bluuebunny/crossref_metadata_2025 huggingface dataset:
XIX.—Some Triple Trigonometrical Series Equations and their Application
Authors: Parihar K. S.
Published: July 2016

Synopsis

Closed 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

Synopsis

A 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

Synopsis

The 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.
10.—Understanding Physics Through Its History
Authors: Devons Samuel
Published: July 2016

It is a great pleasure to join in this tribute to Professor Norman Feather: My recollections of him as teacher, friend and colleague extend back nearly forty years. I recall attending, as a Cambridge undergraduate, his lectures on ‘Properties of Matter’ (what an archaic and nostalgic flavour that title has now!); and hearing him describe his pioneering experiments on the properties of the neutron. He was my doctoral thesis examiner; and later when he was Editor of the Cambridge monographs, it was at his suggestion, and with his help and encouragement, that I made my first essay as an author. I was privileged to succeed my former teachers, Professor C. D. Ellis and Professor Feather, at Trinity College; and in the 1950s I was regularly and warmly welcomed at Edinburgh, the ambivalent benevolence of my role as ‘External Examiner’ notwithstanding.
11.—The Symmetry of Atomic Vibrations in KH2PO4
Authors: Montgomery H. , Paul G. L.
Published: July 2016

Synopsis

The 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

Synopsis

The 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

Synopsis

For 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

Synopsis

Measurement 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

Synopsis

Mirror 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

Synopsis

It 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

Synopsis

This paper is concerned with the asymptotic properties of the eigenvalues and eigenfunctions of the boundary value problem

With 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.
8.—On a Class of Series Expansions in the Theory of Emden's Equation
Authors: Hille Einar
Published: July 2016

Synopsis

This paper deals with the nature of movable singularities of solutions of Emden's equation

at which the solution becomes infinite. If m = 1 + 2/p with p > 1 an integer, then the solution becomes infinite at a given point x = c as

By the general theory of P. Painlevé on movable poles of solutions of non-linear second order differential equations this ‘pseudo-pole’ cannot actually be a pole of order p. Instead of a bona fide Laurent series at x = c we obtain a series expansion of the form

where Pn(t) is a polynomial in t of degree at most [n/(2p + 2)]. The object of this paper is to derive these series and to prove convergence for p = 2. In this case deg [P6m] is strictly equal to m. For other values of p, see Section 8, Addenda.
15.—The Measure of Non-compactness of Some Linear Integral Operators
Authors: Stuart C. A.
Published: July 2016

Synopsis

The measure of non-compactness of linear integral operators on the half-line [0, ∞) of a special type is studied. In particular, a necessary and sufficient condition is established for an operator of this type to define a compact operator from L2(0, ∞) into itself. These results are then used to discuss the spectrum of second-order differential operators. A necessary and sufficient condition for the spectrum to be discrete is established together with estimates for the distance of a point in the resolvent set from the essential spectrum.
2.—Semi-bounded Second-order Differential Operators
Authors: Eastham M. S. P.
Published: July 2016

Synopsis

Differential operators generated by the differential expression My(x) = —y″(x)+q(x)y(x) in L2(0, ∞) are considered. It is assumed that

is bounded for all x in [0, ∞) and some fixed ω > 0. The operators are shown to be bounded below and an estimate for the lower bound is obtained in terms of q(x). In the case where q(x) is LP (0, ∞) for some p ≧ 1, the results are compared with recent ones of W. N. Everitt. Some comments are made on the best-possible nature of the results.
7.—High-frequency Scattering in a Certain Stratified Medium
Authors: Boyd W. G. C.
Published: July 2016

Synopsis

The propagation of scalar waves in a certain stratified medium is studied; the field is due to a line source situated on an opaque plane boundary. The exact field can be expressed in terms of a Fourier integral involving Airy functions. By deforming the real axis, which is the contour of integration of the Fourier integral, only in the neighbourhood of the real axis, it is possible to give a simple but rigorous derivation of the asymptotic nature of the field. Two separate cases are considered: the point of observation lying in the illuminated region, when a steepest descents analysis is appropriate, or lying in the shadow region, when the asymptotic field is given by a residue series.
10.—The Area-function for Non-linear Second-order Oscillations
Authors: Atkinson F. V.
Published: July 2016

This paper is concerned with the rate of growth or decline as t → ∞ of oscillatory solutions of equations of the form

we have in mind equations which resemble in some degree the Lane-Emden-Fowler equation

where n ≧ is integral, and σ is either positive, or if negative is not too large, so that oscillatory solutions exist.
11.—High-Frequency Scattering in a Certain Stratified Medium. The Two-part Problem
Authors: Boyd W. G. C.
Published: July 2016

