bluuebunny/crossref_metadata_2025_split huggingface dataset:
XXIII.—Studies in Practical Mathematics. VII. On the Theory of Methods of Factorizing Polynomials by Iterated Division
Authors: Aitken A. C. 
Published: July 2016

SynopsisThe division of one polynomial by another is studied with the object of ascertaining the errors produced in the coefficients of successive remainders by small errors in the coefficients of the divisor. It is shown that the matrix which effects this transformation of errors is a polynomial in the rational canonical matrix for which the divisor polynomial is characteristic. The theory gives rise to a numerous class of iterative processes for finding an exact factor, such as the extant method based on the penultimate remainder, Bairstow's iterative method of finding a quadratic factor, and many others. Some new suggestions are made for accelerating convergence.
VII—Tidal Charts Based on Coastal Data: Irish Sea
Authors: Doodson A. T. , Rossiter J. R. , Corkan R. H. 
Published: July 2016

SynopsisThe object of this paper is not to produce a chart of co-tidal and co-range lines which is more accurate than an existing one, but to investigate methods of computing such charts on the supposition that there are no observations of tidal streams such as were used to produce the existing chart. Only coastal observations of tidal elevations are supposed to be known, for such conditions would exist in many parts of the world. The methods used are similar to the so-called “relaxation methods”, using finite differences in all variables and attempting to satisfy all the conditions of motion within the sea, proceeding by successive approximations. There are many difficulties, peculiar to the tidal problem, in the application of these methods, due to the very irregular coast-lines and depths, gaps in the coasts, shallow water near the coasts, frictional forces, and the very serious complication due to the fact that the tides are oscillating and thus require two phases to be investigated simultaneously owing to their reactions one upon the other. One very important point in testing the methods is that no use whatever should be made of existing charts in obtaining first approximations of heights to commence the processes, not even where there are wide entrances to the sea. The resulting chart is shown to be very closely the same as the existing chart, thus proving the validity of the method.
IX.—On the Geometry of Ɛ-Matrices
Authors: Ruse H. S. 
Published: July 2016

SynopsisThe two regular representations of quaternions give rise to a classical set of sixteen 4 × 4 matrices that have fairly recently reappeared in a paper by S. R. Milner. He uses them as the basis of a calculus of “Ɛ-numbers”, which he develops for the purpose of making physical applications. The covariantive nature of his calculus is, however, not always fully apparent, and raises some points of interest of which an examination is made in the present paper in terms of 3-dimensional projective geometry. The theory that emerges is the classical one of the collineations of projective 3-space that transform a quadric into itself, but the formulation is different from that of existing theories based on the same set of matrices and having the same or a similar geometrical background. For example, the present theory is quite different from that of 4-component spinors. The constants of multiplication γijk of quaternion algebra make their appearance in a generalized form and in a geometrical setting. In the final section an indication is given of possible generalizations to Riemannian geometry, and of the connection of the present work with the theory of Kähler manifolds of two complex dimensions.
XIV.—Studies in Practical Mathematics. VIII. On the Iterative Methods of Lin and Friedman for Factorizing Polynomials
Authors: Aitken A. C. 
Published: July 2016

SynopsisThe methods of S. N. Lin (1943) and B. Friedman (1949) for approximating to the factors of a polynomial by iterated division are studied from the point of view of convergence. The general theory, hitherto lacking, is supplied. The matrices which transform the errors in coefficients from one iterate to the next are explicitly found, and the criterion of convergence derived. Numerical examples are given. The tentative conclusion is that the methods are less simple in theory and less adaptable than the method of penultimate remainder, which admits of accelerative devices.
XV.—Bounds for the Indices of Nilpotent and Solvable Lie Algebras
Authors: Patterson E. M. 
Published: July 2016

SynopsisBy examining certain connections between the derivatives and the powers of a Lie algebra, bounds are obtained for the indices of nilpotent Lie algebras over an arbitrary field. The results are used to obtain bounds for the indices of solvable Lie algebras over a field of characteristic zero.
XVIII.—Recent Oceanographical Investigations in the Faroe-Shetland Channel
Authors: Tait John B. 
Published: July 2016

