bluuebunny/crossref_metadata_2025 huggingface dataset:
Adiabatic second-order energy derivatives in quantum mechanics
Authors: Brown W. Byers , Higgins H. C. Longuet
Published: November 2008

The general equation for the adiabatic second-order derivative of the energy En of an eigenstate with respect to parameters λ and λ′ occurring in the Hamiltonian ℋ is

The applications of this equation to molecules (λ, λ′ = nuclear position coordinates) and to enclosed assemblies of interacting particles (λ = λ′ = volume) are discussed, and the classical analogue of the equation for a micro-canonical ensemble is derived.
On certain types of derivations
Authors: Patterson E. M.
Published: November 2008

Let R be a ring, not necessarily associative. A derivation of R is a mapping D:R → R, not identically zero, such that

for all x, y ε R. If R is a linear algebra over a field F, then D also satisfies

for all x ε R and α ε F.
Dilatation properties of regular local rings
Authors: Northcott D. G.
Published: November 2008

The results and methods of algebraic geometry, when analysed in terms of modern algebra, have revealed on several occasions algebraic principles of surprising generality. Recently it has become apparent that the geometric theory of infinitely near points has, as it were, an abstract form which forms part of the ideal theory of commutative rings, but there are many details which have yet to be worked out. Roughly speaking, one may say that what corresponds to the theory of the sequence of points on a curve branch is now known in some detail, and forms a substantial addition to our knowledge of the properties of one-dimensional local rings†; but the construction of an abstract theory similarly related to the theory of neighbourhoods in n-dimensional projective space can hardly be said to have been started. A number of necessary preliminary steps were taken by the author in (3)—in the process of providing algebraic foundations for certain applications of dilatation theory—and later some applications were made to 2-dimensional problems. However, the present paper should be regarded as an attempt to initiate a dilatation theory of regular local rings to run parallel to the general theory of infinitely near points in n-dimensional space.
Self-consistent fields without exchange for Pr3+ and Tm3+
Authors: Ridley E. Cicely
Published: November 2008

Self-consistent fields without exchange have been calculated for the ground state of Pr3+ and Tm3+. The spin-orbit coupling constant ζ(4f) is found to have the values 785 and 2742 cm−1 in Pr3+ and Tm3+, respectively. The corresponding values of are 29·4 and 77·5 Å−3. Values of the Slater integrals F2(4f, 4f), F4(4f, 4f) and F6(4f, 4f) are also given for each structure.
The propagation of shock waves of constant strength
Authors: Mackie A. G. , Weir D. G.
Published: November 2008

Conditions are examined under which plane shook waves of constant strength can propagate through a gas in a given homentropic motion. Since the entropy change across the shock is taken to be constant homentropic flow exists behind the shock also. The motion behind the shock is determined by solving a Cauchy problem with data given on the back of the shock. The theory is illustrated by two examples, one of which generalizes a result obtained by Copson.
Generalizations of Wald's fundamental identity of sequential analysis to Markov chains
Authors: Tweedie M. C. K.
Published: November 2008

An extension of Bellman's analogue of Wald's identity is proved for Markov processes with a finite number of states, in which (a) the diagonal matrices of the state functions and the transition matrices may be heterogeneous but are so related that the product of the two matrices which apply at any stage has at least one modal vector in common with all of the corresponding products which apply at other stages, and (b) the probability distribution of the duration of any one realization of the process satisfies certain conditions. The condition (a) is called equimodality. The conditions for (b) are satisfied when there is asymptotically a sufficiently abrupt exponential upper bound to the probability distribution of the duration, and extends simply to validate analogues involving latent roots other than the one which appears in Bellman's formula. A certain generalization of Wald's stopping rule produces a bound of this kind.
The Stokes phenomenon and certain nth-order differential equations
Authors: Heading J.
Published: November 2008

In papers I(1) and II(2), the author has investigated the Stokes phenomenon associated with the arbitrary constants that occur in the asymptotic expressions of solutions of the nth-order differential equation

and has provided rules for tracing a given linear combination of n such asymptotic expressions across certain defined lines, designated as Stokes lines, anti-Stokes lines and branch-cuts. The object of these investigations is to apply the results to the approximate solution of more complicated linear differential equations of the nth order, and to this end it is an advantage to develop the theory using matrix techniques. In the present paper, the equations are recast into a special matrix form, and it is shown how the results obtained in II may be used in a systematic manner.
Single-centre Expansions for the Hydrogen Molecular Ion
Authors: Cohen M. , Coulson C. A.
Published: November 2008

It was shown by Baber and Hassé(1) that the Schrödinger equation for the single electron in H+2 with fixed nuclei is separable in spheroidal coordinates. The separated equations were integrated by Bates, Ledsham and Stewart (2) and their solutions may be regarded as exact. However, several attempts have been made to derive approximate solutions for this, which is the simplest of all molecular problems, generally with the hope that it might be possible to extend the analysis to larger diatomic systems. Recently a single-centre expansion has been much favoured, since it avoids the difficult two-centre integrals; but unfortunately this advantage has been offset by the fact that the rate of convergence for the lowest states has been rather slow since the deviations of diatomic H+2 from the united-atom H+e are serious. But successful approximations to some excited states suggest that a single-centre approach may still be useful, provided that the trial functions are suitably chosen.
Bending of Curvilinear and Rectilinear Polygonal Plates Symmetrically Loaded Over a Concentric Circle
Authors: Bassali W. A. , Hanna N. O. M.
Published: November 2008

