bluuebunny/crossref_metadata_2025 huggingface dataset:
X.—A New Way of Measuring the Velocity of Light
Authors: Houstoun R. A.
Published: July 2016

During the past four years I have been developing a new method of measuring the velocity of light. I had intended to defer publication until the method was perfected, but progress is becoming increasingly difficult owing to war conditions. So I give here an account of the progress already made. I am certain the method has a great future before it. There is a notice of it in a letter to Nature (Houstoun, 1938), but except for this no description of the method has yet been given.
XI.—Quantum Theory of the Chemical Bond
Authors: Coulson C. A.
Published: July 2016

Wave mechanics is able to describe with some precision the motions of electrons in atoms, but when we study molecules we have to use more approximate descriptions. It turns out that what the chemist is accustomed to call a single bond is in reality a pair of electrons, having opposed spins, describing equivalent orbits which have symmetry about the line joining the two nuclei concerned; this may be called a localised bond. The tetrahedral character of the bonds from saturated Carbon atoms are easily fitted into this scheme.

In Ethylene, however, another type of orbit appears; this is the double-streamer orbit, and two electrons in this orbit convert a normal single bond into a double bond. Again the bond is a localised bond, with a characteristic energy and length.

In more complex molecules, such as Benzene, there is a framework of single bonds, and the remaining electrons have orbits that embrace all six of the Carbon atoms; these mobile electrons give the aromatic and conjugated molecules their characteristic properties, but as a result the bonds are neither pure single bonds nor pure double bonds, but a hybrid of the two, and the electrons in these bonds are no longer localised in the region between any two particular nuclei. The energies of these molecules can be calculated in fair agreement with experiment, and from a knowledge of the wave function it is possible to define an order, which is usually fractional, for these bonds. In Benzene all the C-C links are equivalent, and their order is I⅔.

A curve which connects the fractional order with the length of the bond enables us to predict the lengths of these bonds, and, where experimental comparison is available, agreement is found. These mobile electrons are important in a study of vibration frequencies, restricted rotation about C-C bonds, and in polymerisation.
III.—On the Matrix Representation of Complex Symbols
Authors: Rutherford D. E.
Published: July 2016

1. In a recent paper Professor G. Temple has given a matrix representation of the Clebsch-Aronhold symbols by means of which a homogeneous form of degree m in the n variables x1 …, xn may be written (a1x1 + … + anxn)m. The present paper is concerned with an extension of Temple's method to include Weitzenböck's complex symbols which have proved so potent in the treatment of linear and higher complexes. A slight rearrangement of Temple's matrices is suggested which displays more clearly the nature of the representation.
IV.—A Note on Karl Pearson's Selection Formulæ
Authors: Lawley D. N.
Published: July 2016

It is a well-known fact that Karl Pearson's formulae expressing the effect of selection on the means, variances and covariances of a multivariate population hold when the variates are such as to be normally distributed both before and after selection. It is not, however, generally known that the formulae are true under much more general conditions, and in view of a recent controversy it has been thought desirable to establish precisely what these conditions are. In dealing with the problem we shall adopt the shortened vector and matrix notation introduced by Aitken (I). This notation is reproduced below with but slight modifications.
VII.—Quantum Mechanics of Fields. I. Pure Fields
Authors: Born Max , Peng H. W.
Published: July 2016

The difficulties met in the usual treatment of quantised field theories seem to us somewhat similar to those which occurred in Bohr's semi-classical quantum mechanics of particles. In this theory the orbits were described by Fourier series in the time; there was no exact correspondence between the periodic terms of this series and quantum transitions, but only an approximate one for terms of high order. Matrix mechanics considers not the Fourier series, but the single terms which are generalised into matrix elements having not one but two indices. This generalisation is founded on Ritz's combination principle.
XIII.—A Problem in the Random Distribution of Particles
Authors: Eggleton Philip , Kermack William Ogilvie
Published: July 2016

1.

An n : l is defined as a group of n particles which can be contained within a seeker length l moving around a closed line of length L on which N particles are distributed at random. An expression has been obtained for the average number of distinct n : l's per closed line.

2.

An expression has also been derived for the average numbers of n : l's in the corresponding problem where the line of length L containing the N particles is open and not closed.

3.

Analogous problems in two dimensions are considered, in which the particles are arranged at random on a plane and the place of the seeker line is taken by an orientated rectangle. Exact expressions are given for the desired averages.

4.

The extension of the methods used to analogous problems in three dimensions is discussed. Exact expressions have not been obtained, but approximations are given which hold when n is much greater or much smaller than x, the average number found within the seeker area.
XVII.—The Regraduation of Clocks in Spherically Symmetric Space-times of General Relativity.
Authors: McVittie G. C.
Published: July 2016

The changes in his description of events brought about by an arbitrary regraduation of an observer's clock are examined, taking the axioms of general relativity as fundamental. It is shown that regraduation does not imply a change from one Riemannian space-time to another but merely a coordinate transformation within space-time. A generalisation of the “dynamical time” of kinematical relativity is a by-product of the investigation.
XXIII.—The Number of the Elements
Authors: Feather N.
Published: July 2016