SynopsisThe Faroe-Shetland Channel is the threshold from the north-eastern Atlantic Ocean to north-west European seas. Through it passes the main bulk of the oceanic water-mass which is the predominant influx, among several other water-masses, to these seas.The following research into the dynamics and general hydrography of the region is based on numerous observations of temperature and salinity, from surface to bottom, taken mainly on two vertical cross-sections of the Channel between the years 1927 and 1952 inclusive, excepting the war years 1940 to 1945.The research reveals very large scale seasonal and long-term variations in the northeastward volume-transport of oceanic water, suggests the existence on occasions of what appear to be horizontal tortional currents within the oceanic water-mass, and demonstrates (a) the intrusion of Gulf of Gibraltar (extra-Mediterranean) water into this mass over a period of years, (b) the formation of heavy oceanic water and (c) of a sub-oceanic watermass. The last-mentioned may sometimes almost entirely displace the bottom Norwegian Sea water-mass which normally underlies the oceanic mass.One or other, or both, of two types of Arctic water may also sometimes displace bottom Norwegian Sea water as the bottom water-mass of the region, the process, like that of the above-mentioned Gulf of Gibraltar water influx, waxing and waning over a term of years and thus exemplifying the phenomenon of marine climatic change.
XXIII. —Marine Research—The Work of the National Institute of Oceanography
Authors: Deacon G. E. R. 
Published: July 2016

SynopsisThe National Institute of Oceanography owes its formation to a growing need for more detailed and systematic knowledge of the physics as well as the biology of the oceans, and the feeling that the United Kingdom, in co-operation with the Commonwealth countries, should play a part in marine science more in keeping with its tradition of interest in the oceans and their navigation. The Institute is controlled by the National Oceanographic Council, which was granted a Royal Charter in 1949 and operates with the aid of Government grants.The emphasis of the programme is on long-term research under the main headings:Interchange of energy between the atmosphere and oceansResponse of the sea surface to wind and pressure changesGeneral circulation of the oceansFree and forced oscillations in the oceansDistribution of marine organismsOrganic production in the oceansBionomics of whales.There are many other activities and as much as possible is done to work out the practical implications of the research, and to help other marine scientists. The headquarters of the Institute are at Wormley, near Godalming, Surrey, and its research ship, the R.R.S. “Discovery II”, is based at Plymouth.
II.—Spontaneous Fission Rates and the Role of the “Excess” Neutrons in Heavy Nuclei
Authors: Feather N. 
Published: July 2016

SynopsisEmpirical regularities are sought amongst the experimental data relative to the spontaneous fission rates and the nucleon-pair binding energies of the heavy even-even nuclei. Certain regularities are found which have not hitherto been noted; in particular it appears that the isotopic number D is a significant parameter in relation to these various quantities. The speculation is made that possibly the saddle-point deformation of the protons in the nucleus is greater than that of the neutrons. Predictions are made concerning the spontaneous fission rates of the even-even isotopes of thorium.
V.—A Search for n-γ Coincidences from a Po-Li Neutron Source
Authors: Galloway R. B. 
Published: July 2016

SynopsisNemilov and Pisarevskii (1957) have reported a 390 keV γ-ray in coincidence with neutrons from a Po-Li source, and interpret this as evidence for a level in B10 at 390 keV. A search for confirmatory evidence of this γ–ray has been unsuccessful, and it is estimated that such n–γ coincidences would have been detected in the experiment here described if they involved 8 per cent or more of the neutrons from the Po-Li source.
VI.—Endomorphisms of Abstract Algebras
Authors: Evans Trevor 
Published: July 2016

SynopsisA study is made of the generalization of the entropic law (xy)(zw)—(xz)(yw) to an identity connecting two operations. It is shown that such an identity is equivalent “in the large” to the condition that the set of endomorphisms with respect to one operation is closed with respect to the other. Furthermore, for such entropic operations, each may be regarded as a generalized endomorphism of the other and various generalizations of elementary properties of endomorphisms are obtained. Examples of entropic pairs of operations are quite common in mathematics and a number of these are discussed. An important aspect of the paper is the matrix notationintroduced to facilitate what would otherwise be extremely cumbersome computations with entropic operations.
VIII.—Theoretical Expressions for Relaxation Effects in the Thermal Conductivity of Gases as Measured in the Modified Co-axial Cylinder Apparatus of Callear and Robb
Authors: Wright P. G. 
Published: July 2016

