bluuebunny/crossref_metadata_2025 huggingface dataset:
Angular terms in the electron density for the phosphine molecule
Authors: Ballinger R. A. , March N. H.
Published: November 2008

An attempt is made to calculate the first few angular terms in an expansion of the electron density for the phosphine molecule in Legendre polynomials. Such an expansion is appropriate for a model in which the three hydrogen nuclei are smeared to form a circular line charge. The Thomas–Fermi approximation has been used in conjunction with the variational method. The variational density employed includes p and f angular terms. An approximate charge density map is constructed for a plane containing the molecular axis in order to demonstrate the effect of the angular terms.
Transverse bending of infinite and semi-infinite thin elastic plates. I
Authors: Bassali W. A.
Published: November 2008

In recent years several authors have treated the fundamental problems of two-dimensional statical elasticity for isotropic and aeolotropic materials by the use of functions of a complex variable; references are given at the end of (7). In this paper Stevenson's notation (8,9) is adopted. Dawoud (2) has expressed the continuity conditions across a curve between two differently loaded regions in terms of the complex potentials and particular integrals for the two regions. A form of the boundary condition defining certain types of boundary constraint, including the rigidly clamped and hinged boundaries, has been introduced by the author and Dawoud (1). The introduction of this boundary condition is of practical importance, since neither rigidly clamped nor simply supported conditions can be realized fully under actual physical conditions and thus any case met in practice must lie somewhere between these two limiting cases.
A suggested experiment on the clock paradox
Authors: Cochran W.
Published: November 2008

An experiment is suggested to settle the question whether two clocks which are synchronized and subsequently separated must necessarily show identical readings on being brought together again. Charged π-mesons play the role of a clock travelling with a velocity comparable with c, this clock being read by passing the mesons through a scintillation counter or counters. It is pointed out that the experiment can be designed in a way which makes it technically feasible for a laboratory possessing an accelerating machine capable of producing mesons in the 100 Me V. energy range.

Some considerations suggested by the proposed experiment are examined, and it is concluded that only the orthodox result—that in general the two clocks will record the passage of different time intervals between successive coincidences—is consistent with the result of an experiment which has already been made, or with the special theory of relativity.
Equivalence relations in diffraction theory
Authors: Karp S. N. , Williams W. Elwyn
Published: November 2008

Certain equivalence relationships between different problems occurring in diffraction theory are established. The method used is essentially Schwarz's principle of analytic continuation by reflexion across a straight boundary. An explicit solution is also obtained by this method for a plane wave normally incident on a T-shaped structure.
Arc Lengths in Special Relativity
Authors: MacDuffee C. C.
Published: November 2008

There has lately been an unusual amount of interest in a relativistic phenomenon which has come to be called the clock paradox. It may be simply stated in the following way. If a man were to leave the earth on a powered space ship for an extended trip into space, he would find upon returning to earth that he was younger than he would have been if he had remained on earth.
A note on rank and direct decompositions of torsion-free Abelian groups
Authors: Corner A. L. S.
Published: November 2008

Fuchs((1), Problem 22) has asked the following question: Given positive integers ri (1 ≤ i ≤ 4) such thatr1 + r2 = r3 + r4, r1 ≠ r3, r1 ≠ r4, do there exist indecomposable torsion-free Abelian groups Gi (1 ≤ i ≤ 4) such that, where Gi is of rank ri (1 ≤ i ≤ 4)?* We shall show in this note that there do always exist groups Gi with the desired properties; in fact, we shall prove the following stronger result
Resolutions for extensions of groups
Authors: Wall C. T. C.
Published: November 2008

Let G be an extension of the normal subgroup K by its quotient group H. Suppose we are given free resolutions for H, K (see below for definition). We shall show how to construct from them, by a ‘twisted tensor product’, a free resolution for G, which may also be used to give the spectral sequence of the extension.
The real solutions of a certain non-linear system of equations
Authors: Lebovitz N. R.
Published: November 2008

In a recent paper on the behaviour of a system of disk dynamos(1), a problem of a purely algebraic character arose. The problem is to find all the real solutions of the non-linear, inhomogeneous system of 2n equations

where x0 = xn and the parameter ρ is real and not zero, but otherwise arbitrary. The real solutions were correctly given in (1), but the proof that they are the only real solutions was incomplete. A different, and complete, proof is given here.
A mathematical problem in population genetics
Authors: Kingman J. F. C.
Published: November 2008

A genetical population in which different genotypes have different viabilities will undergo, by selection, changes in its genetical constitution which can be expressed in a simple mathematical form. We here consider a system of m alleles A1, A2, …, Am segregating at one locus, and we assume that the genotype AiAj has viability aij, where aij = aji ≥ 0. We assume discrete, non-overlapping generations and random mating.
Topological embeddings of lens spaces
Authors: Vranceanu G. , Ganea T.
Published: November 2008

Let m ≥ 0 and p, p0, …, pm be integers such that p, pj are relatively prime and 1 ≤ Pj < p for 0 ≤ j ≤ m. In the (2m + 2)-dimensional Euclidean space R2m+2 with points (x0, …, x2m+1), the (2m + l)-sphere S2m+1
Scattering of plane elastic waves at rough surfaces. I
Authors: Abubakar Iya
Published: November 2008