Complex variable methods are applied to obtain exact solutions for the complex potentials and deflexions of thin isotropic slabs bounded by regular curvilinear polygonal contours withnsides and subject to symmetrical loading distributed over a concentric circle. The supported boundary is either clamped or has equal boundary cross-couples. The plates taken in thez-plane are conformally mapped on the unit circle in the ζ-plane by the mapping function. Polynomial approximations to the Schwarz—Christoffel transformations are then used to discuss the bending of clamped and simply supported rectilinear plates symmetrically loaded over a concentric circle or acted upon by a central point load.
Proof of Poisson's Formula
Authors: Åkerberg Bengt
Published: November 2008

Let f(z)=u(x, y)+iv(x, y) be regular in a domain containing the unit circle C. If where r < 1, then is regular in |z| ≤ 1 and we have and taking real parts and imaginary parts we obtain Poisson's formulae for the harmonic functions u and v.
The potential of a circular current
Authors: Pringle G. E.
Published: November 2008

The 3-dimensional potential due to a charged circle can be derived in various ways which lead to the same expression in terms of the toroidal coordinate σ through Legendre functions ofs(= cosh σ). Applying the same methods to the magnetic potential of a circular current we find various results: (a) Legendre's equation yields no solution (§4·1). (b) The appropriate transformation from cylindrical to toroidal coordinates gives a Fourier series (equation (48)) the coefficientsof which, though not Legendre functions, are related to them by the identity (§4·2)By this method we also derive expressions in elliptic integrals (equations (56), (58)). (c) Direct integration of the solid angle gives an alternative expression in elliptic integrals (equation (59)).
Vertex generated waves outside metallic wedges
Authors: Williams W. E. , Rosenhead L.
Published: November 2008

A study is made of the waves generated by a magnetic line source placed at the vertex of a wedge of high conductivity and arbitrary angle. The boundary-value problem is reduced to the solution of a difference equation and an exact solution obtained. The method is also applied to the case of dielectric coated wedges where the surface reactance and resistance are arbitrary, and the propagation of surface waves along such surfaces is considered briefly. The forms of the solution for large and small values of the surface impedance are obtained and show complete agreement with the known results available for a right-angled wedge and a plane.
Waring's problem in algebraic number fields
Authors: Birch B. J.
Published: November 2008

Let K be a finite algebraic number field, of degree R. Then those integers of K which may be expressed as a sum of dth powers generate a subring JK, d of the integers of K (JK, d need not be an ideal of K, as the simplest example K = Q(i), d = 2 shows. JK, d is an order, it in fact contains all integer multiples of d!; it also contains all rational integers). Siegel(12) showed that every sufficiently large totally positive integer of JK, d is the sum of at most (2d−1 + R) Rd totally positive dth powers; and he conjectured that the number of dth powers necessary should be independent of the field K—for instance, he had proved(11) that five squares are enough for every K. In this paper, we will show that, as far as the analytic part of the argument is concerned, Siegel's conjecture is correct. I have not been able to deal properly with the problem of proving that the singular series is positive; but since Siegel wrote, a good deal of extra information about singular series has been obtained, in particular by Stemmler(14) and Gray(4). The most spectacular consequence of all this is that if p is prime, then every large enough totally positive integer of JK, p is a sum of (2p + 1) totally positive pth. powers.
On manifolds containing a submanifold whose complement is contractible
Authors: Kelly G. M.
Published: November 2008

The problem discussed here arose in the course of some reflections on the critical point theory of Lusternik and Schnirelmann (4). In (4) it is shown how it is possible to associate, with a suitably differentifiable real-valued function f defined on a compact manifold M, a set of real numbers λ1 ≤ λ2 ≤ … λc, which are critical levels of f and which in certain respects are analogous to, and indeed generalizations of, the eigenvalues of a quadratic form. The number c depends on M and is called the category of M. If Rn is Euclidean n-space, Sn the unit sphere of Rn+1, and Pn the real projective n-space obtained from Sn by identifying opposite points, then a quadratic form φ in the (n + 1) coordinates of Rn+1 defines a real function on Sn and, by passage to the quotient, on Pn. Pn has category n + 1, and the numbers λ in this case are just the eigenvalues of the quadratic form.
Two classical lattice point problems
Authors: Gangadharan K. S. , Ingham A. E.
Published: November 2008

Let r(n) be the number of representations of n as a sum of two squares, d(n) the number of divisors of n, and

where γ is Euler's constant. Then P(x) is the error term in the problem of the lattice points of the circle, and Δ(x) the error term in Dirichlet's divisor problem, or the problem of the lattice points of the rectangular hyperbola.
Error bounds for the Liouville–Green (or WKB) approximation
Authors: Olver F. W. J.
Published: November 2008

Error bounds are derived and examined for approximate solutions in terms of elementary functions of the differential equations

in which u is a positive parameter, the functions f and p are free from singularities and p does not vanish. Bounds are also obtained for the remainder terms in the asymptotic expansions of the solutions in descending powers of u. The variable x ranges over a real interval, finite or infinite or over a region of the complex plane, bounded or unbounded.