SynopsisThe apparent thermal conductivity of a polyatomic or isomerizing gas (as measured in a given apparatus) may decrease at low pressures for two distinct reasons. There may be accommodation effects at boundary surfaces, and there may be relaxation effects arising because molecules with excess energy do not yield it up fast enough to maintain local “chemical” equilibrium. If the apparatus is such that the temperature measured at points in the gas and not in the walls, relaxation effects may be observed directly, and accommodation effects are (in theabsence of relaxation effects) absent.A detailed analysis is made of the apparent thermal conductivity measured in such an apparatus with cylindrical symmetry. Expressions are obtained in closed form. Numerical calculations show that, for a gas of relatively long relaxation time in an apparatus of reasonable dimensions, the apparent thermal conductivity would decrease appreciably at readily-attained pressures.
XIV.—On Eigenvectors of Boolean Matrices
Authors: Blyth T. S. 
Published: July 2016

SynopsisThroughout this paper we shall be concerned with n × n matrices X=[xij] whose elements xij belong to a given Boolean algebra B(≤, ∩, ∪, ′). In tne first part of the paper we show that the set S(λ, x) of matrices with a given eigenvector x and eigenvalue λ is a subgerbier (i.e.), γ-semi-reticulated sub-semigroup of Mn(B), the Boolean algebra of all n × n matrices over B. We also determine the greatest element M(λ, x) of S(λ, x) and consider some of its properties. In the second part of the paper we consider the structure of matrices which possess a given primitive eigenvector x and show in particular that, for any given λ ∈ B, there is a matrix, namely M(λ, x), having x as the maximum primitive eigenvector associated with the eigenvalue λ. We also determine necessary and sufficient conditions under which the matrix M(λ, x) is the same for all λ in any given sublattice of B.
XXIII.—Autocorrelation in Black Body Radiation and in Density Fluctuation Waves in Liquids
Authors: Fürth R. 
Published: July 2016

SynoposisGeneral formulae are derived from first principles for the temporal and spatial autocorrelation functions of stochastic parameters which are defined in terms of superposed, uncorrelated waves with known spectral density distribution. These formulae are first used for obtaining expressions for the autocorrelation functions of the components of the electromagnetic field strength and the electromagnetic energy density in black body radiation fields. The general theory is further applied to compression waves in liquids, and expressions are derived for the temporal and spatial autocorrelation of thermal density fluctuations in liquids, in particular near their critical point. Finally the spectrum of the fluctuations in the total radiation emitted by a thermal source, owing to the fluctuations in the energy supply to the source, is obtained from the appropriate Langevin equation, and the temporal autocorrelation function of the radiation intensity due to this cause is derived from the spectrum.
VIII.—Some 9-Fluorenyl and 9,9′-Bifluorenyl Derivatives.
Authors: Birnie J. M. , Campbell Neil 
Published: July 2016

Summary9-Carbamoylfluorene with lithium aluminium hydride, 9-cyanofluorene with this reagent and aluminium trichloride, or (on one occasion) treatment of the oximes of 9-formylfluorene with thionyl chloride yield 9,9′-dicyano-9,9′-bifluorenyl. 9-Bromofluorene and ethanolic potassium cyanide yield 9-cyano-9,9′-bifluorenyl, and 9-formylfluorene when kept in ether for a month gives 9,9′-diformyl-9,9′-bifluorenyl. The so-called α-oxime of 9-formylfluorene described in the literature contains about 33 per cent of the higher melting β-oxime. Reduction of the oximes with zinc and acetic acid yields di(9-fluorenylidenemethyl)amine, previously obtained by other methods. A new method for the preparation of 9-aminomethylenefluorene is described and its structure has been confirmed. Many 9-substituted and 9,9′-disubstituted fluorenes exhibit characteristic absorption at 1920–1880 and 1960–1940 cm.−1.
XII.—The Hexapole Magnet as a Velocity Selector and Polarizer for Neutral Spin-½ Particles
Authors: Brash H. M. , Campbell D. M. , Farago P. S. , Rae A. G. A. , Siegmann H. Chr. , Wykes J. S. 
Published: July 2016