An approximate solution of the two-dimensional problem of reflexion of plane harmonic P and S V waves by an irregular boundary is obtained using a modification of Rice's perturbation method of approximation on the assumption that the curvature of the surface is everywhere small. The case of a periodic surface is treated in more detail. It is found that the reflected waves are composed of specularly reflected waves and various diffracted waves, propagating in both horizontal directions if the wavelength of the incident waves is long compared with that of the surface. If the wavelength of incident distortional waves is long compared with that of the surface, the amplitudes of some of the scattered waves decrease exponentially with depth. In general the phases of the waves change on reflexion and the phase angles of the reflected waves are functions of the wavelength of the corrugation and the angle of incidence. It is verified, in the case of zero angle of incidence, that the energy going into the scattered radiation is obtained at the expense of the energy of the specularly reflected waves.
Reducible Vector Bundles on a Quadric Surface
Authors: Schwarzenberger R. L. E.
Published: November 2008

Let k be an algebraically closed field, and Pn the protective space of dimension n defined over k. The quadric surface is the product variety Q = P1 × P1 and can be embedded as a primal in P3.
On Contractible Open Manifolds
Authors: McMillan D. R. , Zeeman E. C.
Published: November 2008

By an open manifold we mean a non-compact space, that is triangulable by a countable complex which is a combinatorial manifold without boundary (see next section).
Non-Relativistic Dispersion Relations for a Two Channel Scattering Process
Authors: Underhill J.
Published: November 2008

In (l) Khuri has proved the validity of a dispersion relation for non-relativistic potential scattering. More precisely, he has shown that if the potential V(r) is central and satisfies: then the scattering amplitude f(k, τ) = M(E, τ) (where E = k2 is the energy) satisfies the following dispersion relation for fixed momentum transfer τ ≤ 2α: In (2), Rj(τ) is the (real) residue of the scattering amplitude at the bound state Ej and is the Fourier transform of the potential .
Partial orderings on real linear topological spaces
Authors: Simons S.
Published: November 2008

The following properties of the real numbers are well known:

If (xn) is a sequence of real numbers, xn → x and xn ≥ 0 for each n then x > 0. (1)

If (xn) is a sequence of real numbers, xn → x and x > 0 then, for all sufficiently large n,xn > 0.
Some explicit results for an asymmetric two-dimensional random walk
Authors: Barnett V. D.
Published: November 2008

Three distinct methods are used to obtain exact expressions for various characteristics of a particular asymmetric two-dimensional random walk. The results obtained include, for the transient unrestricted walk, the probability of return to the starting-point and the average number of arrivals at the general lattice point; and, for a walk restricted within a rectangular absorbing barrier, the average number of arrivals at any accessible point and the absorption probabilities on the boundary. Whilst there is some duplication of results by using the three different methods of analysis, this is not extensive and provides a useful check on the results. Also the methods are of some general interest in themselves.
On a denumerable partition problem of Erdős
Authors: Davies Roy O.
Published: November 2008

Solving a problem of Erdős(5), I recently proved (4) that for n = 1, 2, …, the hypothesis implies that if the lines in the Euclidean space Ek (k ≥ 2) are distributed into n + 2 disjoint classes Li(i = 1, …,n + 2), then there exists a decomposition of Ek into sets Si such that the intersection of each line of Li with the corresponding set Si is finite. Without assuming any hypothesis other than the axiom of choice, I have also recently proved (3) that if di (i = 1,2,…) is any infinite sequence of mutually non-parallel lines in the plane, then the plane can be decomposed into N0 sets Si such that each line parallel to di intersects the corresponding set 8i in at most a single point.
On the Rees-Suschkewitsch structure theorem
Authors: Pym John S.
Published: November 2008

It was remarked in (1) that many of the results valid for compact (this term is taken to imply Hausdorff) semigroups with jointly continuous multiplication also hold if the multiplication is only separately continuous. It is the purpose of this note to show that this is true of the structure theorem for compact simple semigroups. Any unsubstantiated facts about semigroups which we use may be found in the second section of (1) (the results given there do not depend on the assumption that a semigroup should contain an identity).
The elastic stresses produced in the mid-plane of a slab by pressures applied symmetrically at its surface
Authors: Greenwood J. A.
Published: November 2008

Formulae for the elastic stress distribution in an infinite slab due to symmetrical pressure distributions on the upper and lower faces have been given by Filon (two dimensions) and by Sneddon (axial symmetry). Few numerical values have been published except for the limiting cases of line and point loads. The stresses in the mid-plane are given here for a uniform pressure acting over circles and bands of different sizes, and the published values for a point load are shown to be wrong.
Order continuous measures
Authors: Roberts G. T.
Published: November 2008

1. Objective. It is possible to define order convergence on the vector lattice of all continuous functions of compact support on a locally compact topological space. Every measure is a linear form on this vector lattice. The object of this paper is to prove that a measure is such that every set of the first category of Baire has measure zero if and only if the measure is a linear form which is continuous in the order convergence.
A central limit theorem for processes defined on a finite Markov chain
Authors: Keilson J. , Wishart D. M. G.
Published: November 2008

