bluuebunny/crossref_metadata_2025_split huggingface dataset:
8.—On a Class of Series Expansions in the Theory of Emden's Equation
Authors: Hille Einar 
Published: July 2016

SynopsisThis paper deals with the nature of movable singularities of solutions of Emden's equationat 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 asBy 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 formwhere 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

SynopsisThe 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

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

SynopsisThe 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 formwe have in mind equations which resemble in some degree the Lane-Emden-Fowler equationwhere 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

SynopsisThe 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 equationwhich is limit-circle at b1 into an equation of the formwhich 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

SynopsisAn 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 byLet 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

SynopsisThis 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

SynopsisFor 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

SynopsisThe 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

SynopsisIn 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

SynopsisIn 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

SynopsisThe 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.
24.—The Origin of the Short-range Alpha Particles of Fission
Authors: Feather N. 
Published: July 2016

SynopsisThe 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

SynopsisThe 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

SynopsisIn 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.
XXI—The Evaluation of a Definite Integral Which Occurs in Asymptotic Partition Theory
Authors: Robertson M. M. 
Published: July 2016

SYNOPSISIn 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 integralwherefor 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

SUMMARYThe 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

SynopsisIf 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

SynopsisThe 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

SynopsisThe 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.
25.—On the Eigenfunction Expansion associated with a Singular Complex-valued Fourth-order Differential Equation
Authors: Chaudhuri Jyoti , Kumar V. Krishna 
Published: July 2016

SynopsisThe 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

SynopsisDual 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

SynopsisIntegral 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

SynopsisWe 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

SynopsisThe 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

SynopsisAn 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.
21.—Runge's Theorem for Parabolic Equations in Two Space Variables
Authors: Colton David 
Published: July 2016

SynopsisLet 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

SynopsisLet S be a compact subset of the open unit disc in C. Associate to S the setLet 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

SynopsisLet 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

SynopsisThe 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 equations0 ≤ x, ≤ 1, r = 1, …, k, subject to the Sturm-Liouville boundary conditionsr = 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].
15.—The Propagation of Second-order Lipschitz Discontinuities in Quasi-linear Hyperbolic Systems with Discontinuous Coefficients
Authors: Jeffrey Alan , Inan Esin 
Published: July 2016

SynopsisThis 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

SynopsisThe paper gives asymptotic estimates of the formas 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.
18.—An Inequality of C. R. Putnam involving a Dirichlet Functional
Authors: Sears D. B. , Wray S. D. 
Published: July 2016

SYNOPSISAn 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.
On a class of Dubreil-Jacotin regular semigroups and a construction of Yamada
Authors: Blyth T. S. 
Published: July 2016

SynopsisIn 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.