The development of the idea of the chemical element is traced from its early beginnings, and the importance for this development of the Newtonian concept of invariable mass is emphasised. The emergence of the nuclear atom model is outlined, and the discovery of the complex (isotopic) nature of the majority of known chemical elements is described. Nuclear charge (Z) and mass (A) numbers are defined. Previously recognised regularities concerning mass and charge numbers of existing stable species are shown to have exact counterparts in regularities relating to the degree of instability (as measured by the energy of disintegration) of β-active species (“naturally” and “artificially” radioactive species). Naturally occurring a-active species are regarded as the analogues of the stable species for charge numbers greater than 83, and for charge numbers both greater and less than this value the limitation to the number of stable or quasi-stable isotopes of a given element (limitation of A values for a given Z) is established as essentially a question of nuclear stability as against β-emission (positive and negative electron emission). Finally, reasons are given for supposing that the number of possible chemical elements is limited (limitation to Z in the direction of Z increasing) by the susceptibility to spontaneous nuclear fission of species of sufficiently high nuclear charge.
XXXIII.—Applications of Elliptic Functions to Wind Tunnel Interference
Authors: Milne-Thomson L. M.
Published: July 2016

A general formula is obtained for the interference velocity when an aerofoil with elliptically distributed circulation is in a closed or open wind tunnel of any cross-section. The mapping of the section on the interior of a circle is given in terms of the Jacobian elliptic functions appropriate to the ellipse and rectangle. The result is worked out for an aerofoil which spans the focal distance in a tunnel whose section is an ellipse.
XXXV.—Graphite Crystals and Crystallites. I. Binding Energies in Small Crystal Layers
Authors: Bradburn Mary , Coulson C. A. , Rushbrooke G. S.
Published: July 2016

Calculations are made of the resonance energy, bond order and bond length in a series of graphitic layers of varying size. Carbon–carbon bond lengths appear to vary very little in size with increasing number of carbon atoms, in agreement with experiment. But variations in resonance energy are significant, and indicate clearly that resonance, by itself, favours an approximately square, rather than oblong, shape. But in the case of such layers in equilibrium in the presence of molecular hydrogen, the most stable layer containing a given number of carbon atoms is of the long, thin polyphenyl type. Some tentative calculations suggest that polymerisation of smaller groups to larger ones should be endothermic, in agreement with the experimental fact that the formation of larger graphitic crystallites during carbonisation occurs, with emission of hydrogen, only at high temperatures.
III.—The Simplest Form of Second-Order Linear Differential Equation, with Periodic Coefficient, having Finite Singularities
Authors: Cambi Enzo
Published: July 2016

Synopsis

A differential equation of the second order, arising in problems of disturbed oscillation, such as occur in frequency modulation, is considered. The nature of its solutions is examined by the method of continued fractions. The cases in which the solutions are periodic, and the regions of stability and instability (lability), are determined according to the values taken by the two parameters involved.
IX.—A Measurement of the Velocity of Light
Authors: Houstoun R. A.
Published: July 2016

Synopsis

The author has already described a new method of measuring the velocity of light, which replaces Fizeau's toothed wheel by a piezo-quartz. This acts as an intermittent diffraction grating, and it interrupts the beam 200 times more rapidly than Fizeau's wheel did.

The present paper describes the application of the method to the measurement of the velocity of light in air. The total length of path was 78 metres. The frequency of interruption was measured by comparison with the Droitwich radio station. The result reduced to vacuum is 299,775 kilometres per second and is in agreement with other recent determinations, but, as a result of the experience gained, it will be possible to increase the accuracy at least ten times.
XII.—Clebsch-Aronhold Symbols and the Theory of Symmetric Functions
Authors: Turnbull H. W. , Wallace A. H.
Published: July 2016

Synopsis

A square matrix A = (aij) is expressed symbolically in terms of Clebsch-Aronhold equivalent symbols aij = aiaj = βibj = …, and the symbolic expressions for symmetric functions of the latent roots of A are considered, the relation between these functions and projective invariants of the bilinear form uAx being noted. The Newton and Brioschi relations between the symmetric functions are obtained by reduction of symbolic determinants and permanents respectively, and the Wronskian relations are shown to be equivalent to certain identities between determinants and permanents due to Muir. Also the fundamental theorem of symmetric functions is obtained symbolically as a consequence of the first fundamental theorem of invariants. The paper concludes with a note on the symbolization of the h-bialternants, that is of the traces of irreducible invariant matrices of A.
XXI.—Index Polynomials and Bifurcating Root-Trees
Authors: Minc H.
Published: July 2016

A study is made of faithful representations of the free cyclic (non-associative) groupoid by means of bifurcating root-trees and by means of index polynomials in two indeterminates emphasizing the intimate connection between these representations. The properties of trees and index polynomials are investigated and the concept of lattice of trees is introduced.
VII.—Jacobians of Transformations involving Orthogonal Matrices
Authors: Jack Henry
Published: July 2016

Synopsis

Using a technique due to Macbeath (Jack and Macbeath 1959) this paper gives what the author hopes is a shorter and easier presentation of the evaluation of certain Jacobians of matrix transformations which have occurred in statistics and the theory of quadratic forms.
VI.—On Solvability of Some Two-Parameter Eigenvalue Problems in Hilbert Space
Authors: Bohte Z.
Published: July 2016

Synopsis

This paper studies two particular cases of the general 2-parameter eigenvalue problem namely

where A, B, B1, B2, C, C1, C2 are self-adjoint operators in Hilbert space, all except A being bounded. The disposable parameters λ and μ have to be determined so that the equations have non-trivial solutions x, y.