Applications are made to parabolic cylinder functions of large orders, and modified Bessel functions of large orders.
A method of evaluating the complex zeros of polynomials using polar coordinates
Authors: Bodmer W. F.
Published: November 2008

Methods for finding complex roots numerically may broadly be classified into the following three types.

(a) Direct methods. These require no knowledge of an initial approximation. The best known are Graeffe's ‘root-squaring’ method (see, for example, Whittaker and Robinson(17)) and Bernoulli's method extended to complex roots (Aitken(1)). Fry (8) gives an interesting method for finding the characteristic roots of a matrix, originally due to Duncan and Collar (5), (6), which contains the above-mentioned methods as special cases.
The exact enumeration of self-avoiding walks on a lattice
Authors: Martin J. L.
Published: November 2008

No adequate theory of self-avoiding walks on a regular lattice exists, and the only effective approach at present is numerical in character. An electronic computer has been used to enumerate prescribed sets of walks on the usual regular lattices; the method is described and some results are given.
Proofs of some ‘binomial’ identities by means of MacMahon's ‘Master Theorem’
Authors: Good I. J.
Published: November 2008

MacMahon (5) used his “Master Theorem”, which is stated in the preceding note (4), in order to obtain an elegant proof of Dixon's identity
The effect of queue discipline on waiting time variance
Authors: Kingman J. F. C.
Published: November 2008

It is usual in the theory of queues to assume that customers are served in the order of their arrival. In some applications (e.g. telephone engineering), however, other forms of queue discipline are more realistic. The precise effect of any such change on the waiting-time distribution of a customer will depend on the procedure envisaged (random service, “last come, first served”, etc.), but it is possible to make certain general statements. Thus it is well known that, under certain conditions, the mean is independent of the queue discipline. The purpose of the present note is to consider the variance of waiting time, and we shall prove that this is a minimum when the customers are served in order of arrival. Thus this is, in a sense, the “fairest” queue discipline. This does not, of course, mean that other procedures may not be justified when different criteria are taken into account.
Wreath Powers and Characteristically Simple Groups
Authors: Hall P.
Published: November 2008

A group G is called characteristically simple if it has no characteristic subgroups other than itself and the unit subgroup. For brevity, we call such groups -groups; we also use to denote the class of all characteristically simple groups.
Sufficient Statistics and Orthogonal Parameters
Authors: Mitchell Ann F. S.
Published: November 2008

Let be, for a set of n real continuous parameters the probability density function of a random variable x with respect to a σ-finite measure μ on a σ-algebra of subsets of the sample space . If x; is a continuous random variable, μ will be Lebesgue measure on the Borel sets of a Euclidean sample space and, if x is discrete, μ will be counting measure on the class of all sets of a countable sample space. The parameters αi are said to be orthogonal (Jeffreys (3), pp. 158,184) if .
Thermo-elastic Rayleigh waves in a prestressed medium
Authors: Flavin J. N.
Published: November 2008

We consider the propagation of thermo-elastic Rayleigh waves in a half space, which has been subjected to large uniform extensions, at constant temperature, in three mutually perpendicular directions, one being perpendicular to the surface, and the extension ratios in the other two directions being equal. Rayleigh waves, assuming (a) isentropic conditions and (b) isothermal conditions, are studied, and the relation of the approximation (a) to the exact thermo-elastic theory is considered.
The propagator for a particle in the presence of a one-dimensional square-well potential
Authors: Beeby J. L.
Published: November 2008

The propagator for a particle in the presence of a one-dimensional square-well potential is calculated. The related t-function is written down in Fourier transform. The analytic properties of the propagator and t-function are discussed.
Groups in which normality is a transitive relation
Authors: Robinson Derek J. S.
Published: November 2008

A group is said to have theproperty Tor to be aT-groupif every subnormal subgroup is normal. Thus the class of T-groups is just the class of all groups in which normality is a transitive relation. Finite T-groups have been studied by Best and Taussky (l), Gaschütz (4) and Zacher (11). Gaschütz has shown that ifGis a finite soluble T-group andG/Lis the unique maximal nilpotent quotient group ofG, thenG/Lis Abelian or Hamiltonian andLis an Abelian group of odd order prime to |G:L| ((4), Satz 1). Our aim is to study infinite T-groups and more especially infinite soluble T-groups with a view to extending Gaschütz's results. One of the simplest results on soluble T-groups is THEOREM 2.3.1.Every soluble T-group is metabelian.
On connexion, invariance and stability in certain flows
Authors: Desbrow D.
Published: November 2008

1. Suppose that f is a homeomorphism of the Euclidean plane E2 onto itself. The set M ⊂ E2 is said to be invariant if f(M) = M and minimal if it is non-void, closed, invariant and irreducible with respect to these properties. In general, invariant and minimal sets in E2 can have a finite or infinite number of components.
On the numerical evaluation of some definite integrals
Authors: Barrett W.
Published: November 2008

An account is given of the application of Gaussian quadrature formulae to the numerical evaluation of certain definite integrals appearing in a recent paper by Craggs (2). Suitable changes of variable are used to effect a substantial reduction in the order of formula required. In comparing methods, use is made of an expression for the remainder as a contour integral, whose value is estimated with the aid of certain asymptotic formulae.
Symmetry properties of Harrison space-times
Authors: Collinson C. D.
Published: November 2008

All possible symmetries of the non-degenerate Harrison space-times are found. It is shown that if such a space-time admits an r-parameter group of motions, with r > 1, then it must admit a 2-parameter Abelian subgroup.
Studies in harmonic analysis
Authors: Varopoulos N. Th.
Published: November 2008

In this paper we shall be mainly concerned with the following three apparently widely differing questions.

