bluuebunny/crossref_metadata_2025_split huggingface dataset:
XIV.—On Substitutional Equations
Authors: Rutherford D. E. 
Published: July 2016

SummarySubstitutional equations of the type considered by the late Alfred Young are shown to be intimately related with the theory of idempotents. Any equation LX = o possessing solutions other than X = o is shown to have the same solutions as another equation MX = o, where M is obtained from L by premultiplying the latter by a suitably chosen expression A and where the minimum equation of M is xψ(x) = o, ψ(x) being prime to x. The expression ψ(M) is then idempotent, and it is shown that the most general solution of LX = o is X = ψ(M)Y, where Y is an arbitrary expression. The number of linearly independent solutions of LX = o is X = ψ(M)Y, where Y is an arbitrary expression. The number of linearly independent solutions of LX = o is kn!, where k is the coefficient of the unit permutation in ψ(M) when that expression is expressed in terms of the permutations of the symmetric group Sn.Corresponding results are obtained for the equation LX = R, and methods are given for solving sets of simultaneous equations of both types.
XXI.—Tables of Chebyshev Polynomials
Authors: Jones C. W. , Miller J. C. P. , Conn J. F. C. , Pankhurst R. C. 
Published: July 2016

1. Introduction.—The object of this paper is twofold: firstly, to present a table of the Chebyshev polynomials Cn(x) = 2 cos (n cos−1 ½x) for n = I(I)12 and x = o(o·o2)2, values being exact or to 10 decimals; secondly, to provide a working list of coefficients and formulæ relating to these and allied functions.Valuable accounts of applications and properties will be found in Van der Pol and Weijers, 1933, in Lanczos, 1938, and in Szego, 1939. Further applications are indicated in the following paper, by J. C. P. Miller, which also suggests methods of reducing the inconvenience caused by the present lack of tables of the allied polynomials Sn(x). It is hoped that suitable tables will be prepared later.
XXXVII.—The Van der Waals Force between a Proton and a Hydrogen Atom. II. Excited States
Authors: Coulson C. A. , Gillam C. M. 
Published: July 2016

SummaryThe interaction energy, or Van der Waals force, between a proton and a hydrogen atom in any one of its allowed quantum states is calculated in terms of the internuclear distance R by an expansion of the formAll the coefficients up to and including E5 are obtained in closed form. For values of R for which the expansion is valid, the coefficients are determined absolutely, no approximations being introduced.
XLI.—Problems in Factor Analysis
Authors: Lawley D. N. 
Published: July 2016

SummaryA set of variables is assumed to depend upon a number of common factors and specifics. Formulae are then derived for the sampling variances and covariances of the residual covariances obtained by removing the effect of the factors. The variances and covariances of the set of estimated loadings are also found. It must, however, be noted that the results obtained are valid only when an efficient method of estimation is used.
XVI.—Some Continuant Determinants arising in Physics and Chemistry—II
Authors: Rutherford D. E. 
Published: July 2016

SynopsisSome of the formulae obtained in this paper are likely to find application in problems concerning a rectangular lattice of “atoms”, each of which is under the influence of its near neighbours. Some of the determinants considered apply to cases in which both the nearest and the next nearest neighbours are operative. The inverses of certain types of matrices are found, and these may prove to be of value either in solving systems of linear equations such as arise in relaxation problems, or in determining the latent roots of matrices which may occur in problems in applied mathematics.
XXII.—Artificial Holograms and Astigmatism
Authors: Rogers G. L. 
Published: July 2016

SynopsisExperiments in diffraction microscopy, previously described, are here continued. Special emphasis is now laid on verifying the theory by the production of an “artificial” hologram, by non-diffractive means, from data calculated for a relatively simple object. The assumed object is then reconstructed in the usual apparatus.A type II linear zone plate of limited width is studied as a particular case of an artificial hologram. It gives rise to an unexpected black artefact, which is explained by a detailed analysis of this particular zone plate, and is shown to be due to its limited extent.Experiments on twisting the linear zone plate skew to the reconstructing beam show that the effective focal length is affected astigmatically by a factor proportional to cos2θ, where θ is the angle of twist, for lines parallel to the axis of twist. Lines perpendicular to the axis of twist are unaffected.The production of a hologram in an astigmatic pencil and its subsequent reconstruction while skew to a parallel beam is described. It is found that the focal length differences can be corrected in this way, but that the lateral scale factors are only partially rectified.
XI.—Theory of Indices for Non-Associative Algebra
Authors: Etherington I. M. H. 
Published: July 2016