SynopsisThe general paraxial particle-optic properties of the hexapole field are summarized and lens formulae derived. Formulae are given for the transmission as a function of particle velocity and system geometry for geometries appropriate to the case of neutral atom beams and of slow neutron beams respectively. These formulae are used to evaluate the hexapole magnet as a velocity selector and polarizer for atomic beams and as a spin polarizer for neutron beams. Some experimental observations on potassium beams are quoted in support of the theory.
IV. A Modified Hilbert Transform
Authors: Jones D. S. 
Published: July 2016

SummaryThe solution of certain integral equations, which are related to the Hilbert transform, is discussed from the viewpoint of generalized functions. The behaviour of the corresponding transforms in the complex plane is examined for generalized functions which effectively vanish at infinity. A space of generalized functions which need not be small at infinity is introduced and the integral equations solved in this space. Various possible extensions of the theory are also mentioned.
V.— On the Spectrum of Second Order Partial Differential Operators.
Authors: Brown K. J. , Michael I. M. 
Published: July 2016

SynopsisIn a recent paper, Jyoti Chaudhuri and W. N. Everitt linked the spectral properties of certain second order ordinary differential operators with the analytic properties of the solutions of the corresponding differential equations. This paper considers similar properties of the spectrum of the corresponding partial differential operators.
XI.—Correlations in alpha-Particle-accompanied fission
Authors: Feather N. 
Published: July 2016

SynopsisThe attempt is made to exhibit the necessary correlations between fragment excitation (or prompt neutron number), fragment kinetic energy, and a-particle energy, in α-particle-accompanied ternary fission. Treating a single mode of mass and charge division on the assumption that the ternary process develops directly out of an intermediate binary phase, and using the mutual electrostatic potential energy of the nascent binary fragments of this phase and the additional kinetic energy developed at the moment of α-particle emission as independent variables, various formal results are obtained and discussed in the light of the experimental evidence.In terms of a classical description, it appears likely that (for a given mode of division) the nuclear configuration at a-particle release in ternary fission is subject to much smaller variations than is the nuclear configuration at scission in binary fission in the corresponding mode. Possible inadequacies of this classical description are very briefly discussed.
XIX.—Some Triple Trigonometrical Series Equations and their Application
Authors: Parihar K. S. 
Published: July 2016

SynopsisClosed form solutions of some triple equations involving trigonometric series have been obtained, in each case, by reducing them to a single integral equation. The results have been applied to determine the distribution of stress in the interior of an infinitely long elastic strip containing a pair of Griffith cracks situated on a line perpendicular to the bounding lines of the strip.
XXIV.—Separation of the Light Particle in the Ternary Fission Process (II).
Authors: Feather N. 
Published: July 2016

SynopsisA previous discussion (Feather 1969a) of the energetics of the separation of an α-particle in the ternary fission of an (e, e) nucleus is generalised to cover the case of a light particle of arbitrary mass and charge separating from a fissioning nucleus the nucleon-parity type of which may be any one of the four possible types. A ‘profitability index’ is defined in terms of which certain broad differences in the light-particle spectra from different types of fissioning nucleus are predicted, and the results of detailed calculations of the minimum energy expense of separation are given for various light particles emitted in the fission of and, for comparison, for an α-particle emitted in the fission of , an (o, o) nucleus.
7.—Beta-spectra Shape Factors in the Decay of 140La and 72Ga
Authors: Connor R. D. , Goldman T. J. , Fairweather I. L. 
Published: July 2016

SynopsisThe shape factors of the partial β-spectra of highest end-point energy in the decay of 72Ga and 140La have been calculated and compared with the only previously reported determination of these quantities. For both isotopes the shape factor is consonant with the view that the Blj matrix element controls the decay. Shape factor analysis alone is insufficient to determine the cause of the predominance of a particular matrix element.
11.—The Symmetry of Atomic Vibrations in KH2PO4
Authors: Montgomery H. , Paul G. L. 
Published: July 2016