Synopsis

The propagation of scalar waves in a certain two-dimensional medium is considered. The incident field, which is due to the presence of a line source, is scattered by two coupled half-planes on each of which the impedance takes a constant value. The Wiener-Hopf technique is used to find a solution which is then examined asymptotically for high frequency. It is found that there is an illuminated region in which the solution is expressed in terms of geometrical optics rays, and a shadow region in which the solution is described by creeping modes. The point of impedance discontinuity may be regarded as producing secondary radiation. The nature of this secondary radiation is quite different according as the point of impedance discontinuity lies in the illuminated or shadow region of the geometrical optics field produced by the source.
13.—On a Problem of Transformations between Limit-circle and Limit-point Differential Equations
Authors: Neuman F.
Published: July 2016

1. In this article the following problem proposed by W. N. Everitt is considered: ‘Under what conditions is it possible to transform a differential equation

which is limit-circle at b1 into an equation of the form

which is limit-point at b2?'

A differential equation of the above type on [a, b) is said to be limit-circle at b iff b is a singular point and every solution . If b is singular and there exists a , then the equation is said to be limit-point at b. See [3] or [4] page 501, also for further details.
14.—The Effect of Radiation due to a Moving Source on a Vortex Sheet
Authors: Jones D. S.
Published: July 2016

Synopsis

An acoustic line source is moving parallel to the vortex sheet dividing two fluids in relative motion and the lifetime of the source is finite. The resulting phenomena are compared with those created by a fixed pulsed source. It is found that the waves produced by the source in motion are not so singular as, though qualitatively similar to, those due to a source which remains in one position but Helmholtz instability still has a dominant part to play. This statement is true whether the source be moving subsonically or supersonically so long as it has non-zero velocity.
16.—A Note on Adjoint Operators
Authors: Dijksma A. , de Snoo H. S. V.
Published: July 2016

In the Hilbertspace with (,) as inner product we consider the linear operator L with domain D(L) and the sesqui-linear form 〈, 〉 defined by

Let the symmetric operator L0 be the restriction of L to D(L0), where
17.—The Limit-point and Limit-circle Cases for Polynomials in a Differential Operator
Authors: Zettl Anton
Published: July 2016

Synopsis

This paper is concerned with the L2 classification of ordinary symmetrical differential expressions defined on a half-line [0, ∞) and obtained from taking formal polynomials of symmetric differential expression. The work generalises results in this area previously obtained by Chaudhuri, Everitt, Giertz and the author.
27.—Generating Sets for Fuchsian Groups
Authors: Chalk J. H. H. , Kelly B. G. A.
Published: July 2016

Synopsis

For a class of Fuchsian groups, which includes integral automorphs of quadratic forms and unit groups of indefinite quaternion algebras, it is shown that the geometry of a suitably chosen fundamental region leads to explicit bounds for a complete set of generators.
I.—On the Use of Stress Functions for Solving Problems in Linear Viscoelasticity Theory that Involve Moving Boundaries
Authors: Graham G. A. C.
Published: July 2016

Synopsis

The traditional method of solution to problems in linear viscoelasticity theory involves the direct application of the Laplace transform to the relevant field equations and boundary conditions. If the shape of the body under consideration or the type of boundary condition specified at a point or both vary with time then this method no longer works. In this paper we investigate the applicability of stress function solutions to this situation. It is shown that for time-dependent ablating regions a generalization of the Papkovich Neuber stress function solution of elasticity holds. As an example the stress and displacement fields are calculated for the problem of an infinite viscoelastic body with a spherical ablating stress free cavity and prescribed time-dependent stresses at infinity.
III.—The Eigenvalue Problem for Boolean Matrices
Authors: Rutherford D. E.
Published: July 2016

Synopsis

In the case of Boolean matrices a given eigenvector may have a variety of eigenvalues. These eigenvalues form a sublattice of the basic Boolean algebra and the structure of this sublattice is investigated. Likewise a given eigenvalue has a variety of eigenvectors which form a module of the Boolean vector space. The structure of this module is examined. It is also shown that if a vector has a unique eigenvalue λ, then λ satisfies the characteristic equation of the matrix.
16.—Waves over Obstacles on a Shallow Seabed
Authors: Jeffrey Alan , Tin Saw
Published: July 2016

Synopsis

In this paper we consider the effect of the passage of shallow water waves over vertical walled objects on a flat seabed. The effect of reflections at the successive walls is taken into account when determining the place of breaking of the wave. The results are obtained by the introduction of special reflection and transmission coefficients at each wall and recurrence relations are formulated connecting their values at successive walls. The effect of this is to reduce the calculation of the place of breaking of the wave over a stepped seabed to an equivalent one for a wave over a flat seabed, the result of which is well known.
20.—Some Results for Two-phase Elastic Planes
Authors: Aderogba K.
Published: July 2016