SynopsisThe logarithmetic L of a non-associative algebra or class of algebras S has been previously defined as the arithmetic of the indices of powers of the general element when indices are added (non-associatively) and multiplied by certain conventions similar to those of ordinary algebra. With respect to addition, L is a homomorphic image of the “most general” logarithmetic B, the free additive groupoid with one generator 1, and in the case of algebras of one operation is essentially the same as the free algebra in one variable on S. The definition is now extended so that L is defined when S is any subset of an algebra or class of subsets of algebras, with the result that every homomorph of B is a logarithmetic ; but a distinction has then to be drawn between closed logarithmetics in which as before both addition and multiplication are defined, and other logarithmetics in which there is only addition. L is its own logarithmetic (taken with respect to addition) only if L is closed. For subsets of palintropic algebras, L is necessarily closed.
XXV.—The Determination of Temperature Gradients over the Greenland Ice Sheet by Optical Methods
Authors: Hamilton R. A. 
Published: July 2016

SynopsisThe temperature gradient in the lower atmosphere can be directly determined by measuring the optical refractive index of the air. This method is suitable for use on the Greenland ice sheet where errors introduced by water vapour are small, and where the strong solar radiation reflected by the snow surface makes it difficult to measure temperature differences over height differences of about I metre.The refraction was measured by observing the apparent vertical angle of each of a set of targets at distances up to 4 km. from a theodolite. The refraction was found to vary linearly with the distance of the target. The true vertical angle to the targets was determined when a second theodolite was available and reciprocal sights could be taken with it from the site of target to the fixed theodolite. The true vertical angle varied with time due to slow descent of the theodolite as the firn slumped; a correction for this was made. The standard error of the temperature gradient measurements was about 1.5 × 10−2 C.°	per metre. It is considered that the method could be developed and improved so that over a range of only 100 metres temperature gradients could be measured to an accuracy of about 0·1° C. per metre.
IV.—On the Radicals of Rings of Row-finite Matrices
Authors: Patterson E. M. 
Published: July 2016

SynopsisLet R be a ring and let ΓR be the Jacobson-Perlis radical. It is shown that the radical of the ring of row-finite matrices over R is the ring of row-finite matrices over ΓR if and only if ΓR is right-vanishing. This is done by extending the results of an earlier paper. One interesting consequence of the theorem is noted.
V—The Brownian Movement of a One-Dimensional Non-Harmonic Oscillator
Authors: Allnutt A. J. 
Published: July 2016

SynopsisThe Langevin equation for the harmonic oscillator is solved by a different method from that normally used. The approximate solution for the case of the slightly anharmonic oscillator is then obtained by an iterative procedure and the results are illustrated by a numerical example based on a simple model of a crystalline solid.
XIII.—The Graph-like State of Matter. I. Statistical effects of correlations due to substitution effects, including steric hindrance, on polymer distributions.
Authors: Gordon M. , Parker T. G. 
Published: July 2016

SynopsisModels which treat molecules as graphs provide the first approximation to practically all chemical and physical theories of polymers. This paper, worded so as to aim at intelligibility by both chemists and mathematicians, deals with statistical effects of correlations on distributions of molecules in their graph-like states. The correlations are formally equivalent to fertility correlations in family trees, and the theory covers the range of correlation to the point where a man's fertility expectations depend on how many brothers he has. Chemically, this describes the ‘second shell substitution effect’ which includes certain steric hindrance situations; mathematically, the problem is solved by restoring the Markovian nature of a conventional Galton-Watson process. Long-range correlations are an inevitable and troublesome aspect of the structure of gels and this work represents a step towards improving models for real gels.
XXIII.—On an Extension to an Integro-differential Inequality of Hardy, Littlewood and Polya
Authors: Everitt W. N. 
Published: July 2016