(a) What are the possible group topologies on an Abelian group that have a given, fixed continuous character group?

In developing our theory, we are very strongly motivated by the duality theory of linear topological spaces and in particular by Mackey's theorem of that theory. This important result gives a complete characterization of all locally convex topologies on a linear space that have a given, fixed, separating dual space. The analogue of Mackey's theorem for groups, together with related results, is examined in sections 1 and 2 of part 2 of the paper.

(b) What are the properties of topological groups that are denumerable inductive limits of locally compact groups? (See section 1 of part 1 of the paper for definitions.)

Our aim here is to extend results known for locally compact groups to this larger class of groups. The topological study of these groups is carried out in section 3 of part 1 of the paper and the really deep results about their characters are proved in section 5 of part 3 of the paper, as applications of the theory developed in that part of the paper, which is a type of harmonic analysis for these groups.

(c) What are the properties of certain algebras of measures of a locally compact group G, that strictly contain L1(G), and share most of the pleasing properties of L1(G), that is, they do not have any of the pathological features of the full measure algebra M(G) such as the Wiener–Pitt phenomenon or asymmetry?
On the one-region two-centre expansions of a generalized Coulomb field in spherical polar and parabolic coordinates
Authors: Edelstein L. A.
Published: November 2008

1. Introduction and summary. There has been considerable interest in the problem of expanding a Coulomb potential l/r12 in spherical polar coordinates defined relative to two centres A(0, 0, − d) and B(0, 0, + d) where 2d = D = rAB.
The stability of a system of queues in series
Authors: Loynes R. M.
Published: November 2008

1. Introduction. The stability of a system of queues in series was investigated in (1) but the results obtained were not quite complete, for the situation in which all component queues are substable and at least one is critical was scarcely touched upon. The only reference to this situation, in fact, was an example on page 513 showing that necessary and sufficient conditions are not obvious. (The ideas and terminology of (1) will be used freely throughout this paper.)
Covering a sphere with congruent great-circle arcs
Authors: Conway J. H. , Croft H. T.
Published: November 2008

1. Mikusiński ((2), Problem 84) asks: can we express the surface of a sphere (in Euclidean 3-space) as the union of non-overlapping congruent great-circle arcs? We give a partial answer to this question.
The torsional rigidity of tubes of constant normal thickness
Authors: Westbrook D. R.
Published: November 2008

The purpose of this note is to obtain an improved approximation to the torsional rigidity of tubes of constant normal thickness which may be applied when Bredt's first approximation(l) is not sufficiently accurate. (The reference to Bredt's approximation is taken from(2), page 299.)
In-groups, coverings, and imbeddings
Authors: Neuwirth Lee
Published: November 2008

Introduction. Let Mn+1 denote a closed, orientable, combinatorial (n + 1)-manifold. Let Kn denote an n-dimensional subcomplex of Mn+1. If Mn+1 – Kn is connected, then by ‘cutting’ Mn+1 along Kn we define in a natural way a fundaments domain of a certain regular covering of Mn+1. In case Kn is a spine ((5)) of Mn+1 the any covering space of Mn+1 may be constructed. This construction is quite precis and is combinatorial in spirit.
On the degree of smoothness and on singularities in distributions of statistical functions
Authors: Mulholland H. P.
Published: November 2008

Introduction. When the distribution of a statistical function could not be determined exactly, a suitable form for an approximate formula has sometimes been found by the use of information about the order of smoothness of the distribution and about any singularities it might have (see, for example, Geary ((4))). The object of the present paper is to develop a general theory that will provide information of this kind. For the singularity in the distribution of a function at an end of its range Hotelling ((8)) obtained formulae for cases where the original distribution is uniform over a curved space. However, some special sampling distributions are known to have singularities in the interior of the range, for example, the distribution of √ b1 (see (4)) and those of various serial correlation coefficients (see (6)). In view of this I shall deal below with orders of smoothness and with singularities over the whole range. For simplicity I shall assume that the original distribution has a fairly high order of smoothness: attention will therefore be concentrated on the singularities (if any) that arise from the form of the statistical function.
Wald's identity and absorption probabilities for two-dimensional random walks
Authors: Barnett V. D.
Published: November 2008

Suppose a particle executes a random walk on a two-dimensional square lattice, starting at the origin. The position of the particle after n steps of the walk is Xn = (Xl, n, X2n), where

and we will assume that the Yi are independent bivariate discrete random variables with common moment generating function (m.g.f.)

where a, b, c and d are non-negative. We assume further that (i) pi, j is non-zero for some finite positive and negative i, and some finite positive and negative j (− a ≤ i ≤ b, − c ≤ jd), such values of i and j including – a, b and – c, d, respectively, whenever a, b, c or d is finite, and (ii) the double series defining Φ(α, β) is convergent at least in some finite region D, of the real (α, β) plane, which includes the origin.
Magneto-thermo-elastic plane waves
Authors: Purushothama C. M.
Published: November 2008

The combined effects of uniform thermal and magnetic fields on the propagation of plane waves in a homogeneous, initially unstressed, electrically conducting elastic medium have been investigated.