Synopsis

The first part of this paper is concerned with the elastic response of an isotropie infinite plane containing a central circular inhomogeneity to a deformation such that in the homogeneous infinite plane the associated elastic displacement is the gradient of a harmonic function. Based upon the Papkovich stress-function approach, it is shown that it is possible to represent all solutions to such problems by means of only one family of equations. The second part of the paper is concerned with the elastokinetics of two bonded dissimilar half-planes subjected to any uniformly moving body force. It is again shown that there exist simple relations between components of the elastic field in the composite plane and those of a particular field in the homogeneous infinite plane.
21.—Truncation Error in the Solution of Integral Equations
Authors: Jones D. S.
Published: July 2016

Synopsis

A way is suggested of modifying the kernel of an integral equation of the second kind so that truncation of the algebraic system of equations corresponding to the new kernel is subject to less error than that for the original kernel. A computable bound for the error is derived.
24.—The Origin of the Short-range Alpha Particles of Fission
Authors: Feather N.
Published: July 2016

Synopsis

The suggestion is made, and rendered plausible by detailed calculation, that the short-range α-particles of fission, identified by Kugler and Clarke [1], are emitted, in competition with ‘prompt’ γ-rays, from highly excited post-neutron-emission fragments of even neutron number N and, predominantly, of mass number A in the range 140 ≦ A ≧ 145. A similar process is energetically forbidden in relation to triton emission.
25.—Further calculations, being an Addendum to the paper ‘Calculations relative to the Experimental Observation of the Products of Disintegration of 5He liberated as a Light Particle in Ternary Fission’.
Authors: Feather N. , Vass D. G.
Published: July 2016

Synopsis

The analysis of the previous paper is extended, taking account of the spread of Q-values in the disintegration 5He→He+1n. Conclusions in respect of the experimental results of Cheifetz et al. (1972) remain essentially unchanged.
II.—On Certain Moduli of Rectangular Matrices
Authors: Marathe C. R.
Published: July 2016

Synopsis

In this paper are considered certain numbers called moduli associated with a rectangular matrix with complex elements. These moduli have to satisfy a set of conditions analogous to those satisfied by the modulus of a complex number. For a complex rectangular matrix A = (aij) it is shown that R(A), C(A) and | A |° are moduli of A where:

where cmin(H) and cmax(H) denote, respectively, the minimum and maximum characteristic values of the hermitian matrix H, A* being the transpose conjugate of A. Using the various properties of a modulus of a matrix and taking R(A), C(A) and | A |° as the moduli, a number of known results about the characteristic values of a matrix are obtained and extended. Relations between |A|°, |A |° R(A), p(A) and C(A), ɣ(A) are also studied. These relations provide a number of results about estimates of bounds of characteristic values of sums and products of matrices.
VI.—Complex Four-dimensional Lie Algebras
Authors: Wallace E. W.
Published: July 2016

Synopsis

Canonical forms of the four-dimensional complex Lie algebras are obtained by considering the roots of certain well-defined vectors of the algebras. A complete set of characters of the algebras is also given, enabling any given four-dimensional complex Lie algebra to be identified with one of the canonical forms.
XIII.—A Remark on Coefficients in Certain Asymptotic Expansions
Authors: Morrison E. E.
Published: July 2016

SYNOPSIS

This note explains a “numerical coincidence” observed by Professor A. C. Aitken in a recent article in these Proceedings.
XXI—The Evaluation of a Definite Integral Which Occurs in Asymptotic Partition Theory
Authors: Robertson M. M.
Published: July 2016

SYNOPSIS

In order to estimate the number of partitions of a multi-partite number, the components of which are all large and of approximately the same order of magnitude, it is necessary to evaluate for ℛ(z1) > o(l=1, 2, …,j — 1) the integral

where

for o < u < 2π min (1, |z1|−1, …, |zj−1|−1) and Asymptotic expansions are obtained for I when the z1 are small. Simple expressions give an approximate value of I when every zl is real and exact formulæ are derived when every Zl is real and rational.
XXVII.—Hydrogenated 1-Naphthoic Acid Derivatives
Authors: Campbell Neil , Kidd David , Pollock Alexander J.
Published: July 2016

The preparation of 1-benzhydrylnaphthalene, 1, 2, 3, 4-tetrahydro-1-benzoylnaphthalene, and 1-diphenylmethylenetetralin is described.