We shall be concerned in this paper with a class of temporally homogeneous Markov processes, {R(t), X(t)}, in discrete or continuous time taking values in the space

The marginal process {X(t)} in discrete time is, in the terminology of Miller (10), a sequence of random variables defined on a finite Markov chain. Probability measures associated with these processes are vectors of the form

where

We shall call a vector of the form of (0·2) a vector distribution.
Polynomials which are near to k-th powers
Authors: Matthews K. R.
Published: November 2008

1. Let f(x) be a polynomial of degree n ≥ 2 with integral coefficients, the highest coefficient being positive. It is well known that if f(x) is an exact k-th power for all sufficiently large integers x, where k ≥ 2, then f(x) = g(x)k identically, where g(x) is another polynomial with integral coefficients. (See Pólya and Szegö (4), section 8, problems 114, 190; also Davenport, Lewis and Schinzel(1), where other references are given.) The main purpose of this note is to prove that if we suppose only that
Note on the abstract group (2,3,7;9)
Authors: Leech John
Published: November 2008

The abstract group

is finite for n = 4,6,7,8, and the relations are incompatible for n = 1,2,3,5. A criterion of Coxeter ((1)) suggests that (2,3,7; n) should be infinite for all n ≥ 9, but its applicability to these groups is unproved, and it is not known whether there are any further examples of finite groups (2,3,7; n). However, (2,3,7; 9) has been proved infinite by Sims ((3)), and it follows at once that (2,3,7; n) is infinite whenever n is a multiple of 9 as it then has an infinite factor group.
On convex sets and measures
Authors: Larman D. G. , Ward D. J.
Published: November 2008

If α is a positive real number then, for each set E in R3, we define

where U(ρ, E) is any countable collection of convex sets, each of diameter at most ρ, whose union covers E, and Aα is a positive real number. Then the convex α-dimensional measure Λα(E), of E, is . In this paper we shall only be considering the cases α = 1,2 where, as is usual, we take A1 = 1 and A2 = ¼π. The symbols ,Λs(E) will denote the spherical (circular) measure of E, i.e. when the coverings are restricted to being collections of spheres (circles), and E(z) is the intersection of a set E (in R3) with the plane (x, y) at z.
On a conjecture of Conway
Authors: Milner E. C.
Published: November 2008

Small letters denote non-negative integers and capital letters denote sets. The set theoretic difference of A and B is A − B. The empty set is ø and |A| denotes the cardinal of A. We say that A is an immediate extension of B if B ⊂ A and |A − B| = 1. If C is a set of sets we put C(n) = {X: X ∈ C, |X| = n}. A set of sets C, has the property P(k, n) if (i) ø ∈ C and (ii) if X ∈ C(i) and i < n, then C(i + 1) contains at least k immediate extensions of X. We assume throughout that k ≥ 1.
On tubular neighbourhoods of manifolds. II
Authors: Hirsch Morris W.
Published: November 2008

Introduction. This paper is a continuation of (1), where definitions may be found. All manifolds, maps, bundles, etc., are to be piecewise linear. Recall that cells are always closed.
A note on the monomer dimer problem
Authors: Bondy J. A. , Welsh D. J. A.
Published: November 2008

In this note we determine a lower bound for the number of different configurations of monomers and dimers on the hypercubical lattice. This combinatorial problem has important applications in statistical mechanics ((1)) and solid state chemistry.
Complete subsets of mappings over a finite domain
Authors: Schofield P.
Published: November 2008

Improved sufficient conditions are developed for the functional completeness of a set F of functions with variables and values ranging over N = {1, 2, …, n}, where n ≥ 3. In particular, F is complete if it generates a triply transitive group of permutations of N and contains either (i) only a single function, or (ii) at least one function satisfying the Slupecki conditions, the latter apart from certain exceptional cases. These are shown by a detailed investigation to occur only when n = 3 or when n is a power of 2 and all functions of F are linear in each variable, relative to some representation of N as a vector space over Z2.
On generalized inverses of matrices
Authors: Pearl M. H.
Published: November 2008

The notion of the inverse of a matrix with entries from the real or complex fields was generalized by Moore (6, 7) in 1920 to include all rectangular (finite dimensional) matrices. In 1951, Bjerhammar (2, 3) rediscovered the generalized inverse for rectangular matrices of maximal rank. In 1955, Penrose (8, 9) independently rediscovered the generalized inverse for arbitrary real or complex rectangular matrices. Recently, Arghiriade (1) has given a set of necessary and sufficient conditions that a matrix commute with its generalized inverse. These conditions involve the existence of certain submatrices and can be expressed using the notion of EPr matrices introduced in 1950 by Schwerdtfeger (10). The main purpose of this paper is to prove the following theorem:

Theorem 2. A necessary and sufficient condition that the generalized inverse of the matrix A (denoted by A+) commute with A is that A+ can be expressed as a polynomial in A with scalar coefficients.
A combinatorial proof of a theorem of Tutte
Authors: Halton John H.
Published: November 2008