On the assumption that the solution is known for ∊ = o, solutions are constructed in the form of series for λ, μ, x, y as power series in ∊ with finite radius of convergence.
X.—Determinants for Matrices over Lattices
Authors: Chesley D. S. , Bevis J. H.
Published: July 2016

In the literature concerning matrices whose co-ordinates are elements of a Boolean lattice, one may find three different definitions for the determinant of a matrix. We shall call these the first, second and third determinant and will denote the value of the ith determinant of a matrix A by |A |i for i = 1, 2, 3. The first determinant may be defined for square matrices over an arbitrary lattice. The second and third determinants may be defined for square matrices over any lattice L with a greatest element I, a least element o and an orthocomplementation′: L→L, that is a′ is a complement of a, a = a″ and a ≤ b implies that b′ ≤ a′ for all a, b in L. In this paper we obtain some elementary properties of these determinants in this general setting and in the particular case where L is an orthomodular lattice, that is a lattice with o, 1 and an orthocomplementation' such that
XIII.—The Spectrum of a Fourth-Order Differential Operator
Authors: Chaudhuri Jyoti , Everitt W. N.
Published: July 2016

Synopsis

This paper considers the properties of the spectrum of a differential operator derived from differential expressions of the fourth order. With certain conditions on the coefficients of the differential expression the spectrum of the operator is discrete and an estimate is obtained of the number of eigenvalues lying in a given bounded interval of the real line. The results are compared with those obtained by alternative methods. Additional restrictions on the coefficients give special cases previously considered by other authors.
XV.—Separation of the light particle in the “ternary” fission process
Authors: Feather N.
Published: July 2016

Synopsis

It is shown that the essential condition, energy-wise, for the emission of a light particle from a binary fission fragment, in the immediate post-scission stage, is that the fragment deformation must be capable of sudden partial collapse so as to release energy equal to the binding energy of the particle in the undeformed (unexcited) fragment together with an amount cancelling the mutual (electrostatic) potential energy of the particle and the residual deformed (excited) fragment at the instant of separation. On this basis the mean energy required for alpha-particle release in the thermal-neutron-induced ternary fission of 235U is estimated as 16·5 MeV, and it is concluded that those binary fissions which develop into alpha-particle-accompanied ternary fissions in this case are those in which, on the average, the total deformation-excitation energy of the fragments is some 10 MeV greater than the most probable value for all binary fissions. A qualitative discussion of the relative yields of other light-particle-accompanied modes is also given.

An Appendix contains a summary of comparable calculations based on the one-stage view of the ternary fission process.
I.—A class of symmetric polynomials with a parameter
Authors: Jack Henry
Published: July 2016

Synopsis

In an attempt to evaluate the integral (5) below, using a decomposition of an orthogonal matrix (Jack 1968), the author is led to define a set of polynomials, one for each partition of an integer k, which are invariant under the orthogonal group and which depend on a real parameter α. An explicit representation of these polynomials is given in an operational form. When α = − 1, these polynomials coincide with the augmented monomial symmetric functions. When α = 1, a systematic way of taking linear combinations of these polynomials is explained and it is shown that the resulting polynomials coincide with the Schur functions from the representation theory of the symmetric group. A similar procedure in the case α = 2 then appears to give the zonal polynomials as defined by James (1964, p. 478).
VIII.—The Schrödinger Equation with a Periodic Potential
Authors: Eastham M. S. P.
Published: July 2016

Synopsis

The differential equation

in N dimensions is considered, where q(x) is periodic. When N = 1, it is known that the conditional stability set coincides with the spectrum and that these also coincide with two other sets involving eigenvalues of associated eigenvalue problems. These results have been proved by means of the Floquet theory and the discriminant. Here an alternative proof is given which avoids the Floquet theory and which applies to the general case of N dimensions.
XVI.—Creeping Waves in an Inhomogeneous Medium
Authors: Boyd W. G. C.
Published: July 2016

Synopsis

This paper is concerned with high-frequency scattering in a medium, the square of whose refractive index varies linearly with height from a plane boundary. Two asymptotic methods are examined, namely the method of stationary phase and evaluation by residue series. The first of these corresponds to geometric optics and gives the high-frequency field in the illuminated region, while the second complements the first in the sense that if thsre is no point of stationary phase, the residue series is an asymptotic expansion of the field. The Airy functions in the residue series can be replaced by their asymptotic developments in terms of exponentials, and when this is done only the first term or first creeping wave is of genuine significance.
XVII.—Interaction of Toluenesulphinic Acid, Aldehydes and Secondary Amines
Authors: Hayes B. T. , Stevens T. S.
Published: July 2016

Synopsis

p-Toluenesulphinic acid, with formaldehyde or an aromatic aldehyde, attack the nucleus of a secondary aromatic amine, giving a 4-alkylamino-benzyl (or -benzhydryl) p-tolyl sulphone.
XX.—A Coördinatisation of Lattices by One-sided Baer Assemblies
Authors: Blyth T. S. , Hardy W. C.
Published: July 2016