Thanks are expressed to the Department of Scientific and Industrial Research for a maintenance grant (to D. K.); to the Royal Society of London for a grant (to A. J. P.); to Miss E. W. Robertson for help with the experimental work; and to the Education Committee of Stirlingshire for permission to use laboratory facilities.
IX.—The Propagation of Thermal Stresses in Thin Metallic Rods
Authors: Sneddon I. N.
Published: July 2016

Synopsis

If the temperature in an elastic rod is not uniform and if it varies with time, dynamic thermal stresses are set up in the rod. This paper is concerned with the calculation of the distribution of temperature and stress in an elastic rod when its ends are subjected to mechanical or thermal disturbances. Simple waves in an infinite rod are first discussed and then boundary value problems for semi-infinite rods and rods of finite length. The paper concludes with an account of an approximate method of solving the equations of thermoelasticity.
X.—The Dynamic Stresses Produced in Elastic Bodies by Uneven Heating
Authors: Eason G. , Sneddon I. N.
Published: July 2016

Synopsis

The presence of a non-uniform distribution of temperature in an elastic solid gives rise to an additional term in the generalized Hooke's Law connecting the stress and strain tensors and to a term involving the time rate of change of the dilatation in the equation governing the conduction of heat in the solid. The present paper is concerned with the effects produced by these additional terms in two simple situations. In the first, the elastic solid is regarded as being of infinite extent and the distribution of temperature in the solid is produced by heat sources whose strength may vary with time. In the second, the solid is supposed to be semi-infinite and to be deformed by prescribed variations in the temperature of the bounding plane and by heat sources within itself.
XI.—The Free Commutative Entropic Logarithmetic
Authors: Minc H.
Published: July 2016

Synopsis

The commutative and entropic congruence relations determine a homomorphism on the free logarithmetic , the arithmetic of the indices of powers of the generating element of a free cyclic groupoid. A necessary and sufficient condition that two indices should be concordant (i.e. congruent in the free commutative entropic logarithmetic) is that the bifurcating trees corresponding to these indices should have the same number of free ends at each altitude. It follows that the free commutative entropic logarithmetic can be represented faithfully by index ψ-polynomials (or θ-polynomials) in one indeterminate.

In the concluding section enumeration formulæ are obtained for the number of non-concordant indices of a given altitude and for the number of indices concordant to a given index.
The English Hospice Of St. George at Sanlucar De Barrameda
Authors: Williams Michael E.
Published: July 2016

NOT FAR from Cadiz there is an English property that has remained Catholic for close on five hundred years. Its history goes back to pre-reformation days, indeed to the thirteenth century when the port of Sanlucar de Barrameda was recaptured from the Moors by the Guzman family who later became the Dukes of Medina Sidonia. Strategically Sanlucar was an important port because it was at the mouth of the Guadalquivir and as well as capturing the Seville trade it also commanded the traffic from the Mediterranean to Northern Europe and eventually it was the point of departure for ships leaving for the New World. Among the various nations using the port the English were conspicuous and their merchants were granted various privileges by the Dukes of Medina Sidonia during the fifteenth century. By the early sixteenth century there is evidence of a sizeable colony in the town; in fact the English were the largest single group of foreigners and many English names appear in the baptismal registers as both parents and godparents. At least one of them held high public office in the town. On the accession of Henry VIII to the throne of England, the situation further improved as he abandoned the neutrality of his father and allied himself with Spain against France. So it was that in 1517 a new charter of privileges for the English merchants in Sanlucar was drafted. A grant of land by the river was made so as to provide a chapel and a burial place for Englishmen. The chapel was dedicated to St. George and it was to be looked after by a confraternity. The chaplain was to be appointed by the Bishops of London, Winchester and Exeter, since it was from these dioceses that most of the merchants came. Although there have been rebuildings, this site has remained English ever since.
On the quadratic invariant of binary sextics
Authors: DUNAJSKI MACIEJ , PENROSE ROGER
Published: July 2016

We provide a geometric characterisation of binary sextics with vanishing quadratic invariant.
On the discrete approximation of eigenvalue problems with holomorphic parameter dependence
Authors: Jeggle Hansgeorg , Wendland Wolfgang
Published: July 2016

Here eigenvalue problems A(λ)u = 0 and their approximations Ai(λ)νi = 0 are studied where the densely denned closed semi-Fredholm operators A and Ai depend holomorphically on the parameter λ. Two different kinds of approximations are established. One is based on a generalisation of the spectral projection and the other on a suitable linearisation of the problem. To this end generalised eigenvectors and suitable product spaces are introduced which provide a representation formula for the principal part of A -1 and A -1, respectively, in the neighbourhood of poles. The convergence of the methods is shown in the framework of discrete convergence theory. The results generalise the corresponding results for linearly dependent A, Ai in two directions: both methods are available for arbitrary holomorphic dependence on the parameter λ, and the first method provides convergent approximations to the whole generalised eigenspace which works also in cases of linear parameter dependence when the usual method fails.
23.—The Linear Transport Equation. The Degenerate Case c = 1. II. Half-range Theory
Authors: Lekkerkerker C. G.
Published: July 2016