We refer to a beautiful and important result of Tutte(1), in the theory of graphs; that a linear graph G is prime if and only if it contains a set ∑ of vertices, such that u(G∑) > n(∑); where n(∑) is the number of vertices in ∑, G∑ is the graph obtained from G by deleting the star of ∑ (all the vertices of G in ∑, together with all the edges of G meeting vertices of ∑), and u(G∑) is the number of connected components of G∑ having an odd number of vertices.
Some cyclic and other inequalities. II
Authors: Diananda P. H.
Published: November 2008

We continue below the note ‘Some cyclic and other inequalities’ (3).
Addenda to processes defined on a finite Markov chain
Authors: Keilson Julian , Wishart David M. G.
Published: November 2008

Introduction. Two previous papers (1,2) have dealt with additive processes defined on finite Markov chains. Such a process in discrete time may be treated as a bivariate Markov process {R(k), X(k)}. The process R(k) is an irreducible Markov chain on states r = 1, 2, …, R governed by a stochastic transition matrix B0 with components brs. The marginal process X(k) ‘defined’ on the chain R(k) is a sum of random increments ξ(i) dependent on the chain, i.e. if the ith transition takes the chain from state r to state s, ξ,(i) is chosen from a distribution function Drs(x) indexed by r and s. The distribution of the bivariate process may be represented by a vector F(x, k) with components

These are generated recursively by the relation

where the increment matrix distribution B(x) has components brsDrs(x). We denote the nth moment of B(x) by Bn = ∫xndB(x), so that B0 = B(∞).
Lyapunov functions and a control problem
Authors: Kayande A. A. , Muley D. B.
Published: November 2008

1. One of the most important techniques in the study of non-linear differential systems is the Lyapunov's second method and its extensions. One of the extensions of the method depends upon the fact that the function satisfying a differential inequality can be majorized by the maximal solution of the corresponding differential equation. This method was used extensively by V. Lakshmikantham and others for obtaining results, in a unified way, on stability and boundedness of systems of differential equations.
Abstract Köthe Spaces. I
Authors: Fremlin D. H.
Published: November 2008

The purpose of this paper and the next is to demonstrate that the ‘perfect Riesz spaces’ of (1) are an effective abstraction of the ‘espaces de Köthe’ of (2). I shall follow the ideas of (1), with certain changes in notation:

If L is a Riesz space and x, y ∈ L, let us denote sup (x, y) by x ∧ y and inf (x, y) by x ∧ y. I shall use the convenient if informal notation xr↓ ((1), section 16·1) and shall in this usage assume that 0 ∈ {r} and that x0 ≥ xτ for all τ. A set A ⊆ L is solid if x ∈ A and |y| ≤ |x| together imply that y ∈ A; A is then balanced. The solid hull of A is the set {y: ∃ x ∈ A, |y| ≤ |x|}; this is the smallest solid set containing A. An ‘ideal’ ((1), section 17) is then a solid subspace.
A criterion for the planarity of the total graph of a graph
Authors: Behzad Mehdi
Published: November 2008

Two well-known numbers associated with a graph G (finite and undirected with no loops or multiple lines) are the (point) chromatic and the line chromatic number of G (see (2)). With G there is associated a graph L(G), called the line-graph of G, such that the line chromatic number of G is the same as the chromatic number of L(G). This concept was originated by Whitney (9) in 1932. In 1963, Sedlâček (8) characterized graphs with planar line-graphs. In this note we introduce the notions of the total chromatic number and the total graph of a graph, and characterize graphs with planar total graphs.
A note on integral integer-valued functions of several variables
Authors: Baker A.
Published: November 2008

1. The consequences of assuming that an integral function or its derivatives take integer values for some or all integer values of the variable have been widely investigated. The best-known result of this kind was obtained by Pólya(9,10) and Hardy(6) in 1920 and states that among all transcendental integral functions which assume integer values for all non-negative integer values of the variable that of least increase is the function 2Z(cf. also (7)). Analogous results relating to integers in the Gaussian field were proved by Fukasawa(3)in 1926 and Gelfond(4)in 1929, and theorems relating to the function and its derivatives are included in the works of Gelfond (5), Selberg(14) and Straus (15). (For further results and references see (1,2, 11).) The earlier of these investigations may be regarded as the genesis of the celebrated Gelfond-Schneidersolution to the seventh problem of Hilbert and the much important work arising therefrom.
Uniform approximation by Fourier–Stieltjes transforms
Authors: Ramirez Donald E.
Published: November 2008

Let G be a locally compact Abelian group; Γ the dual group of G; CB(Γ) the algebra of continuous, bounded functions on Γ C0(Γ) the algebra of continuous functions on Γ which vanish at infinity; M(G) the algebra of bounded Borel measures on G; M(G)^ the algebra of Fourier–Stieltjes transforms; and M(G)^− the completion of M(G)^ in the sup-norm topology on Γ. The object of this paper is to study the natural pairing between M(G)^ and M(Γ).
Dispersion of soluble matter in the hydromagnetic laminar flow between two parallel plates
Authors: Gupta A. S. , Chatterjee A. S.
Published: November 2008