When the magnetic field is parallel to the direction of wave propagation, the compression wave is purely thermo-elastic and the shear wave is purely magneto-elastic in nature. For a transverse magnetic field, the shear waves remain elastic whereas the compression wave assumes magneto-thermo-elastic character due to the coupling of all the three fields—mechanical, magnetic and thermal. In the general case, the waves polarized in the plane of the direction of wave propagation and the magnetic field are not only coupled but are also influenced by the thermal field, once again exhibiting the coupling of the three fields. The shear wave polarized transverse to the plane retains its magneto-elastic character.

Notation.

Hi = primary magnetic field components,

ht = induced magnetic field components,

To = initial thermal field,

θ = induced thermal field,

C = compression wave velocity.

S = shear wave velocity,

ui = displacement components,

cv = specific heat at constant volume,

k = thermal conductivity,

η = magnetic diffusivity,

μe = magnetic permeability,

λ, μ = Lamé's constants,

β = ratio of coefficient of volume expansion to isothermal compressibility.
Spectral synthesis on spheres
Authors: Varopoulos N. Th.
Published: November 2008

Summary of results. In this paper we shall study some properties of the algebra A(Rn) for n ≥ 1 relative to the unit sphere
The expansion of Lamé functions into series of associated Legendre functions of the second kind
Authors: Sleeman B. D.
Published: November 2008

Introduction. In this paper, a study is made of the solutions of Lamé's differential equation as series of associated Legendre functions. The particular feature studied is the representation of the second solution corresponding to the case when the first solution is a Lamé polynomial.
Rotating fluid masses in general relativity. II
Authors: Boyer R. H.
Published: November 2008

We describe some properties of a stationary, isolated, axially symmetric, rotating body of perfect fluid, according to general relativity. We first specialize to the case of constant specific entropy and constant angular velocity. The latter condition is equivalent to rigidity in the Born sense; both conditions are consequences of a simple variational principle. The hydrodynamic equations can then be integrated completely. Analogous first integrals are given also for the case of differential rotation. No use is made of the full field equations.
Note on the scattering of water waves by a vertical barrier
Authors: Williams W. E.
Published: November 2008

Introduction. In this note an alternative approach is presented to the problem of the scattering of small amplitude two-dimensional water waves by a fixed barrier, one edge of the barrier lying in the free surface of the water. This problem was first solved by Ursell ((1)) and generalizations of the problem have been considered by John ((2)) and Lewin ((3)).
General study of a heat transmission problem of a channel-gas flow with Neumann-type thermal boundary conditions
Authors: Tyagi V. P.
Published: November 2008

In this paper a fundamental problem of thermal transport in a channel-gas flow has been studied. The most general solution in terms of integral formulae has been obtained when the duct is bounded by an arbitrary closed contour, the prescribed normal temperature gradient at the wall varies arbitrarily along the periphery, and the specified additional heat generation varies in an arbitrary manner from point to point in a cross-section. The power series solution is also discussed, and is given in a separate section. One of the objects of the present study is to look forward to the qualitative as well as quantitative effects of viscous dissipation and work of compression. Several results and all the graphs of Tao ((13)) have also been corrected.
On subsets of a partially ordered set
Authors: McCartan S. D.
Published: November 2008

In this paper we discuss the structure of a partially ordered set X by studying in detail certain subsets of X.
On the infinitesimal generator of the Poisson operator
Authors: Okikiolu G. O.
Published: November 2008

Throughout this paper we shall denote the space Lp(− ∞, ∞) by Lp, and the numbers p and p′ will be related by 1/p + 1/p′ = 1.

The Poisson operator Pa is defined by
The attenuation of a Rayleigh wave in a half-space by a surface impedance
Authors: Gregory R. D.
Published: November 2008

A time harmonic Rayleigh wave, propagating in an elastic half-space y ≥ 0, is incident on a certain impedance boundary condition on y = 0, x > 0. The resulting field consists of a reflected surface wave, scattered body waves, and a transmitted surface wave appropriate to the new boundary conditions. The elastic potentials are found exactly by Fourier transform and the Wiener-Hopf technique in the case of a slightly dissipative medium. The ψ potential is found to have a logarithmic singularity at (0,0), but the φ potential though singular is bounded there. Analytic forms are given for the amplitudes of the reflected and transmitted surface waves, and for the scattered field. The reflexion coefficient is found to have a simple form for small impedances. A uniqueness theorem, based on energy considerations, is proved.
The diffraction of an obliquely incident surface wave by a vertical barrier of finite depth
Authors: Faulkner T. R.
Published: November 2008