SynopsisThis paper considers an extension of the following inequality given in the book Inequalities by Hardy, Littlewood and Polya; let f be real-valued, twice differentiable on [0, ∞) and such that f and f are both in the space fn, ∞), then f′ is in L,2(0, ∞) andThe extension consists in replacing f′ by M[f] wherechoosing f so that f and M[f] are in L2(0, ∞) and then seeking to determine if there is an inequality of the formwhere K is a positive number independent of f.The analysis involves a fourth-order differential equation and the second-order equation associated with M.A number of examples are discussed to illustrate the theorems obtained and to show that the extended inequality (*) may or may not hold.
27.—The Zeros of Optical Coherence Functions and their Utility in the Study of Spectra
Authors: Thomas G. M. 
Published: July 2016

SynopsisThis paper shows that the coherence function associated with a band-limited optical spectrum is expressible as an infinite product involving its zeros. Only a finite number of zeros are physically significant and these can be determined from measurements of the visibility of interference fringes; but an ambiguity remains in the sign of the imaginary part of each zero. If the spectrum is to be recovered using the visibility measurements, an auxiliary experiment is needed to supply the necessary signs, but it need not be especially accurate. If the signs are not known there is only a limited number of different spectra that are compatible with the visibility measurements. Finally it is suggested that wavelength measurements made on an asymmetric spectral line with a Michelson interferometer may yield differing results when used with long and short delays.
2.—Asymptotic Behaviour of an Aperture Integral
Authors: Jones D. S. 
Published: July 2016

SynopsisThe asymptotic behaviour of double integrals over a portion of the plane is investigated when the integrand contains an exponential factor with a large parameter. The exponent can have a stationary point which may or may not be close to the boundary of the domain of integration. Results are first derived for a rectangular region with a particularly simple exponent in the integrand and shown to be uniformly valid under certain conditions. In some circumstances the asymptotic terms can be evaluated by means of a universal function. The theory is then generalised to cover more complicated exponents and arbitrary wedge-shaped domains; it is found that the asymptotic behaviour can still be expressed in terms of the same universal function.
4.—The Principal Chords of an Elliptic Quartic
Authors: Edge W. L. 
Published: July 2016

SynopsisThe curve Γ common to two quadric surfaces has 24 principal chords; they are the sides of six skew quadrilaterals each of which has for its two diagonals a pair of opposite edges of that tetrahedron S which is self-polar for both quadrics. These three pairs of quadrilaterals serve to identify the three pairs of quadrics through Γ that are mutually apolar.The vertices of the quadrilaterals lie four on each edge of S. Both nodes of the plane projection of Γ from such a vertex are biflecnodes.
12.—The Instability due to Acoustic Radiation striking a Vortex Sheet on a Subsonic Stream
Authors: Hardisty N. 
Published: July 2016

SynopsisThe propagation of sound across a vortex sheet separating two fluids in relative subsonic motion is examined for both harmonic and impulsive sources, the sound source being in the moving fluid. It is found that instability waves arise in a certain region depending on M, the Mach number.
14.—An Example concerning the Separation Property of Differential Operators
Authors: Everitt W. N. , Giertz M. 
Published: July 2016

SynopsisThe differential expression M[f] = −f″+qf, on a half-line [a, ∞), is said to be ‘separated’ in L2(a, ∞) if the collection of all functions f ∈ L2(a, ∞) such that M[f] is defined and also in L2(a, ∞), has the property that both terms f″ and qf ar separately in L2(a, ∞). When q is positive and differentiable on [a, ∞) it is known that separation holds for M[·] if q satisfies the condition |q| ≦ on [a, ∞)(*) provided the constant c satisfies 0 < c < 2. This paper constructs a class of examples of the coefficient q to show that (*) does not necessarily yield separation if c > 4/√3>2. The precise upper bound of c for which (*) yields separation is not known.
12.—On an Inequality of Hardy and Littlewood
Authors: Brodlie K. W. , Everitt W. N. 
Published: July 2016