The paper presents an analytical solution for the dispersion of a solute in an electrically conducting fluid flowing between two parallel plates in the presence of a transverse magnetic field. It is shown that the solute is dispersed relative to a plane moving with the mean speed of the flow with an effective Taylor diffusion coefficient which decreases with increase in magnetic field.
The Boltzmann-Landau transport equation
Authors: Mitra Arun K. , Mitra Sunanda
Published: November 2008

The first-order Chapman-Enskog (CE) approximation has been used to linearize the Boltzmann-Landau (BL) equation primarily in the binary collision approximation and a linear integral equation with a non-symmetric kernel is obtained. The solubility conditions are discussed on the basis of conservation theorems. The formal solutions and the transport coefficients have been obtained in a subsequent paper.
The Boltzmann-Landau transport equation
Authors: Mitra Arun K. , Mitra Sunanda
Published: November 2008

The linearized Boltzmann-Landau transport equation given in a previous paper with a non-symmetric Kernel has been formally solved and the solution has been used to calculate the coefficients of shear viscosity (η) and thermal conductivity (λ) up to first-order density corrections.
Erratum
Authors: Segal S. L.
Published: November 2008

Mr K. A. Jukes has kindly pointed out that on page 398 line 21 and page 400 line 12 of my paper ‘Summability by Dirichlet Convolutions’, this journal 63 (1967), 393–400, Theorem 56 of Hardy and Riesz, The general theory of Dirichlet series, is misquoted.
Bounds for the solutions of the hyperelliptic equation
Authors: Baker A.
Published: November 2008

The purpose of this note is to extend the result which I established recently (see (3)) on the Diophantine equation

to some further equations of a similar kind. The following theorems will be proved.
On certain soluble groups
Authors: Gupta C. K.
Published: November 2008

In (2), Hall considered the question: for what varieties of soluble groups do all finitely generated groups satisfy max-n (the maximal condition for normal subgroups)? He has shown that the variety M of metabelian groups and more generally the variety of Abelian-by-nilpotent-of-class-c (c ≥ 1) groups has this property; whereas on the contrary, there are finitely generated groups in the variety V of centre-by-metabelian groups (i.e. defined by the law [x, y; u, v; z]) which do not satisfy max-n. One naturally raises the question: for what subvarieties of V do all finitely generated groups satisfy max-n?
Rarefied gas flow between parallel plates
Authors: Williams M. M. R.
Published: November 2008

The flow of a rarefied gas between parallel plates has been studied via the linearized Boltzmann transport equation. Using a general boundary condition, which includes an arbitrary ratio of specular to diffuse reflection from the wall, we have derived an integral equation for the mass flow velocity. The integral equation is solved by using a replication property of the kernel and application of the method of Muskelishvili.

The total volumetric flow rate is obtained and a slip boundary condition is deduced for use with the hydrodynamic equations.

Certain aspects of the eigenvalue spectrum associated with the Boltzmann equation are discussed.
On sunny collapsing
Authors: Hudson J. F. P.
Published: November 2008

The technique of sunny collapsing was introduced by Zeeman in (1) in order to prove the unknotting of PL ball pairs with co-dimension ≥ 3. Since then the technique has been used a number of times. In each case the method was used to prove special cases of the Theorem in this paper. Apart from providing a more unified treatment of various applications of this one technique, the theorem has important applications to the theory of relative regular neighbourhoods.
On ǵroups consisting mostly of involutions
Authors: Wall C. T. C.
Published: November 2008

The following problem arose in recent work of Stewart Robertson on finite subsets of Eucidean space admitting transitive groups of isometries (1): Determine the greatest number of involutions that can be possessed by a finite group of given ordern.
On the analyticity of static vacuum solutions of Einstein's equations
Authors: Hagen H. Müller zum
Published: November 2008

It is shown that around each point of a C3static vacuum space-time there is a neighbourhood in which the metric is, in a certain suitable coordinate system, analytic. If two of these neighbourhoods overlap, then the transformation betweea these suitable coordinate systems is analytic (in the overlapping domain).
Transient free convection flow due to the arbitrary motion of a vertical plate
Authors: Sinha P. C. , Singh Punyatma
Published: November 2008

The paper deals with the free convection flow along a vertical plate moving arbitrarily in its own plane. The basic equations of the boundary-layer flow and heat transfer are linearized and the first two approximations are considered. The first approximation is the case of steady-state free convection flow while the second approximation is the response of the fluid velocity and temperature fields to the motion of the plate for which limiting solutions are obtained by the Laplace transform technique in two regions; namely, for large times and for small times. The particular case when the plate is given an impulsive start at t = 0 is investigated in detail. It is shown how the skin friction and the rate of heat transfer at the plate respond to the motion of the plate.
On subgroups of amalgamated free products
Authors: Ordman Edward T.
Published: November 2008

In 1934 Kurosh(9) proved that ‘a subgroup of a free product of groups is again a free product’. Many other proofs of this, and attempts to generalize it to amalgamated free products, have appeared (e.g. (7), (1), (10) and (8)). Recently the theory of groupoids has been applied to this area with increasing success. In 1966 Higgins (6) used groupoids to prove the generalization of Grushko's Theorem (3) due to Wagner (14).
Absolute total-effective (N, pn) means
Authors: Dikshit H. P.
Published: November 2008