We use the notations and results of Part I of this paper [see 7]. Sections and formulae of that paper will be referred to as section I.4, formula (1.5), etc. In particular, T, A, TN0, K, σ, F and Λ(λ) will have the same meaning as in [7].
25.—On the Eigenfunction Expansion associated with a Singular Complex-valued Fourth-order Differential Equation
Authors: Chaudhuri Jyoti , Kumar V. Krishna
Published: July 2016

Synopsis

The direct convergence theory of eigenfunction expansions associated with boundry value problems, not necessarily self-adjoint, generated from complex-valued fourth-order symmetric ordinary differential expressions on semi-infinite intervals, is discussed. An admissible class of functions for the expansion is characterised. Also a generalisation of Stieltjes representation theorem for analytic functions discussed in [13, §§ 22.23 and 24] is obtained.
Upper and lower bounds for solutions of linear operator problems with unilateral constraints
Authors: Collins W. D.
Published: July 2016

Synopsis

Dual extremum principles characterising the solutions of problems for a positive-definite self-adjoint operator on a Hilbert space which involve unilateral constraints are formulated using a Hilbert space decomposition theorem due to Moreau. Various upper and lower bounds to these solutions are then obtained, these bounds involving the solutions to subsidiary problems with less restrictive conditions than the solution to the original problem.
4.—Comparison Theorems for a Class of Second-order Non-linear Ordinary Differential Equations
Authors: Benson Donald C.
Published: July 2016

Synopsis

Integral inequalities are used to obtain comparison theorems for a class of second-order differential equations which includes the Emden-Fowler equation, certain Liénard equations, and linear equations of the form d2y/dx2+f(x)y = 0. For these linear equations the results below imply Sturm's classical comparison theorem.
10.—Applications of the Todd-Coxeter Algorithm to Generalised Fibonacci Groups
Authors: Campbell C. M. , Robertson E. F.
Published: July 2016

Synopsis

We use programmes for the Todd-Coxeter coset enumeration algorithm and the modified Todd-Coxeter coset enumeration algorithm to investigate a class of generalised Fibonacci groups. In particular we use these techniques to discover a finite non-metacyclic Fibonacci group and to study its structure.
15.—Differentiability on the Boundary of the Solutions of Linear Elliptic Boundary Value Problems
Authors: Schappel Renate
Published: July 2016

Synopsis

The present paper is concerned with the problem of regularity of weak solutions of boundary value problems. We shall present a new method to prove differentiability on the boundary. This method was developed in our thesis [12] within the theory of abstract Sobolev spaces, introduced by Stummel [14]. Here, we shall describe it by applying it to elliptic boundary value problems. It will be seen that the advantage of this method consists in the fact that it is based on functional analysis only and therefore may be used for other types of differential equations as well.
17.—Absolutely Convergent Sturm-Liouville Expansions
Authors: Wray S. D.
Published: July 2016

Synopsis

An analogue of full-range Fourier series is introduced in the Sturm-Liouville setting and a theorem generalising Wiener's theorem for functions with absolutely convergent Fourier series is proved. The Banach algebra structure of the theory is examined. Use is made of second-order asymptotic formulae for the Sturm-Liouville eigenfunctions.
18.—Linear-quadratic Optimal Control
Authors: Coppel W. A.
Published: July 2016

Synopsis

New proofs are given for some main results in the theory of linear-quadratic optimal control.
21.—Runge's Theorem for Parabolic Equations in Two Space Variables
Authors: Colton David
Published: July 2016

Synopsis

Let u be a real valued strong solution defined in a cylindrical domain of a linear second-order parabolic equation in two space variables with entire coefficients. Then it is shown that on compact subsets of its domain of definition u can be approximated arbitrarily closely in the maximum norm by an entire solution of the parabolic equation.
6.—Rationally Convex Hulls and Potential Theory.
Authors: Basener Richard F.
Published: July 2016

Synopsis

Let S be a compact subset of the open unit disc in C. Associate to S the set

Let R(X) be the uniform algebra on X generated by the rational functions which are holomorphic near X. It is shown that the spectrum of R(X) is determined in a simple wayby the potential-theoretic properties of S. In particular, the spectrum of R(X) is X if and only if the functions harmonic near S are uniformly dense in the continuous functions on S. Similar results can be obtained for other subsets of C2 constructed from compact subsets of C.
8.—A Rule relating the Deficiency Indices of Lj to those of Lk.
Authors: Kauffman Robert M.
Published: July 2016