Synopsis

In an earlier publication [1] we introduced the notion of a Baer assembly and applied it to obtain a coördinatisation theory for semilattices. This was achieved by considering the semigroup of quasi-residuated (i.e. ℴ-preserving and isotone) mappings on a bounded semilattice. In the present paper we consider the semigroup of quasi-residuated ∪-homomorphisms (or hemimorphisms) on a bounded lattice and thus show how a particular type of one-sided Baer assembly can be used to provide a coördinatisation theory for lattices; and in particular for complemented, modular and distributive lattices.
1.—Introduction—Personal Reflections
Authors: Dee P. I.
Published: July 2016

The invitation, made to me, to write the introduction to this volume of the Proceedings of the Royal Society of Edinburgh, which is dedicated to Norman Feather, suggested that I should place on record some of my reminiscences which might convey a picture of his character and personality. I feel, however, that first I should make some reference, albeit brief, to the distinction of his contributions to science over the 45 years which have elapsed since he first started to research in the Cavendish Laboratory at Cambridge.
18.—A Kinetic Approach to the Irreversible Thermodynamics of Membrane Transport
Authors: Hogg J. , Williams E. J.
Published: July 2016

Synopsis

The problem of the simultaneous transport of solute and water through a cell membrane is examined from a kinetic standpoint. It is shown that the results obtained by this approach are in accord with those obtained conventionally by irreversible thermodynamics. Stress is laid on the fact that the kinetic theory has certain advantages over the conventional approach in that it provides, among other things, a form of membrane model and at the same time its range of validity is indicated.
28.—Comments on Ternary Fission: Post-scission α-Particle Emission
Authors: Vass D. G.
Published: July 2016

Synopsis

The possibility of α-particle emission from fully accelerated fission fragments is considered. The results of trajectory calculations for such α-particles emitted during the fission of 252Cf are presented, and their significance in relation to the observed energy and angular distributions of the α-particles is discussed.
1.—On Measurable Selections
Authors: Robertson A. P.
Published: July 2016

Synopsis

Measurable selection theorems are proved, for a compact-valued measurable multifunction into a Hausdorff space that is the continuous image of a separable metric space, and for a closed-valued measurable multifunction from a suitable measure space to a regular Souslin space. The connection between Polish spaces and certain subsets of the real line is related to a measurable selection theorem for multifunctions into a Polish space.
20.—On the Limit-point and Limit-circle Theory of Second-order Differential Equations
Authors: Ong K. S.
Published: July 2016

Synopsis

In this paper the Weyl limit-point and limit-circle theory of second-order differential equations is extended to the case that the weight function is allowed to take on both positive and negative values—the polar case. This extension is achieved using Weyl's limit circle method.
30.—Acoustic Diffraction by an Absorbing Semi-infinite Half Plane in a Moving Fluid
Authors: Rawlins A. D.
Published: July 2016

Synopsis

The diffraction of a line source of sound by an absorbing semi-infinite half plane in the presence of a fluid flow is examined. It is found that the radiated sound intensity, in the half space in which the source is located, can be considerably reduced by a suitable choice of the absorption parameter. For subsonic flow the system exhibits no acoustic instabilities.
18.—Singular Linear Differential Operators with many Parameters
Authors: Sleeman B. D.
Published: July 2016

Synopsis

It is the purpose of this paper to make a study of the solutions of the following k-formally self-adjoint differential equations

where ar, br), r = 1, 2, …, k, denote k semi-open intervals in which ar is finite and br is arbitrary and the λs, s = 1, 2, …, k, are spectral parameters.

The main theme of the paper is that of extending the Hermann Weyl limit-point, limit-circle theory to the multi-parameter case. That is we consider under which circumstances there exist, for each r, one or two solutions yr(xr) of (*) which are square integrable in a suitably defined Hilbert space Hr. This is then generalised to consider the problem of investigating the possibility of the product

of solutions of (*) being square integrable in H, the tensor product of the separate spaces Hr. The analyticky of the corresponding generalised Hermann Weyl coefficients mr(λ1, λ2,…, λk), r = 1,…, k, is also investigated. Some examples illustrating the theory are given and an alternative formulation of the problem is suggested.
28.—Simultaneous Emission of two Light Charged Particles in Fission
Authors: Feather N.
Published: July 2016

Synopsis

The suggestion is made that in certain cases the light particles emitted in ternary fission may be released otherwise than in the ground state—and, in particular, that low-lying particle-unstable states may be involved. This suggestion is examined quantitatively in respect of the release of 8Be in the first excited short-lived state of 2·9 MeV excitation, and is shown to provide a satisfactory interpretation of the observation of quaternary fission in the α,α mode recently reported by Kataria et al. (1973). It appears probable that the other modes of quaternary fission observed by these authors should be similarly interpreted.
30.—The Chord Locus of a Certain Curve in [n]
Authors: Edge W. L.
Published: July 2016

Synopsis

The sharing of a common self-polar simplex by n−1 quadrics in [n] confers special features on their curve of intersection Γn. The three-dimensional locus Mn of chords of Γn. has certain singularities, some of which are described. In conclusion, a few comments refer to the case n = 4 when Mn is defined by a single equation.
III.—On the Moments and Regression Equations of the Fourfold Negative and Fourfold Negative Factorial Binomial Distributions
Authors: Wiid A. J. B.
Published: July 2016