The effect of a vertical barrier, fixed in an infinitely deep sea, on normally incident surface waves of small amplitude was first considered by Ursell (1) and generalizations which retain the two-dimensional aspects of the problem have subsequently been considered by John (2) and Lewin (3). The fluid motion due to the flexural vibrations of a barrier of finite depth has been considered by Alblas (4), the motion in this case being three-dimensional.
Applications of the theory of the general hodograph equation
Authors: Mackie A. G.
Published: November 2008

The solutions of the equation

obtained by the author in (3) are applied to determine the Kirchhoff-Helmholtz flow past a wedge in which the free stream Mach number has any value from 0 to 1 inclusive. For small wedge angles the results are developed further and expressions are found for the drag coefficient and for the asymptotic shape of the free streamlines far downstream. While general formulae are obtained for an arbitrary functionk(σ) more detailed results are given for the particular function k(σ) which corresponds to the case of a polytropic gas.
A new proof of the Routh-Hurwitz stability criterion using the second method of Liapunov
Authors: Parks P. C.
Published: November 2008

The second method of Liapunov is a useful technique for investigating the stability of linear and non-linear ordinary differential equations. It is well known that the second method of Liapunov, when applied to linear differential equations with real constant coefficients, gives rise to sets of necessary and sufficient stability conditions which are alternatives to the well-known Routh-Hurwitz conditions. In this paper a direct proof of the Routh-Hurwitz conditions themselves is given using Liapunov's second method. The new proof is ‘elementary’ in that it depends on the fundamental concept of stability associated with Liapunov's second method, and not on theorems in the complex integral calculus which are required in the usual proofs. A useful by-product of this new proof is a method of determining the coefficients of a linear differential equation with real constant coefficients in terms of its Hurwitz determinants.
A note on lemmas of Green and Kondo
Authors: Morris A. O.
Published: November 2008

Let R be the field of rational numbers, {x} = {x1, z2, …}, {y} = {y1, y2, …} be two countably infinite sets of variables and t an indeterminate. Let (λ) = (λ1, λ2, …, λm) be a partition of n. Then Littlewood (5) has shown that

can be expressed in the form

where Qλ(x, t) and Qλ(y, t) denote certain symmetric functions on the sets {x} and {y} respectively. In addition

where is the partition of n conjugate to (λ). In fact, Littlewood (5) showed that

where the summation is over all terms obtained by permutations of the variables xi (i = 1, 2, …) and

.
Asymptotic formula in a generalized Waring's problem
Authors: Thanigasalam K.
Published: November 2008

Introduction. In the classical Waring's problem, the following asymptotic formula was established by Vinogradov (see Vinogradov (2), Chapter VII).
Flow due to a point source of momentum in a viscous fluid contained in a sphere
Authors: Ranger K. B.
Published: November 2008

It is argued that the zero Reynolds limit of the steady incompressible axially symmetric viscous fluid motion interior to a sphere due to a Landau source at the centre is a Stokes flow. The first three terms of the perturbation expansion are determined and the order of magnitude of the general term not derivable from the Landau source is established. Comparison of the convection terms with the diffusion terms for each order of the Reynolds number demonstrates self consistency at each stage of the expansion.
On a conjecture of B. L. van der Waerden
Authors: Marcus Marvin , Minc Henryk
Published: November 2008

Let Ωn be the set of all n-square doubly stochastic matrices, i.e. matrices with non-negative entries all of whose row sums and column sums are 1. If A = (aij) is an n-square matrix let per (A) denote the permanent of A, that is.
Topological degree of non-compact mappings
Authors: Taylor J. G.
Published: November 2008

We extend the notion of topological degree to a suitable class of continuous mappings of a locally convex Hausdorff topological vector space into itself which are not completely continuous. This extension and the resulting existence theorems for non-linear equations are further extended to product spaces. Our extension is developed with application to quantum field theory in mind; this application is made elsewhere.
Symmetric products of the circle
Authors: Morton H. R.
Published: November 2008

The nth symmetric product of a topological space, X, is defined to be the quotient of the Cartesian product Xn by the action of the symmetric group which permutes the factors. Even if X is a manifold, this product is, in general, not a manifold. The purpose of this note is to determine these products when X is the circle, S1, and to show that they are manifolds with boundary.
Stability of an elastico-viscous liquid film flowing down an inclined plane
Authors: Gupta A. S. , Rai Lajpat
Published: November 2008

An analysis is made of the stability of a layer of an elastico-viscous liquid flowing down an inclined plane in the presence of two-dimensional disturbances. The modified Orr-Sommerfeld equation is solved by a regular perturbation technique for disturbances of large wavelengths. It is shown that in the absence of surface tension, the layer is more unstable as compared with that for an ordinary viscous liquid if Q1 > Q2, Q1 and Q2 being stress relaxation and strain retardation parameters respectively.
Reflected and diffracted fields within a medium formed by a circular cylinder and two planes
Authors: Maulik Tapendra Nath
Published: November 2008

Green's function for SH-line source inside a wedge shaped medium with its apex removed by a circular cylindrical surface has been obtained. Exact location of the image sources facilitates the determination of the reflexion and diffraction at the circular and also at the plane boundary of the multiple phases clearly. The scattered field from the corner is indicated. The connexion of this solution with that of complete wedge has been shown with suitable contour integrations.
The Borel structure of a hyperspace
Authors: Smith D. Hammond
Published: November 2008