Synopsis

Let d(L) denote the deficiency indices (which are equal) of a formally symmetric differential expression L with real coefficients. Then it is shown that

(a)if k>1, d(Lk)−d(Lk−1) ≤ order L, and

(b)if j ≥ k > 1, d(Lj)−d(Lj−1) ≥ d(Lk)−d(Lk−1).
11.— A Left Definite Multiparameter Eigenvalue Problem in Ordinary Differential Equations
Authors: Källström A. , Sleeman B. D.
Published: July 2016

Synopsis

The main result of this paper is to establish the completeness of the eigenfunctions for the multiparameter eigenvalue problem defined by the system of ordinary differential equations

0 ≤ x, ≤ 1, r = 1, …, k, subject to the Sturm-Liouville boundary conditions

r = 1, …, k. In addition it is assumed that the coefficients ars of the spectral parameters λs, satisfy the ellipticity condition , s = 1, …, k, for all xrɛ[0, 1], r = 1, …, k, and some real k-tuple μ1, …, μk and where is the co-factor of asr in the determinant . The theory developed here contrasts with the results known when …k is assumed non-vanishing for all xrɛ[0,1].
12.—Sequences of Deficiency Indices
Authors: Read T. T.
Published: July 2016

Synopsis

An explicit characterisation is given of those sequences of positive integers which occur as the deficiency indices associated with the sequence of powers of some formally symmetric 2nth order real differential expression on [0, ∞) which is regular at 0.
15.—The Propagation of Second-order Lipschitz Discontinuities in Quasi-linear Hyperbolic Systems with Discontinuous Coefficients
Authors: Jeffrey Alan , Inan Esin
Published: July 2016

Synopsis

This paper develops the general theory of the propagation of Lipschitz discontinuities in first- and second-order partial derivatives of the initial data for a conservative quasi-linear hyperbolic system with discontinuous coefficients. After establishing that such weak discontinuities propagate along characteristics the appropriate transport equations are derived. The effect on this form of wave propagation of the strong discontinuity associatedwith the discontinuous coefficients is then studied and the transmission and reflection characteristics of the resulting waves are analysed. In conclusion, an application of this general theory is made to the propagation of plane shear waves through two different continuous hyperelastic solids to determine the transmitted and reflected waves.
18.—Asymptotic Estimates for the Lengths of the Gaps in the Essential Spectrum of Self-adjoint Differential Operators
Authors: Eastham M. S. P. , Everitt W. N.
Published: July 2016

Synopsis

The paper gives asymptotic estimates of the form

as λ→∞ for the length l(μ)of a gap, centre μ in the essential spectrum associated with second-order singular differential operators. The integer r will be shown to depend on the differentiability properties of the coefficients in the operators and, in fact, r increases with the increasing differentiability of the coefficients. The results extend to all r ≧ – 2 the long-standing ones of Hartman and Putnam [10], who dealt with r = 0, 1, 2.
19.—Asymptotic Stability for Some Functional Differential Equations
Authors: Walther Hans-Otto , Everitt W. N.
Published: July 2016

Synopsis

For a non-linear functional differential equation from population biology, a result on asymptotic stability is obtained by investigating the zeros of the characteristic equation of the linearised functional differential equation.
18.—An Inequality of C. R. Putnam involving a Dirichlet Functional
Authors: Sears D. B. , Wray S. D.
Published: July 2016

SYNOPSIS

An inequality of C. R. Putnam involving a Dirichlet functional in the singular theory of the Sturm-Liouville differential equation is generalised. The corresponding result for the Sturm-Liouville theory over a compact interval is established and this is extended to the singular case by means of a Tauberian argument.
Solutions d'une classe de systèmes symétriques
Authors: Benachour Said
Published: July 2016

Synopsis

Let Ω be an open set of ℝm; we consider some symmetric singular systems:

We study the existence and uniqueness of a solution of the problem Lu = f plus boundary conditions when the characteristic matrix is of constant rank only on the boundary.
On a class of Dubreil-Jacotin regular semigroups and a construction of Yamada
Authors: Blyth T. S.
Published: July 2016

Synopsis

In the publication [2] we obtained some structure theorems for certain Dubreil-Jacotin regular semigroups. A crucial observation in the course of investigating these types of ordered regular semigroups was that the (ordered) band of idempotents was normal. This is characteristic of a class of semigroups studied by Yamada [5] and called generalised inverse semigroups. Here we specialise a construction of Yamada to obtain a structure theorem that complements those in [2], The important feature of the present approach is the part played by the greatest elements that exist in each of the components in the semilattice decompositions involved.
Limit-point criteria for polynomials in a non-oscillatory expression
Authors: Read T. T.
Published: July 2016