1. Definitions and notations. Let be a given infinite series with the sequence of its partial sums {sn}. Let {pn} be a sequence of constants, real or complex, and let us write
The scattering of surface waves by two submerged cylinders
Authors: Schnute Jon T.
Published: November 2008

1. Introduction. The historical basis for the work in this paper lies in a remarkable fact discovered by Dean in 1948. He found that time-periodic surface waves in an ideal fluid experience no reflexion when they encounter normally a fixed, submerged, right-circular cylinder. We might reasonably ask if a similar non-reflective property carries over to different geometrical configurations of submerged objects, for example, two or more cylinders. This question motivates the investigation which follows.
On the Rayleigh assumption in scattering by a periodic surface. II
Authors: Millar R. F.
Published: November 2008

The Rayleigh assumption, which concerns the domain of validity of a representation for the field scattered by a periodic surface under time-harmonic plane-wave excitation, is re-examined. The analysis employs a technique developed to locate singularities of solutions to the Helmholtz equation. When applied to the surface profile v = b cos κu (-∞ < u < ∞) considered originally by Lord Rayleigh, it is found that the Rayleigh assumption is valid if 0 ≤ κb < 0·448 and is not valid if κb > 0·448; this is more precise than an earlier result. It is shown how these findings may be reconciled with the work of others who have suggested, or concluded, that the Rayleigh assumption is never valid.
Some remarks on the Kakeya problem
Authors: Davies Roy O.
Published: November 2008

Besicovitch's construction(1) of a set of measure zerot containing an infinite straight line in every direction was subsequently adapted (2, 3, 4) to provide the following answer to Kakeya's problem (5): a unit segment can be continuously turned round, so as to return to its original position with the ends reversed, inside an arbitrarily small area. The last word on Kakeya's problem itself seems to be F. Cunningham Jr.'s remarkable result(6)‡ that this can be done inside a simply connected subset of arbitrarily small measure of a unit circle.
Arcs with increasing chords
Authors: Larman D. G. , McMullen P.
Published: November 2008

Let f:[0, 1]→R2 be a Jordan arc, and for t, u ∈ [0, 1] let d(t, u) = d(f(t), f(u)) denote the Euclidean length of the chord between f(t) and f(u), and l(t, u) = l(f(t), f(u)) the corresponding arc-length, when this is defined. We say that f has the increasing chord property if d(t2, t3) ≤ d(t1, t4) whenever 0 ≤ t1 ≤ t2 ≤ t3 ≤ t4 ≤ 1. In connexion with a problem in complex analysis, K. Binmore has asked (private communication, see (1)) whether there exists an absolute constant K such that

.
A note on orthogonality and permutability of subnormal subgroups
Authors: Lennox John C.
Published: November 2008

Let H and K be groups with derived sub groups H´ and K´. Following (5) we say that H and K are orthogonal, and write H ⊥ K, if and only if the tensor product H/H´ ®K/K´ is trivial. If H and K are sub groups of a group we say that H and K permute if and only if HK = KH. We recall finally that the subgroup H of the group G is said to be subnormal in G, written H sn G, if and only if there exists a finite chain of subgroups

connecting H to G.
The use of Riemann problems in solving a class of transcendental equations
Authors: Burniston E. E. , Siewert C. E.
Published: November 2008

A method of finding explicit expressions for the roots of a certain class of transcendental equations is discussed. In particular it is shown by determining a canonical solution of an associated Riemann boundary-value problem that expressions for the roots may be derived in closed form. The explicit solutions to two transcendental equations, tan β = ωβ and β tan β = ω, are discussed in detail, and additional specific results are given.
Polynomial approximations to the solution of the heat equation
Authors: Scraton R. E.
Published: November 2008

An approximation is found to the solution of the partial differential equation

in the region −1 ≤ x ≤ 1, t > 0, where u satisfies a general linear boundary condition on x = ± 1. This approximation is a polynomial in x, and is an exact solution of a perturbed form of the differential equation. By choosing the perturbation appropriately, this approach is mathematically equivalent to some recent methods for solving the differential equation in the form of a Chebyshev series. Better approximations to the required solution (and particularly to the eigenvalues) are obtained by choosing the perturbation to satisfy a least squares criterion.
Some large classes of n-valued Sheffer functions
Authors: Muzio J. C.
Published: November 2008

In 1967, Singer (11) gave 3 classes of n-valued two-place functors and proved that all these functors were Sheffer functions. Out of the n possible assignments needed to define a functor completely, Singer showed that it was sufficient to define 3n − 2, 3n − 2, and 2n assignments respectivelyfor the 3 classes. We shall enlarge Singer's classes to give functors of type Ia, type II and type III. For types Ia and III, it will be shown that it is sufficient to define 2n − 1 assignments and for type II we require 2n − 1 assignments to be defined and conditions on a further n/p1 assignments (where P1 is the least prime factor of n). These classes of functors include all of Singer's classes. We also introduce functors of type Ib, similar to those of type Ia, and show that for these itis sufficient to define 2n − 1 assignments to ensure the functor is a Sheffer function.
Zeeman's Catastrophe Machine
Authors: Poston T. , Woodcock A. E. R.
Published: November 2008