SynopsisThe space group for paraelectric KH2PO4 is described and tables of irreducible representations are presented. These are used to derive the symmetry of the vibrational modes for both the heavy atoms and the hydrogen atoms, the latter being treated by a simple pseudo-spin model. Recent experiments on KH2PO4 and related materials are analysed in terms of the mode symmetries, and the light these experiments throw upon the nature of the phase transitions is briefly discussed.
16.—Fast Neutron Polarisation
Authors: Galloway R. B. 
Published: July 2016

SynopsisThe state of knowledge of fast neutron polarisation due to elastic scattering is reviewed in relation to the optical model of the process. There is clearly a substantial lack of polarisation data at neutron energies and concerning nuclei appropriate to the optical model description. The situation for the extremes of small and of large angle scattering is also considered.
20.—Energy in the Future—The Role of Nuclear Fission and Fusion
Authors: Bennett Lewis W. 
Published: July 2016

SynopsisFor comfort the total future world population faces a limit. Three or more educated generations seem likely before the population growth will be checked. It seems prudent to envisage a world population of about 15 thousand million in a century or two. Their energy needs for food, water, warmth, clean air, locomotion and management of wastes may be assessed at 5 to 50 thermal kilowatts per capita. That amounts to 0·05 per cent, to 0·5 per cent, of the energy supply from the sun. Generating this amount by nuclear fission would cause no severe pressure on world resources of uranium and thorium for hundreds of centuries, if near-breeders such as CANDU reactors are used exploiting the uranium-thorium fuel cycle. Neither the cost of fuel supply nor of waste management to keep the environment unburdened by extra radioactivity need be as high as prices now paid for coal and oil. The addition of breeders or other means such as nuclear fusion or spallation for increasing the neutron supply would significantly lower the use of uranium and thorium but would not necessarily reduce the cost of power.
25.—Soft-mode Spectroscopy of NdA103
Authors: Scott J. F. 
Published: July 2016

SynopsisMeasurement of the frequencies, linewidths and cross-sections of optical phonons in neodymium aluminate by means of inelastic light scattering have permitted the prediction of a trigonal-to-cubic phase transition below the melting point of the crystal. This phase transition was subsequently directly measured via differential thermal analysis.
29.—The Possibility of Second-Class Currents in Gamow-Teller β-Decay
Authors: Wilkinson D. H. 
Published: July 2016

SynopsisMirror Gamow-Teller β-decay takes place from members of isospin multiplets to a common final state or to final states that are mirrors of each other. It is found that in these cases the intrinsic speed of the positon emission is systematically lower, by 10–15 per cent, than that of the mirror negaton emission. This difference could be due to a fundamental weak interaction effect, a second-class term in the weak hadronie current, or to a failure of exact symmetry in the nuclear structure. The influence of binding energy differences is studied in the latter context and it is concluded that they cannot be responsible for the empirical asymmetry. The effects of changes of configurational mix across the multiplets is also considered; very few estimates have ben explicitly made but it would be surprising if this effect turned out to be of constant sign and magnitude over the wide range of A involved. No present firm conclusion can be made but none of the empirical evidence is inconsistent with the reality of a largely momentum-transfer-independent second-class current.
5.—Mott Scattering and Stern-Gerlach Effect
Authors: Farago P. S. 
Published: July 2016

SynopsisIt is shown that, from an operational point of view, the production and detection of spin-polarised electrons by Mott scattering is equivalent to the performance of a Stern-Gerlach type experiment with heavy spin-one-half particles. The argument is based on standard quantum mechanical formalism using density matrices and Stokes vectors for the description of polarised assemblies of particles.
6.—On the Distribution of the Eigenvalues and the Order of the Eigenfunctions of a Fourth-order Singular Boundary Value Problem
Authors: Chaudhuri Jyoti , Everitt W. N. 
Published: July 2016

SynopsisThis paper is concerned with the asymptotic properties of the eigenvalues and eigenfunctions of the boundary value problemWith suitable restrictions placed on the real-valued coefficient q the spectrum of this problem, with respect to the eigenvalue parameter λ, is discrete; let {λn; n = 1, 2, …} and {ψn; n = 1, 2, …} be the eigenvalues and associated eigenfunctions. Asymptotic formulae are obtained for N(λ), the number of eigenvalues not exceeding the real number λ, and for ψn(x) as n→∞ where x is a fixed, positive real number.