This paper considers a natural extension of the following inequality of Hardy and Littlewoodto an inequality of the formwhere μ>0 and 0<K(μ)<∞. The best possible value of k(μ) is determined, and all the cases of equality.
22.—On the Non-existence of Solutions to the Generalised Swirling Flow Problem
Authors: Bushell P. J. 
Published: July 2016

1. The generalisation of von Kármán's equations of swirling flow studied by Serrin [7] and Hartman [3, 4] is the systemwith the boundary conditionsWhen α = β = ½ this system reduces to von Kármán's equations studied by several authors recently (see [1, 6] for many references).
17.—The Bromination of 1-Nitro-, 1-Bromo-, and 1-Acetamidofluoranthene
Authors: Jagroop C. L. , Charlesworth E. H. 
Published: July 2016

Synopsis1-Nitro-, 1-bromo-, and 1-acetamidofluoranthene undergo bromination to yield 4,9-dibrominated products. 1-Acetamidofluoranthene also yields 1-acetamido-4,8,9-tribrofluoranthene and as a side product 1-acetamido-4,8,9,x-tetrabromofluoranthene, from the former of which 1,4,8,9-tetrabromofluoranthene is obtained and found to be identical with the product resulting from the interaction of 1-bromofluoranthene with three molecules of bromine.
6.—Oscillation Theory for Semilinear Schrödinger Equations and Inequalities
Authors: Noussair E. S. , Swanson C. A. 
Published: July 2016

SynopsisSufficient conditions are derived for every solution of a nonlinear Schrödinger equation (or inequality) to be oscillatory in an exterior domain ofEn. Such results apply in particular to then-dimensional Emden-Fowler equation. The method involves oscillatory behaviour of solutions of a nonlinear ordinary differential inequality satisfied by the spherical mean of a positive solution of the Schrödinger equation.
XVII.—Inherited Measures
Authors: Macbeath A. M. , Swierczkowski S. 
Published: July 2016

SYNOPSISCertain families of measures on coset-spaces, namely inherited, stable, and pseudo-invariant measures, were defined, and shown to exist, in earlier papers, where Jacobians and factor functions, generalizing the idea of Jacobians in theory of functions of several variables, were also denned. In this paper, the existence is established of exact Jacobians and factor functions, which satisfy certain characteristic identities exactly, without an exceptional set of measure zero. A study is made of how properties of a measure are reflected by properties of the Jacobian or the factor function. Necessary and sufficient conditions are found for a function to be an exact Jacobian for some measure.
XXV.—Properties of I-extensions of Anti-commutative Algebras
Authors: Wallace E. W. 
Published: July 2016

SYNOPSISS. T. Tsou and A. G. Walker have defined the I-extension of a given Lie algebra as a certain Lie algebra on the Cartesian product of the given algebra and one of its ideals (Tsou 1955). I-extensions have been studied also in connection with metrisable Lie groups and metrisable Lie algebras. The definition can be applied immediately to any anti-commutative algebra, and in this paper properties of such I-extensions are established. A list of all proper I-extensions of dimension not greater than four over a field of characteristic zero is also given together with a set of characters.
Discussion: Economic and Social Indicators of Rural Development From an Economic Viewpoint
Authors: Waldrop John E. 
Published: July 2016

Drs. Edwards and Coltrane show great courage in agreeing to discuss in one paper all of the variables, data, theories, models, and relationships that might be included according to the title. It was comforting to me to see that their very first endeavor was to set forth some definitions for some often used terms but terms whose meaning is seldom clearly stated by the user.The authors distinguish between economic development and growth and suggest some alternative definitions for rural and rural development. Let us take a closer look at their efforts.
The Distribution of the Costs and Benefits of Public Schooling
Authors: Holland David W. 
Published: July 2016