René Thom(5) has shown that if the state of a system is determined by the local minimization of a potential – that is, if the system is so highly dissipative that transients can safely be ignored–then, though a smooth change in the potential function may give rise to a discontinuous change of state, the ways in which this can happen are quite limited. Infact, if we have at most a four-parameter family of potentials, discontinuities of this kind can occur in only seven ways up to local diffeotype, if they are to be structurally stable. (This latter condition is the requirement that it be im-possible to alter the discontinuity type by an arbitrarily small perturbation of the family of potentials: roughly it requires that the behaviour of the system, considered as a function of possible families ofpotentials, be ‘continuous’ at the family concerned. It bears the same relationship as does continuity to computability: a computer given approximately correct data for which to compute the value of a function or the behaviour of a system will give an approximately correct answer if the function is continuous, the system structurally stable. If not, only analytic methods will serve, and the physical significance of the result will be dubious. Fortunately, in the dimensions with which we are concerned, structural stability is an open dense property in the space of possible families of potentials; thus unstable systems can be ignored for almost all purposes, just as we ignore the ‘possibility’ of balancing a pin on its point.)
The vorticity equation as an angular momentum equation
Authors: Chatwin P. C.
Published: November 2008

In a Newtonian fluid, in which the body forces are conservative, in which the pressure is a function only of density and in which the kinematic viscosity v is uniform, the vorticity ω satisfies the equation

where u is the velocity field, so that ω = ▿ × u. This equation is normally derived by taking the curl of the Navier–Stokes equations. However, the vorticity has many interpretations in terms of the angular velocity of elements of fluid and it is natural to expect that (1) can be derived by equating the rate of change of the angular momentum of a small material volume element about its centre of mass with the moment of the forces acting on the element. Such a derivation is presented here in the hope that it may be of pedagogic interest.
An approximate theorem for Borsuk's conjecture
Authors: Knast Robert
Published: November 2008

In 1933 Borsuk(2) made the following conjecture: Every bounded set of points in Euclidean n-space En can be represented as the union of n + 1 sets of smaller diameter. He proved it for n = 2. Hadwiger (5) proved Borsuk's conjecture assuming the additional condition that the surface of the set is sufficiently smooth. On the other hand, up to now the conjecture has been proved for n ≤ 3 only (3, 4).
The effect of a semi-contaminated free surface on mass transport
Authors: Porter D. , Dore B. D.
Published: November 2008

The mass transport velocity field is determined for surface waves which propagate from a region with a clean free surface into a region beneath an inextensible surface film. The waves are assumed to be incident normally on the edge of the film. Determination of this velocity field requires the investigation of a mixed boundary value problem for the bi-harmonic equation, the solution of which is obtained using the Wiener–Hopf technique. Streamlines for the mean motion of the fluid particles are thus obtained. It is found that considerable vertical displacement of fluid is possible due to the presence of the surface film.
A generalized Kerr-Schild solution of Einstein's equations
Authors: Vaidya P. C.
Published: November 2008

A very general exact solution of Einstein's equation Rik = σξiξk, is given in terms of the Kerr-Schild metric, gik = ηij + Hξiξk, where ξi is a shearfree geodetic null congruence. The Kerr-Schild solution for Rik = 0 is derived as a particular case.
The horse-shoe and spinning diffeomorphisms
Authors: Arrowsmith D. K.
Published: November 2008

In (5), a generalization of a Morse-Smale diffeomorphism known as a Smale diffeomorphism is given and all such diffeomorphisms are shown to be C° dense on a C∞ compact manifold and structurally stable. Also Morse inequalities for Smale diffeomorphisms are given which supersede the classical Morse-Smale inequalities (3).
Integrals with a large parameter and several nearly coincident saddle points; the continuation of uniformly asymptotic expansions
Authors: Martin J.
Published: November 2008

In the contour integral

f and g are regular functions of z in a neighbourhood of the contour C and of the complex parameters (α1, α2, …, αp) = α in a domain of . N is a positive parameter and asymptotic expansions are considered as N → + ∞. The method of steepest descents provides an asymptotic expansion for each fixed value of α, but this is non-uniform with respect to α near critical values α0 at which certain saddle points tend to coincidence. A more complicated expansion, involving Airy functions or generalizations thereof, is valid and uniform for α near α0. This expansion is known to be valid in a neighbourhood of α0 which does not depend on N, in contrast to certain other expansions having regions of validity which contract to a point or surface as N → ∞. It is thereby suggested that the actual region of validity has a definite geometrical shape determined by the functions f and g and the contour C, just as steepest descents expansions are valid in regions bounded by Stokes' lines. In this paper, procedures are derived for the determination of such regions, subject to assumptions concerning f, g and C. The validity of the expansion is established in regions so determined.
The numerical solution of systems of stiff ordinary differential equations
Authors: Cash J. R.
Published: November 2008