Synopsis

Several criteria are given for some or all polynomials in an expression M(y) = –(py′)′+qy which is non-oscillatory on (0, ∞) to be limit-point. One of these states that if M is non-oscillatory and , then every polynomial in M is limit-point.
The non-monotonicity of solutions in swirling flow
Authors: McLeod J. B. , Parter S. V.
Published: July 2016

Synopsis

This paper studies the boundary-value problem arising from the behaviour of a fluid occupying the region 0 ≦ x ≧ 1 between two rotating discs, rotating about a common axis perpendicular to their planes, when the discs, are rotating in the same sense with speeds 0 ≦ Ω0<Ω1. The equations which describe the axially symmetric similarity solutions of this problem are

with the boundary conditions

where ε = v/2Ω1 and v is the kinematic viscosity.

The major result is: There is an ε0 such that for 0<ε≦ε0 there does not exist a solution 〈H(x,ε), G(x,ε)〉 with G′(x,ε)≧0.
The limit-point, limit-circle nature of rapidly oscillating potentials
Authors: Atkinson F. V. , Eastham M. S. P. , McLeod J. B.
Published: July 2016

Synopsis

The Weyl limit-point, limit-circle nature of the equation y″(x)–q(x)y(x) = 0(0≦ x< ∞) is analysed When q(x) has the form q(x) = xαp(xβ), where α and β are positive constants and p(t) is a continuous periodic function of t.
Second-order differential expressions whose squares are limit-3
Authors: Eastham M. S. P. , Zettl A.
Published: July 2016

Synopsis

Second-order expressions –d2/dx2+qwhose squares are of limit-3 type are constructed. The construction is simpler than others in the literature. The functionqcan be chosen to be infinitely differentiable and decreasing and to satisfyq(x)≧ –x2(logx)2+ε(x→∞) for a given ε>0.
Analysis of an adaptive third-order relay control system using non-linear switching surface theory
Authors: Zinober A. S. I.
Published: July 2016

Synopsis

Using a non-linear transformation of the state space, the dynamic behaviour of an adaptive thirdorder relay system is analysed. The control strategy yields state paths close to the time-optimal trajectories.
Strong limit-point and Dirichlet criteria for ordinary differential expressions of order 2n
Authors: Hinton Don
Published: July 2016

Synopsis

Conditions are given which ensure that a weighted 2nth order ordinary differential equation on a half-line satisfy the strong limit-point and Dirichlet conditions. Perturbation terms are permitted which either satisfy certain pointwise bounds or integral type bounds. Not all of the coefficients of the equation are required to be non-negative.
The endomorphism near-ring of an infinite dihedral group
Authors: Meldrum J. D. P.
Published: July 2016

Synopsis

In this paper, we determine the semigroup End D of endomorphisms of the infinite dihedral group D, and give a multiplication table for it. We determine the additive structure of the near-rings E(D) generated by the endomorphisms of D, A(D) generated by the automorphisms of D and I(D) generated by the inner automorphisms of D, and determine their radicals and all their maximal right ideals.
On semilinear elliptic boundary value problems in unbounded domains
Authors: Bose Deb Kumar
Published: July 2016

Synopsis

We consider the possibility of solving semilinear elliptic boundary value problems in unbounded domains. We first treat the case when the non-linear terms are independent of terms involving gradients. Using a monotone iteration scheme, we show that the existence of a weak subsolution v and a weak supersolution w ≧ v, implies the existence of a weak solution u, and v ≦ u ≦ w. We also state conditions which guarantee the existence of a solution when only a subsolution is known to exist. Next, we suppose the non-linear terms can depend on gradient terms. Using a method developed in [4], based on perturbation theory of maximal monotone operators, we prove the existence of a H2(Ω) solution lying between a given H2(Ω) subsolution v and a given H2(Ω) supersolution w ≧ v.
Toxocara Canis infestation with encephalitis
Authors: Mikhael Nadia Z. , Montpetit Vital J.A. , Orizaga Manuel , Rowsell Harry C. , Richard Michael T.
Published: July 2016

The syndrome “visceral larva migrans” produced by migration of nematode larvae (commonly Toxocara species) in the extraintestinal tissues in unusual hosts, including man (particularly children of dirt eating age), with the production of reactive granulomatous lesions, was first described in 1952. About 200 cases have been reported since. Well documented cases are rare due to the difficulty in histological verification. Three cases have been reported in Canada but these were not verified. We believe the present report is the first verified case of visceral larva migrans in Canada and the fifth case of human cerebral involvement by Toxocara in the English literature. It is hoped that this report will emphasize the need for further research into the role of host versus parasite and will underline the potential danger, albeit remote, of household pets to children. Indeed, prevention is the only line of attack, as there is no effective drug against the migrating larvae of Toxocara.
Use of Diazepam in Carotid Angiography
Authors: Jain K.K.
Published: July 2016