Synopsis

The bivariate distribution corresponding to the univariate negative binomial, and the corresponding distribution when sampling is without replacement, are investigated. Formulas are derived for the factorial moment generating functions and for the regression equations, which are linear.
IV.—The Differential Equations Associated with the Uniformization of Certain Algebraic Curves
Authors: Rankin R. A.
Published: July 2016

Synopsis

Every algebraic equation can be uniformized by automorphic functions belonging to a certain group of bilinear transformations. In certain cases, such as for hyperelliptic equations, this group is a subgroup of the monodromic group of a differential equation of the form

where R(z) is a rational function which, in general, contains unknown parameters as coefficients. A conjecture of E. T. Whittaker regarding the values of these parameters for the hyperelliptic case is proved for a wide variety of algebraic equations whose branch points possess certain symmetric properties, and is extended to equations of higher type. In several cases, the uniformizing functions belong to subgroups of the groups of the Riemann-Schwarz triangle functions.
VII.—Entropic Functions for Linear Algebras
Authors: Etherington I. M. H.
Published: July 2016

Synopsis

The function ɸx = log x satisfies the functional equation ɸxy = λɸx + μɸy + ϰ, where in this case x, y are complex variables, λ = μ = i, and ϰ = 2πi, o or − 2πi according as σ < − π, − π < σ < π, π < σ, where σ = arg x + arg y. Generalizing this situation, let A be a linear algebra with basis e1, …, en over the real or complex field and let ɸx be a complexvalued function of the hypercomplex variable x = Σξiei, i.e. of the n real or complex variables ξi. Assume that the gradient ∂ɸx, i.e. the column vector of partial derivatives {∂ɸx/∂ξi}, exists at a general point of A. Then ɸx is called an entropic function if it satisfies a functional equation of the above-mentioned form and obeys certain other postulates, ϰ being a step function of the two hypercomplex variables. Values of the constants λ, μ (complex numbers, not both zero) for which a solution exists are entropic roots of A. They are usually discrete.
XVIII.—Energy Levels of B10: Evidence from the B10 γ-ray Spectrum.
Authors: Galloway R. B. , Sillitto R. M.
Published: July 2016

SYNOPSIS

A study of the γ-rays produced during the bombardment of a thick Be9 target by 600 keV deuterons was made to investigate the possible existence of a level at 2·86 MeV in B10, about which contradictory reports have appeared in the literature.

A spectrum of the γ-rays in coincidence with the 0·72 MeV B10 γ-ray (text-fig. 5) was obtained, and is interpreted as providing evidence for a level in B10 at 2·86 MeV. The relative intensities of the γ-rays in an ungated spectrum, and in spectra gated by the 0·72 and 1·02 MeV B10 γ-rays, were found, and a decay scheme consistent with the observations is deduced (text-fig. 6b). The relative intensities of the transitions in this decay scheme are consistent with the intensities of the neutron groups in a spectrum of the neutrons from this reaction. A spin value of 2 or 3 is suggested for the 2.86 MeV level.
XXII.—Some Nilpotent Algebras with Centres of Given Dimension
Authors: Wallace E. W.
Published: July 2016

SYNOPSIS

Algebras which are nilpotent and anti-commutative are studied. Canonical forms are found for all such algebras of dimension n whose centres have dimension n−r (r < 3), and characters are given which enable any two non-isomorphic algebras to be distinguished.

A metrisable Lie algebra is a Lie algebra for which there is a non-singular, symmetric, adjoint-invariant bilinear form a(λ, μ), and such an algebra is reduced if its centre is contained in its derived algebra. The importance of the reduced algebras follows from the fact that every metrisable Lie algebra is the direct sum of a reduced metrisable Lie algebra and an abelian Lie algebra. Tsou (Thesis 1955) introduced metrisable Lie algebras, and obtained canonical forms for all real reduced metrisable Lie algebras whose derived algebras have dimension 3. We conclude this paper by providing an alternative derivation, two of the algebras being nilpotent.
XXIII.—Representations of a Number as the Sum of a Large Number of Squares
Authors: Rankin R. A
Published: July 2016

SYNOPSIS

An asymptotic formula is given for the number r(s, P; N) of representations of an integer N as the sum of s non-negative squares, where each square does not exceed P2. The numbers s, P and N are large and are subject to certain conditions, one of which is that N is approximately ⅓sP2.
VIII.—The Central Limit Theorem for a Convergent Non-homogeneous Finite Markov Chain
Authors: Mott J. L.
Published: July 2016

Synopsis

The distribution of xn, the number of occurrences of a given one of k possible states of a non-homogeneous Markov chain {Pj} in n successive trials, is considered. It is shown that if Pn → P, a positive-regular stochastic matrix, as n → ∞ then the distribution about its mean of xn/n½ tends to normality, and that the variance tends to that of the corresponding distribution associated with the homogeneous chain {P}.
Food insecurity as a driver of obesity in humans: The insurance hypothesis
Authors: Nettle Daniel , Andrews Clare , Bateson Melissa
Published: July 2016

