremlin D. H. 
Published: November 2008

The purpose of this paper and the next is to demonstrate that the ‘perfect Riesz spaces’ of (1) are an effective abstraction of the ‘espaces de Köthe’ of (2). I shall follow the ideas of (1), with certain changes in notation:If L is a Riesz space and x, y ∈ L, let us denote sup (x, y) by x ∧ y and inf (x, y) by x ∧ y. I shall use the convenient if informal notation xr↓ ((1), section 16·1) and shall in this usage assume that 0 ∈ {r} and that x0 ≥ xτ for all τ. A set A ⊆ L is solid if x ∈ A and |y| ≤ |x| together imply that y ∈ A; A is then balanced. The solid hull of A is the set {y: ∃ x ∈ A, |y| ≤ |x|}; this is the smallest solid set containing A. An ‘ideal’ ((1), section 17) is then a solid subspace.
Dispersion of soluble matter in the hydromagnetic laminar flow between two parallel plates
Authors: Gupta A. S. , Chatterjee A. S. 
Published: November 2008

AbstractThe paper presents an analytical solution for the dispersion of a solute in an electrically conducting fluid flowing between two parallel plates in the presence of a transverse magnetic field. It is shown that the solute is dispersed relative to a plane moving with the mean speed of the flow with an effective Taylor diffusion coefficient which decreases with increase in magnetic field.
The Boltzmann-Landau transport equation
Authors: Mitra Arun K. , Mitra Sunanda 
Published: November 2008

AbstractThe first-order Chapman-Enskog (CE) approximation has been used to linearize the Boltzmann-Landau (BL) equation primarily in the binary collision approximation and a linear integral equation with a non-symmetric kernel is obtained. The solubility conditions are discussed on the basis of conservation theorems. The formal solutions and the transport coefficients have been obtained in a subsequent paper.
The Boltzmann-Landau transport equation
Authors: Mitra Arun K. , Mitra Sunanda 
Published: November 2008

AbstractThe linearized Boltzmann-Landau transport equation given in a previous paper with a non-symmetric Kernel has been formally solved and the solution has been used to calculate the coefficients of shear viscosity (η) and thermal conductivity (λ) up to first-order density corrections.