In a recent paper (1) Effros defined a topology for the hyperspace of nonempty closed sets of a topological space X, and showed that, if X is polonais, the Borel structure of is standard. We shall show that this conclusion will also follow if X is σ-compact and metrizable.
Finite approximations to infinite non-negative matrices
Authors: Seneta E.
Published: November 2008

In applying the theory of infinite Markov chains to practical examples, it is important to know how the ergodic properties defined by the infinite stochastic or substochastic matrix under consideration are related to those of then×n(n= 1, 2, 3, …) truncated corner sub-matrices. In particular, it is of interest whether the relevant eigenvalues and eigenvectors of the truncated matrices in some sense approximate to corresponding quantities for the infinite matrix asn→ ∞.
Non-simplicially collapsible triangulations of In
Authors: Goodrick Richard E.
Published: November 2008

Given any triangulation of a topological n-cell, In, there is an integer k such that the kth barycentric subdivision of the triangulation is simplicially collapsible (3). In the following, we show that given any integer k, we can construct a triangulation of an n-cell, n > 2, whose kth barycentric subdivision is not simplicially collapsible.
On a construction due to P. M. Cohn
Authors: Samuel Pierre
Published: November 2008

In an intersting article, ‘Bezout rings and their subrings’ (2) P. M. Cohn has introduced a construction which may also be applied in a slightly different situation, i.e. the imbedding of a Krull ring in a Dedekind ring.
Paranormed sequence spaces generated by infinite matrices
Authors: Maddox I. J.
Published: November 2008

A paranormed space X = (X, g) is a topological linear space in which the topology is given by paranorm g—a real subadditive function on X such that g(θ) = 0, g(x) = g(−x) and such that multiplication is continuous. In the above, θ is the zero in the complex linear space X and continuity of multiplication means that λn → λ, xn → x(i.e. g(xn − x) → 0) imply λnxn → λx, for scalars λ and vectors x. We shall use the term semimetric function to describe a real subadditive function g on X such that g(θ) = 0, g(x) = g(−x). Two familiar paranormed sequence spaces, which have been extensively studied (3), are l(p) and m(p). For a given sequence p = (gk) of strictly positive numbers, l;(p) is the set of all complex sequences x = (xk) such that and m(p) is the set of x such that sup Throughout, sums and suprema without limits are taken from 1 to ∞. Simons (3) considered only the case in which 0 < pk ≤ 1 so that natural paranorms would seem to be in m(p). In fact Simons showed that g1 was a paranorm for l(p), but that g2 did not satisfy the continuity of multiplication axiom.
The average number of real zeros of a random trigonometric polynomial
Authors: Das Minaketan
Published: November 2008

Let a1, a2,… be a sequence of mutually independent, normally distributed, random variables with mathematical expectation zero and variance unity; let b1, b2,… be a set of positive constants. In this work, we obtain the average number of zeros in the interval (0, 2π) of trigonometric polynomials of the form

for large n. The case when bk = kσ (σ > − 3/2;) is studied in detail. Here the required average is (2σ + 1/2σ + 3)½.2n + o(n) for σ ≥ − ½ and of order n3/2; + σ in the remaining cases.
Capillary–gravity waves against a vertical cliff
Authors: Packham B. A.
Published: November 2008

In considering the problem of waves on a sloping beach, little regard seems to have been given to the effect of surface tension. Wehausen and Laitone (7) tend to attribute this to the fact that the additional force is small. This does not, of course, preclude the possibility that the effect may be appreciable in certain regions, and Longuet-Higgins (3), for example, has shown this to be the case near the crests for waves on the point of breaking. They also add, which is probably rather more pertinent, that difficulties arise when a solid boundary pierces the surface, since an additional boundary condition is required at the intersection, but give no indication as to what the boundary condition should be.
On the invariance of the essential spectrum of an arbitrary operator. III
Authors: Schechter Martin
Published: November 2008

The spectrum of the hydrogen energy operator

(Δ is the Laplacian and r is the distance from the origin) consists of the non-negative real axis and a sequence of negative eigenvalues of finite multiplicities converging to O. In the present study we are interested in finding sufficient conditions on a potential q(x) such that the spectrum of the operator

in En has a ‘hydrogen-like’ spectrum, i.e. a spectrum consisting of

(a) the non-negative real axis,

(b) at most a denumerable set of negative eigenvalues of finite multiplicities having zero as its only possible limit point.
The inductive limit of uncountably many compact Abelian groups
Authors: Salinger D. L.
Published: November 2008

In (2), Varopoulos examined the structure of topological groups that are the inductive limits of countably many locally compact Abelian groups. The purpose of this note is to show that the theory does not extend to the case of uncountably many groups. We give two examples, the first to show that the strict inductive limit of uncountably many compact Abelian groups need not be complete, the second to show it need not be separated by its continuous characters. The treatment of this latter half follows closely that given for topological linear spaces by Douady in (l).
Finite dimensional submodules of G-modules for a compact group G
Authors: Hofmann Karl Heinrich
Published: November 2008

We propose an exceptionally short and effortless approach to the fundamental facts of the representation theory of compact groups on Hilbert spaces and expand our methods to prove some generalizations concerning representations on arbitrary locally convex vector spaces.
Identities involving powers of persymmetric determinants
Authors: Cusick T. W.
Published: November 2008