Integrative explanations of why obesity is more prevalent in some sectors of the human population than others are lacking. Here, we outline and evaluate one candidate explanation, the insurance hypothesis (IH). The IH is rooted in adaptive evolutionary thinking: The function of storing fat is to provide a buffer against shortfall in the food supply. Thus, individuals should store more fat when they receive cues that access to food is uncertain. Applied to humans, this implies that an important proximate driver of obesity should be food insecurity rather than food abundance per se. We integrate several distinct lines of theory and evidence that bear on this hypothesis. We present a theoretical model that shows it is optimal to store more fat when food access is uncertain, and we review the experimental literature from non-human animals showing that fat reserves increase when access to food is restricted. We provide a meta-analysis of 125 epidemiological studies of the association between perceived food insecurity and high body weight in humans. There is a robust positive association, but it is restricted to adult women in high-income countries. We explore why this could be in light of the IH and our theoretical model. We conclude that although the IH alone cannot explain the distribution of obesity in the human population, it may represent a very important component of a pluralistic explanation. We also discuss insights it may offer into the developmental origins of obesity, dieting-induced weight gain, and anorexia nervosa.
ASYMPTOTICS OF DIAGONAL ELEMENTS OF PROJECTION MATRICES UNDER MANY INSTRUMENTS/REGRESSORS
Authors: Anatolyev Stanislav , Yaskov Pavel
Published: July 2016

This article sheds light on the asymptotic behavior of diagonal elements of projection matrices associated with instruments or regressors under many instrument/regressor asymptotics. When the diagonal elements do not exhibit variation asymptotically, certain results in the many instrument/regressor literature lead to elegant solutions and conclusions. We establish conditions when this happens, provide relevant examples, and analyze instrument designs, for which this property does or does not hold.
A Leicestershire Recusant Family: The Nevills of Nevill Holt—111*
Authors: Elliott Bernard
Published: July 2016

WHEN COSMAS NEVILL II died on 1 June 1829 at the ripe old age of 81’ the family were within striking distance of having to quit their traditional home of Nevill Holt in the East Midlands, for only two more generations of Nevills were to reside there. The first of these was Cosmas II'S second son, Charles, the eldest son, Cosmas III, having predeceased him. While his father was alive, Charles Nevill seems to have preferred living in London to residence at Holt. In 1821 he married Lady Georgiana Bingham, youngest daughter of the second Earl of Lucan, and the couple then spent some time in Paris where their first child, Lavinia Elizabeth, was born on 23 January 1824. Subsequently she was conditionally baptised by a Catholic priest at Holt. Their next child, another daughter, Charlotte, was born in London on 10 March 1827 and the Register of Nevill Holt chapel does not mention her baptism. Though Cosmas II died in June 1829, Charles and Georgiana were in no great hurry to settle in the Leicestershire countryside, for their third child and heir, Cosmo George, was born in London on 3 June 1830 and was baptised by the Rev. Mr. Fryer. Their fourth child, Louisa, was also born there, on 21 September 1831, as was their fifth, Richard, on 15 December 1832. Their sixth and last child, William, was likewise born in London, on 21 April 1834.2
Priesthood and Politics in the Nineteenth Century: The Turbulent Career of Thomas Mcdonnell
Authors: F. Champ Judith
Published: July 2016

THE CONSEQUENCE of the Cisapline attempt to ‘grapple with the social and intellectual transformation of the modern world” and to bring about a ‘revision of the pyramidal structure of the Tridentine Church” was the greater assimilation of English Catholics into contemporary society. Encouraged by a new sense of freedom, clergy and laity participated more actively in English public life’ and dismantled much of the closed élite community of the recusant period. This led to a brief phase in which both clergy and laity exercised their new-found freedoms, but which was dogged by disputes. Arguments raged between liberalism and authority, and between sectarian ideals and non-denominational activities. They were eventually resolved in a restoration, by 1850, of the pyramidal structure of the Tridentine Church, in which the role of the laity was subject to the authority and guidance of the clergy.
Some Notes on the ‘Painted Life’ of Mary Ward
Authors: Marmion John P.
Published: July 2016

MARY WARD, foundress of the Institute of the Blessed Virgin Mary (I.B.V.M.) and so indirectly of the Loreto nuns, was born in 1585 and died in 1645. In addition to some autobiographical writings in English and Italian, there are a significant number of Mary Ward's letters preserved. Early lives were written by Winefrid Wigmore in English (there is a seventeenth century manuscript of this in Loreto college, Manchester), by Pageti in Italian in 1662, by Bissel in Latin in 1667 or ‘68 and in German by Lohner in 1689. The Pageti and Lohner Mss. are in the Institute's archives at Nymphembourg, but in addition to these documentary sources the Institute possesses at Augsburg a remarkable collection of fifty large canvases called ‘The Painted Life’. As a series of paintings on the life of a single individual from this period these would seem to be almost unique. There is a short cycle by Rubens on Marie de’ Medici, and some baroque series on the lives on the saints in Roman churches. But there does not seem to be anything comparable to this ‘Painted Life’ from the seventeenth century. Some account of it will therefore be of interest both to the historian and the art critic and, since certain incidents in Mary Ward's life are alluded to in connection with the paintings, a brief outline-biography is provided by way of introduction.
Elementary proofs of relations between Eisenstein series
Authors: Rankin R. A.
Published: July 2016