Diazepam (Valium) has been used in 100 patients as a sedative for cerebral angiography. No serious complications were encountered. The advantages of this drug as well as the precautions for its use are listed.
HL-A Frequencies in Patients with Multiple Sclerosis
Authors: Paty D.W. , Mervart H. , Campling B. , Rand C.G. , Stiller C.R.
Published: July 2016

The histocompatibility antigens (HL-A) have been determined in 100 multiple sclerosis (M.S.) patients and 143 randomly selected controls. In the M.S. group there was a statistically significant increase in the frequency of HL-A 7 and W 18 with an insignificant increase in HL-A 3. The variance from normal HL-A patterns in the M.S. population may play some role in establishing the substrate for this disease. Studies in experimental animals have shown that susceptibility to autoimmune disease and to virus infection is linked to the major histocompatibility locus. This has interesting implications for both the “slow virus” and the “autoimmune” theories of the etiology of multiple sclerosis.
Relative Prognostic Significance of Vasospasm Following Subarachnoid Hemorrhage
Authors: Weir Bryce , Rothberg Charles , Grace Michael , Davis Faye
Published: July 2016

A retrospective analysis of 274 patients with intracranial aneurysms, diagnosed either angiographically or at autopsy between 1968 and 1973 at the University of Alberta, was carried out. One hundred and forty-six patients had intracranial clipping of the aneurysm. Clinical and radiologic data were abstracted from the chart and the angiographic studies. Probability of survival curves were constructed. Associations between various clinical factors and survival at two months were demonstrated. The most important prognostic factors were the clinical grade at angiography or surgery, followed by the presence of preoperative spasm, hematoma or focal edema, elevated blood pressure on admission, time interval from hemorrhage to surgery and age. The data lends some support to the policy of operating on patients in good neurological condition, even if their pre-operative angiogram shows spasm.
The Nature of Primary Vocal Tremor
Authors: Hachinski V.C. , Thomsen I.V. , Buch N.H.
Published: July 2016

Three elderly women with marked progressive voice tremor, without other neurological symptoms, and negative family histories were investigated.

All had a 4-5 Hz respiratory tremor in expiration and, to a lesser degree, in inspiration; and all had vocal tremulousness synchronous with their respiratory irregularity. Articulation of phonemes was normal. In two cases the neurological examination was otherwise normal; in the third case there was a minimal 7½ Hz tremor in the left thumb and index finger.

Simultaneous speech and vocal air pressure recordings, as well as cinematographic studies of the vocal apparatus and diaphragm were carried out.

It is suggested that these cases represent primarily an action tremor of respiration, that they belong in the spectrum of essential tremor, and hence may be amenable to treatment with propranolol.
Laying the Ghost of ‘Muscles Versus Movements’
Authors: Phillips C.G.
Published: July 2016

It is certainly a mark of great wisdom to the organizer of this meeting to have chosen Charles Phillips as our Guest Lecturer for this meeting.

Whilst I am grateful for this opportunity of saying a few words of introduction, I am sure it can be said of Charles Phillips that very little introduction is necessary, since we meet him at the most important meetings in Neurophysiology around the world. Any meeting on the motor system, of course, is not complete without the participation of Dr. Phillips. However, some of the less old-timers than myself, and the students, might be interested in the beginnings of Dr. Phillips’ career in neurophysiology and to know that he was first a clinical neurologist beginning his medical studies in Oxford just before the war, where he came under the influence of Sir Charles Sherrington with Jack Eccles as his tutor.
Clinical Problems of Motor System Disease
Authors: Richardson J.C.
Published: July 2016

This prologue to a symposium of research studies on motor mechanisms is a general commentary by a clinical neurologist. The vast extent and intricacy of modern basic neurological scientific knowledge presents a rather bewildering challenge to reasonable clinical application. In some degree this is being handled by complex and diverse neurological subspecialization. It is recalled that many past advances in the knowledge of neurological disease were achieved by a series of alternating and supporting bedside and laboratory observations and studies.

The varied disorders of movement and muscle tone which signal disordered motor mechanisms will continue to demand explanation and will keep the human model in a leading research position. Clinical and laboratory research leading to part discoveries of mechanisms of disease is sometimes productive of dramatic new means of therapy. The story of Wilson’s disease is briefly reviewed in that context. Some recent studies on hypoxic myoclonus are described with the evidence of a serotonin defect and useful related therapy.