Arthur Okun recently observed that even in economics the times are changing. Whereas the focus of the profession in the early 1960's was on the aggregative issues – mainly lagging economic growth – it has now shifted to a considerable degree to the distributive issues. Particular attention is being given to the problems of the low end of the distribution – “the poverty problem.” In agricultural economics we are witnessing a somewhat similar trend with increasing interest and professional time being allocated to rural poverty and rural development.One of the manifestations of the trend in both general and agricultural economics to examine distributional questions has been a substantial increase in studies of the distributional impact of various public programs [cf. 6, 7, 23, 24]. Both because of the magnitude of expenditure involved and its presumed role in providing equality of opportunity, education has received particular attention. Consequently substantial literature has now been accumulated on the distributive impact of public schooling.
Effects of Alternative Farm Policies on Farm and Nonfarm Sectors of Rural America
Authors: Sonka Steven T. , Heady Earl O. 
Published: July 2016

During the last three decades, a major change affecting rural America has been the dramatic increase in the productivity of agricultural labor. Output per farm worker increased by 237 percent between 1947 and 1970. Although this growth in productivity provided increased income to some farm operators, it also resulted in decreased income opportunities for others. This decrease in opportunities led to migration of people from rural to urban areas. As the farming industry became more mechanized and as more rural residents were forced to migrate to urban areas in quest of jobs, the economic viability of many rural communities declined drastically. Mayer summarizes these changes in economic activity as follows:“… the changing structure of agricultural production has significantly altered the flows of money in rural towns. More money flows to sources in urban areas and less remains to provide jobs in rural towns. As mechanization of agriculture increased and as capital intensification occurred, rural towns have experienced a slow draw-down of economic vitality.”
A Comparative Regional Analysis of the Wholesale Marketing Margin for Crushed Cottonseed
Authors: Ethridge M. Dean 
Published: July 2016

Economic events since the 1972–73 crop year have generated much interest in cottonseed prices. Cotton producers, observing a doubling of wholesale prices of cottonseed oil and meal during the last three years, have wondered whether they are getting an equitable share of this increased income. They have largely stopped thinking of their cottonseed as merely a means of payment for ginning charges and have begun to regard it as a potential source of supplementary income.The limited objective of this paper is to examine, by regions, the wholesale marketing margin for cottonseed during the crop years of 1958–1973 in order to (1) more clearly determine how the marketing margin has behaved and (2) discover differences, if any, among regional marketing margins.
Inflation and Machinery Cost Budgeting
Authors: Watts Myles J. , Helmers Glenn A. 
Published: July 2016

Machinery costs are affected by inflation which, if not correctly considered, will bias cost estimates. Accurate estimates of machinery costs are essential in budgeting costs of crop production in farm policy programs, crop hedging decisions, and general cost of production research. Machinery decisions relating to replacement, size, and custom or leasing alternatives are also improved by realistic budgeting techniques.The objective of this article is to examine appropriate adjustments for inflation in developing machinery costs through budgeting techniques. The authors attempt to demonstrate the adjustments necessary in both capital and traditional budgeting models to place cost estimates on a real basis. Capital budgeting is first compared with traditional budgeting. Next, by the use of a simple machinery example, the necessary adjustments to account for inflation are shown for a capital budgeting model that does not include income tax considerations. Similarly, the inflation adjustments are demonstrated for a traditional budgeting model that does not include income tax considerations. Finally, a capital budgeting model including income tax aspects is examined in reference to real and nominal after-tax cost estimates.
Production, Price and Risk Factors in Channel Catfish Farming
Authors: Hanson Gregory D. , Martin Neil R. , Flynn John B. 
Published: July 2016

AbstractThe effects of several production/management, price and risk factors upon channel catfish profitability are analyzed with a multiperiod mixed-integer linear programming model. Factors analyzed include pond size and optimal stocking rates, alternate levels and trends in catfish prices, pond production losses and level of family consumption withdrawals. Model results indicate that channel catfish offer the potential to significantly increase farm rates of return while providing an avenue of intensive farm growth, without expanding the land base of the farm. However, the long range financial success of the firm was very sensitive to several of the management and risk factors examined.