Algorithms are developed for the numerical solution of systems of first-order ordinary differential equations, the solutions of which have widely different rates of variation. The iterative procedures described use a step length of integration proportional to the rate of change of the required slowly varying solution in a region of integration, where either the transient components of the complete solution have become negligible compared with the chosen working accuracy or in a region where rapidly increasing components of the solution are theoretically possible but are made absent by the initial conditions. Several numerical examples are given to demonstrate the algorithms.
Further transformable nth order differential equations with transition points of particular order m
Authors: Heading John
Published: November 2008

Further consideration is given to properties of the Stokes phenomenon associated with a class of nth order differential equations with transition points of order m (a rational fraction) at the origin. The systematic vanishing of the Stokes multipliers leads to the discovery of sets of equations that are transformable from certain specific values of m to particular lower values of m. A series of theorems in the theory of congruences is first proved, from which the main transformation theorem is then deduced. The theory is ifiustrated by various numerical examples.
The normal and subnormal structure of general linear groups
Authors: Wilson J. S.
Published: November 2008

The problem of locating and classifying the normal subgroups of GLn(R), the general linear group of degree n over a commutative ring R with an identity element, has received considerable attention. The solution when R is a field is well known (of. Dieudonné(5), Artin(1)): unless n is equal to two and R has two or three elements, normal subgroups of GLn(R) either lie in the centre of GLn(R) or contain the special linear group SLn(R). However, if R is not a field, then for each ideal I of R the natural map R → R/I induces a homomorphism

and, if 0 < I < R, the kernel of θ1 is a non-central normal subgroup of GLn(R) which does not contain SLn(R). The most that may be expected is that each normal subgroup determines an ideal I of R, in such a way that all normal subgroups determining the same ideal I lie between suitably defined greatest and smallest normal subgroups of GLn(R) corresponding to I. For example, write ZI for the inverse image of the centre of GLn(R/I) under the homomorphism θI, and write KI for the intersection of SLn(R) with the kernel of θI. Then KI ≽ ZI, and the results of Klingenberg(7) and Mennicke(9) show that if n ≥ 3, and if R is either a local ring or the ring of rational integers, then any normal subgroup H of GLn(R) satisfies

for some uniquely determined ideal I. There are many similar theorems. That the above result breaks down for arbitrary rings, even for large n, follows easily from the negative solution to the congruence subgroup problem for certain rings of algebraic integers (see Bass, Milnor and Serre(3)).
Arrivals associated with a class of self-similar problems in elastodynamics
Authors: Willis J. R. , Bedding R. J.
Published: November 2008

A systematic method is given for finding all of the singularities in the stress fields associated with self-similar elastodynamic problems for half-spaces, directly from representations developed in (1). No particular symmetry is assumed, and both two- and three-dimensional problems are discussed for arbitrarily anisotropic half-spaces. Expressions for body wave arrivals are obtained, reproducing results given in (1), and head wave arrivals, not discussed in (1), are found. Arrivals produced by interfacial dislocations and cracks are found by the same method. As an example, an account of the first arrival produced by a penny-shaped crack expanding under shear on an interface is completed, by finding the head wave that precedes the body wave, already found in (1), in a region exterior to a cone in one of the half-spaces.
Moving forcing effects in a two-phase medium
Authors: Chee-Seng Lim
Published: November 2008

An unbounded two-phase medium is transmitting radiation from a group of uniformly moving, two-dimensional forcing effects. The emission process is governed by two matrix differential equations for both interior regions, accompanied by a common set of boundary conditions along the communicating interface. Anticipating possible hyperbolic modes, in co-existence with elliptic modes during the eventual steady state, a radiation condition is initially applied. This radiation condition, together with a particular boundedness criterion, decides the precise number of boundary equations required. An exact solution is established for moving forcing effects having arbitrary spatial distributions. Those aligned along the interface correspond to sheet sources. Superimposed are additional source sheets virtually induced by the action of internal forcing effects, converting their combined incident field into reflection and refraction. A peculiar feature is the invariant propagation along characteristics of certain hyperbolic modes, under the essential presence of elliptic modes, as real Hilbert transforms (in the sense of principal values) of a net interface source density.
Moduli spaces of algebraic curves with rational maps
Authors: Lange Herbert
Published: November 2008

Let ℳg be the coarse moduli scheme of curves of genus g. For an algebraically closed field k define is a quasiprojective algebraic variety over k, its dimension being 3g – 3 for g ≥ 2, 1 for g = 1, and 0 for g = 0. It can be considered as the moduli variety for the classes of birationally equivalent curves of genus g over k. For 0 < g, g′ and n ≥ 1 let be the subset of those points of whose corresponding curves possess a rational map of degree n into a curve of genus g′ over k.
Maximal matchings in graphs with given minimal and maximal degrees
Authors: Bollobás B. , Eldridge S. E.
Published: November 2008

In this note we determine the greatest lower bound of the number of independent edges in a graph in terms of the number of vertices, the minimal degree and the maximal degree. More generally we examine the greatest lower bound of the number of independent edges in aK-connected (or λ-edge-connected) graph with given number of vertices and given minimal and maximal degrees. These results extend those of Erdös and Pósa (3) and Weinstein (7). Our proof makes heavy use of Berge's slight extension of Tutte's characterization of graphs with 1-factors.