Our main result is that the fourth power of the determinant of an arbitrary m by m persymmetric matrix is identically equal to the determinant of a certain 2m by 2m skew-symmetric matrix. Having proved this, we derive several other identities involving powers of persymmetric determinants.
On Kelly's congruence theorem for trees
Authors: Bondy J. A.
Published: November 2008

A graph G is a pair (V{G), E(G)); V(G) is the set of vertices of G, and E(G) the set of edges of G; E(G) is a subset of the Cartesian product V(G) × V(G) (all pairs (υi, υj) where υi∈ V(G), υj ∈ V(G)). If υ is a vertex of G we shall write υ ∈ G. The graph-theoretic terminology used in this paper is that of Berge (1). We shall consider only graphs which are finite, undirected, and without loops.
Note on the representation of integers as sums of certain powers
Authors: Thanigasalam K.
Published: November 2008

In the paper entitled ‘Asymptotic formula in a generalized Waring's problem’, I established an asymptotic formula for the number of representations of a large natural number N in the form

where x1, x2, …, x7 and k are natural numbers with k ≥ 4 (see (2) Theorem 2).
Transformation of an infinite series of G-function
Authors: Bajpai S. D.
Published: November 2008

In this paper we have obtained a transformation of an infinite series of G-function. In section 3 we have shown how such transformations are useful to sum certain series of G-function and E-function (2).
On the summability |N, pn| of a Fourier series at a point
Authors: Dikshit H. P.
Published: November 2008

Let ∑an be a given infinite series and {sn} the sequence of its partial sums. Let {pn} be a sequence of constants, real or complex, and let us write
The self-similar problem of the expanding elliptical crack in an anisotropic solid
Authors: Burridge R. , Willis J. R.
Published: November 2008

The title problem is solved by a semi-inverse method involving plane wave expansions. Expressions are given for stress singularities near the crack edge and for the radiation patterns produced by the crack. The case of an elliptical crack in an isotropic medium expanding under shear is examined in detail. Finally, some generalizations are discussed. Solutions are outlined for when the centre of the crack is allowed to move and for a self-similar phase transformation problem, in which a phase change takes place in an expanding ellipsoidal region.
(LF)-spaces with absolute bases
Authors: Bennett G. , Cooper J. B.
Published: November 2008

Suppose E is a locally convex space over a field K which can be the real line or the complex plane. Then a basis for E is a sequence (xk) of elements of E such that, if x ∈ E, x can be expressed uniquely as

where ξk ∈K for each k. If this representation converges absolutely, i.e. if

for every continuous seminorm p on E, then (xk) is called an absolute basis for E. If the mappings x → ξk from E into K are continuous for each k, then (xk) is a Schauder basis for E. The purpose of this paper is to prove some results for (LF)-spaces with bases and to use them to extend some theorems due to Pietsch. We recall that an (F)-space is a complete metrizable locally convex space and an (LF)-space the inductive limit of a strictly increasing sequence of (F)-spaces (En, τn) such that τn+1|En = τn for all n.
On obtaining zero-free regions for the zeta-function from estimates of M(x)
Authors: Allison D.
Published: November 2008

We use some of the standard notation of analytic number theory; ψ(x) is defined in (1), page 12, and where μ(n) is the Möbius function.
An alǵebraic technique for the solution of Laplace's equation in three dimensions
Authors: Kunz K. S.
Published: November 2008

In obtaining a solution of Laplace's equation in two dimensions by the method of conformal mapping, one first maps the points (x, y) of the Euclidean plane R2 into the algebra of complex numbers C by means of the real-linear function g: R2→C using the prescription g(x, y) = x + iy ≡ z. One then obtains solutions of Laplace's equation by allowing those mappings of C into itself that are expressed by analytic functions.
On critical pairs of product sets in a certain matrix group
Authors: McCrudden M.
Published: November 2008

1. Introduction. If G is a locally compact Hausdorff topological group, and μ is (left) Haar measure on G, then we denote by ℬ(G) the class of all Borel subsets of G having finite measure, and by VG the set {μ(E): E ∊ ℬ(G)} of real numbers. The product set function of G, ΦG: VG × VG → VG, is defined (see (4) and (5)) by

and, for each u, v ∈ VG, we call a pair (E, F) of Borel subsets of G a critical (u, v)-pair, if μ(E) = u, μ(F) = v, and μ*(EF) = ΦG(u, v). We denote the class of all critical (u, v)-pairs by and we write ℰG for .
A case of thermal boundary-layer separation by a magnetic field
Authors: D'sa E. R.
Published: November 2008

In (1) the velocity and magnetic fields were studied in the stagnation point region of a magnetized blunt body rotating with angular speed Ω. Some familiarity with this paper is assumed here: briefly, the nose section of the body was approximated by a disc of thickness t and conductivity σ′ and a perturbation solution was derived for small values of the diffusivity ratio ε (= ν/λ) and of the magnetic force coefficient N = σB2/4ρa. B is the uniform normal field component at the upperside (z = 0) of the disc, a is the strength of the external flow, p and σ are the density and conductivity of the fluid. The other two governing parameters are ω = Ω/a and β = σ′/σL, where L = (λ/a)½.