Synopsis

Eisenstein series are entire modular forms Ek of even integral weight k≧ 4 with Fourier expansions given by (1.1). There are numerous identities, such as E8 = , relating these series. These are usually proved by arguments making use of the dimensions of vector spaces of modular forms, and not directly. The paper shows how such identities can be proved by elementary methods by studying chains of solutions of Diophantine equations of the form xξ+yη = n.
2.—Barriers bounded by Two Transition Points of Arbitrary Odd Order
Authors: Heading John
Published: July 2016

Synopsis

A generalisation is considered of the potential barrier problem beyond the familiar case in which the barrier is bounded by transition points of order one. Here, the two transition points involved are of arbitrary odd order. The approximate method employed, though formal in character, avoids certain pitfalls often made in the past whereby certain exponentially small terms within the barrier are confounded with inherent error terms. This confusion is avoided in the treatment given here by tracing uniformly approximate solutions round the transition points in the complex plane by means of the Stokes phenomenon, the method not requiring the dubious concept of a subdominant term existing in the presence of a dominant term on a Stokes line. At the same time, the solutions to which the reflection and transmission coefficients may be attached are carefully discussed, so that the appearance of a small exponential term may be seen to be genuine when taken in conjunction with inherent error terms. The resulting formula for the modulus of the reflection coefficient generalizes the more elementary formula.
11.—Limit-n Criteria of Integral Type
Authors: Atkinson F. V.
Published: July 2016

Synopsis

Conditions are obtained which ensure that a 2nth order ordinary differential equation on a halfaxis have at most n linearly independent square-integrable solutions. The emphasis is on bounds on the integrals of the coefficients, and on conditions imposed over a sequence of intervals.
12.—Spectral Theory of Rotating Chains
Authors: Stuart C. A.
Published: July 2016

Synopsis

Two eigenvalue problems associated with steady rotations of a chain are considered. To compare the spectra of these two problems, let σK(n) denote the set of all angular velocities with which a chain of unit length with one end fixed and the other free can rotate in a vertical plane so as to have exactly n nodes on thevertical axis (including the fixed end). In a linearised theory σK(n) is a single point, i.e.

In the full non-linear theory σK(n) is an infinite interval lying to the right of ⍵n. Indeed,

This is established in [1].

Next, let σM(β, n) denote the set of all angular velocities with which a chain, having ends fixed at unit distance apart on the vertical axis, can rotate in a vertical plane so as to have exactly n+l nodes on the vertical axis (including the ends) and so that the tension takes the value β at the lower end. This problem, in which the length of the chain is not prescribed is a model for a spinning process in which the ‘chain’ is continuously created in a rotating configuration. For β>0, we again have in the linearised theory that σM(β, n) is a singleton, i.e.

In the full non-linear theory σM(β, n) lies to the left of λn(β). Although unable to determine exactly σM(β, n) for β>0, we have

where ⍵n λn(β), and are all characterised as the nth zeros of known combinations of Bessel functions.
1.—On Square-integrable Solutions of Symmetric Systems of Differential Equations of Arbitrary Order.
Authors: Kogan V. I. , Rofe-Beketov F. S.
Published: July 2016

Synopsis

The following results are obtained for symmetric differential expressions of arbitrary order r ≧ 1 with matrix coefficients on the half-line, with a non-negative (and possibly identically degenerate) weight matrix W(t), and with a spectral parameter λ: upper and lower bounds for the deficiency indices N(λ) are found; it is proved that N(λ) is independent of λ for Im λ< 0and for Im λ>0; under very general conditions it is proved that the maximum possible values of the deficiency indices in the half-planesIm λ≷0 can only be attained simultaneously; sufficient conditions for first-order expressions to be quasi-regular are derived; and it isshown that a symmetric system of any order reduces to a canonical, first-order system.

Examples are constructed, and the case of the whole line is also touched on.
2.—On Weyl's Function m(λ).
Authors: Wray S. D.
Published: July 2016

Synopsis

A proof is given of a formula connecting Weyl's function m(λ) with a spectral integral, in the setting of the singular theory of the Sturm-Liouville differential equation. In addition, a relatively short treatment is given of the argument introducing the function. The theory applies to both the limitpoint and limit-circle cases.
4.—Perturbations of Non-linear Integral Operators
Authors: Stummel Friedrich
Published: July 2016

Synopsis

Non-linear integral operators are studied under perturbations of the integrals, the domains of integration, the kernels and the inhomogeneous terms. The main theorem establishes the local solubility of the perturbed integral equations, the biconvergence of their solutions and associated two-sided error estimates. This theorem is of interest even for the special class of linear Fredholm integral equations.
5.—The Uniform Asymptotic Stability of Certain Linear Differential-difference Equations
Authors: Datko R.
Published: July 2016

Synopsis

A necessary and sufficient condition is developed for determination of the uniform stability of a class of non-autonomous linear differential-difference equations. This condition is the analogue of the Liapunov criterion for linear ordinary differential equations.
14.—Dynamic Programming applied to Some Non-linear Stochastic Control Systems
Authors: Fuller A. T.
Published: July 2016

Synopsis

Optimal switching curves are investigated for some second-order control systems with random disturbances, saturating control, and a mean-square-error performance index.

It is first shown that the optimal switching curves are asymptotic at infinity to the optimal switching curves for corresponding deterministic systems.

The systems are then discretised in time and state, and ordinary dynamic programming is applied to the resulting Markov chain models. The discretisation techniques are described in some detail.

The same techniques are applied to a deterministic system for which the optimal switching curve is known, and it is found that the resulting discretisation errors are considerable.

Thus it turns out that ordinary discrete dynamic programming gives only a rough approximation to the optimal switching curve. It seems desirable to tackle the equations of continuous dynamic programming by techniques more refined than crude discretisation. One such technique is suggested.
16.—On Uniqueness of Topological Degree for Set-valued Mappings
Authors: Webb J. R. L. , Everitt W. N.
Published: July 2016

Synopsis

It is shown that three independent axioms uniquely determine the topological degree of set-valued maps of the form I – G, where G is a convex-valued, limit compact map. This extends earlier work of Amann and Weiss, Nussbaum, and others, in that, apart from dealing with set-valued maps, a larger class of maps is considered even in the single-valued case.
24.—The Solution of Initial-Boundary Value Problems for Pseudoparabolic Partial Differential Equations
Authors: Rundell William , Sneddon I. N.
Published: July 2016

Synopsis

A fundamental solution for equations of pseudoparabolic type is constructed by a method analogous to Hadamard's construction for elliptic equations. By the use of this fundamental solution we obtain a regularity theorem and integral representations for the solution to the more common initial boundary value problems that are associated with these equations.
Small solutions of a non-linear equation in Banach space for a degenerate case
Authors: Shearer M.
Published: July 2016

Synopsis

The existence of zeroes of an operator in Banach space near a. known zero is examined, when the non-degeneracy condition of Magnus [1] is violated, and replaced by a condition which is called simple-degeneracy. Criteria are given which help determine the number and the structure of curves of zeroes of the operator, passing through the known zero.

These conditions and results are interpreted in terms of bifurcation theory. By considering several different cases, it is shown that the failure of various conditions frequently employed in the Lyapunov–Schmidt method may be overcome using the idea of simple-degeneracy.
On the essential self-adjointness of powers of Schrödinger-type operators
Authors: Evans W. D.
Published: July 2016

Synopsis

Let T be the formally self-adjoint second-order elliptic differential expression

in ℝn, where the coefficients bj, ajk, q are real-valued and ajk ≈ akj. In this paper sufficient conditions for all positive integer powers Tm of T to be essentially self-adjoint on are obtained.
Application des méthodes de convexité et monotonie a l'étude de certaines équations quasi linéaires
Authors: Attouch H. , Damlamian A.
Published: July 2016

Synopsis

Using Hilbert space methods, existence and uniqueness are proved for the solution of some strongly non-linear partial differential equations of elliptic and parabolic type.

They are associated with quasi-linear operators of the form: -div(β(x, grad u)) + β0(x, u) where β (resp β0) is a maximal monotone subdifferential on ℝN(resp ℝ) depending smoothly on x in a bounded domain Ω of ℝN

These operators are shown to be the subdifierentials over Lp(Ω) of convex functional of the following type:

where j is a normal convex integrand over Ω×ℝN+1 satisfying a coerciveness condition.

This method avoids the theory of Sobolev-Orlicz spaces. An application is given also forthe gas-diffusion equation over ℝ+.
The operator equation TS = 1 and representations of l1 of the bicyclic semigroup
Authors: Conway John B. , Morrel Bernard B. , Stampfli Joseph G.
Published: July 2016

Synopsis

The operator equation TS = 1 is studied for power bounded operators T, S on Hilbert space, and its relation to *—representations of the bicyclic semigroup is explored.
Related singular problems and the generalized Hilbert transform
Authors: Dettman John W.
Published: July 2016

Synopsis

Abstract versions of the Cauchy problem for the Euler-Poisson-Darboux equation and the Dirichlet problem for the equation of generalized axially symmetric potential theory are related by an integral transformation. In certain special cases, this leads to abstract versions of (1) the Poisson formula for the solution of G.A.S.P.T. in a half-space, (2) pseudo-analytic functions in a half-space, and (3) a generalized Hilbert transform related to the work of Heywood, Kober, and Okikiolu. Some properties of this generalized Hilbert transform are studied including an inversion theorem.
7.—Acoustic Diffraction by an Absorbing Semi-Infinite Plane in a Moving Fluid, II
Authors: Rawlins A. D.
Published: July 2016

SYNOPSIS

The diffraction of a line source by an absorbing semi-infinite plane in the presence of a subsonic fluid flow is examined. Expressions for the total far field for the leading edge (no wake present) and the trailing edge (wake present) situation are given.

It is found that the presence of fluid flow attenuates the sound level in the shadow region of the absorbent screen. There is greater attenuation of the sound level in the trailing edge situation than there is for the leading edge situation.

Sound level attenuation on the source side of the absorbent screen is found to depend more critically on the absorptive properties of the screen than on the fluid flow. The radiated sound intensity, in the half space in which the source is located, can be considerably reduced by a suitable choice of the absorption parameter.