bluuebunny/crossref_metadata_2025 huggingface dataset:
A unified approach to the solution of four crack problems in plane elastostatics
Authors: Stallybrass M. P.
Published: November 2008

Four crack problems are considered in classical two-dimensional elastostatics. Each problem is equivalent to a mixed boundary-value problem for a quarter-plane. A governing functional relation of the Wiener–Hopf type is obtained for each problem by the application of various Mellin transforms. It is shown that values for the stress intensity factor, and the crack energy, in corresponding ‘interior’ arid ‘exterior’ problems are identical for certain distributions of crack pressure.
Embedding locally convex spaces in nearly exotic spaces
Authors: Peck N. T.
Published: November 2008

It is proved that every locally convex space is linearly homeomorphic to a subspace of a space which admits no non-trivial continuous linear functionals.
Small solutions of the congruence
Authors: Williams Kenneth S.
Published: November 2008

Throughout this paper a0, a1, a2, l1, l2 denote fixed integers with l1 ≥ 2, l2 ≥ 2. We let l = max(l1, l2) and let P be the set of primes Mordell (4) has shown that for any sufficiently large prime p the congruence

is soluble. Thus there are at most a finite number of such p for which (1·1) is insoluble. If there is at least one prime p ∈ P for which (1·1) is insoluble, we let p0 denote the largest of such p, so that (1·1) is soluble for all p ∈ P with p > p0 but not for p = p0. Otherwise (1·1) is soluble for p ∈ P and we let p0 = 1. From the work of Mordell (4) we have

For p ∈ P with p > p0 (1·1) is thus always soluble and any such solution (x1, x2) can be taken to satisfy

Chalk(2) has posed the problem of estimating a ‘small’ solution of (1·1), at least for p sufficiently large; that is a solution for which p in the inequality (1·3) can be replaced by something less than p. Smith(5) has shown that for p sufficiently large there is always a solution satisfying 1 ≤ xi ≪ p¾log p (i = 1, 2). It is the purpose of this paper to prove the following sharper and more precise result.
On multipliers of Fourier transforms
Authors: Pigno Louis
Published: November 2008

In this paper G is a locally compact Abelian group, φ a complex-valued function defined on the dual Γ, Lp(G) (1 ≤ p ≤ ∞) the usual Lebesgue space of index p formed with respect to Haar measure, C(G) the set of all bounded continuous complex-valued functions on G, and C0(G) the set of all f ∈ C(G) which vanish at infinity.
Integrals and derivatives of functions in MacLane's class and of normal functions
Authors: Barth K. F. , Schneider W. J.
Published: November 2008

If f is meromorphic in D = {,|z| < 1}, we say that f has the asymptotic value a at ζ, ζ ∈ C(={|z| = 1}), if there exists an arc Γ such that Γ ends at ζ, (Γ − {ζ}) ⊂ D, and f has the limit a as |z| → 1 on Γ. The class (M) originally defined by G. R. MacLane ((8), p. 8), consists of those functions that are non-constant and holomorphic (meromorphic) in D and that have asymptotic values at a dense subset of C. In (8), p. 36, MacLane has shown the following three conditions are sufficient for a non-constant holomorphic function f to be in :
The Riemann tensor and stress singularities
Authors: Sackfield A.
Published: November 2008

The algebraic invariants of the Riemann tensor for a static axially symmetric gravitational field are examined. Contrary to common expectation they are shown to be well defined in a region of the field known to be physically singular.
The application of a new source potential to the problem of the transmission of water waves over a shelf of arbitrary profile
Authors: Evans D. V.
Published: November 2008

A new source potential is constructed in the linearized theory of water waves. It is shown how this source potential may be used to reduce the problem of the reflexion and transmission of waves by a shelf of arbitrary profile to an integral equation. It is further shown that by suitably restricting the shelf profile, the Fred-holm theory is applicable to the integral equation so that in general a solution of the equation and hence of the problem, exists and is unique except possibly for certain discrete values of the parameters of the problem corresponding to trapping modes over the shelf.
High frequency diffraction by a soft circular disc
Authors: Newby J. C.
Published: November 2008

Initially a general incident field is considered and the equations are split into Fourier components. Each Fourier component gives rise to an integral equation similar to that obtained when investigating diffraction of a normally incident plane wave. After the oblique wave has been specified an analysis similar to that used for the normal wave leads to a solution of the problem containing an infinite number of constants. It is shown, however, that these constants do not affect the leading terms of the high frequency scattering coefficient.
Ground state of a system of many Bose particles
Authors: Lekner J.
Published: November 2008

A many-body variational calculation is carried out using a trial wave function which is a product of pair functions. An exact integro-differential equation for the pair function is derived, valid for any system of more than three Bose particles.
Dimensionally dependent divergences
Authors: Horndeski Gregory W.
Published: November 2008

It is well known that in an even dimensional Riemannian space there always exists a particular scalar density which is such that its associated Euler–Lagrange tensor vanishes identically. In this note we show that each of these scalar densities can be expressed as the ordinary divergence of a contravariant vector density. Furthermore, there exists an entire class of contravariant vector densities which enjoy this property. These results are discussed with particular reference to the general theory of relativity.
On the short-wave asymptotic motion due to a cylinder heaving on water of finite depth. III
Authors: Robinson-Rhodes P. F.
Published: November 2008

This paper concludes the investigation into the short-wave asymptotic motion due to a cylinder heaving on water of finite constant depth, and we now present a rigorous method for an arbitrary smooth cylinder which intersects the free surface normally. The aim is to justify those assumptions made in a previous paper concerning the asymptotic values on the cylinder of the two related auxiliary potentials–only one of which need be considered–which were introduced to enable an evaluation of the descriptive wave-making and virtual-mass coefficients. This is done by constructing an integral equation of the second kind whose kernel is small and deducing the leading term in the iteration solution. The method is an extension of the method for infinite depth which is treated first and it is found that there is a simple relationship between the kernels. The argument depends to some extent on results established in another previous paper–which is generalized herein–on the rigorous method for the special case of a half-immersed circular cylinder.
Semi-automorphisms of Hadamard matrices
Authors: Hall Marshall
Published: November 2008

A Hadamard matrix Hn is an n by n matrix H = [hij], i, j = 1, …, n in which every entry hij is + 1 or − 1, such that

It is well known that possible orders are n = 1, 2 and n = 4m. An automorphism α of H is given by a pair P, Q of monomial ± 1 matrices such that

Here P permutes and changes signs of rows, while Q acts similarly on columns.
A further result on the existence of periodic solutions of the equation
Authors: Ezeilo J. O. C.
Published: November 2008

1. In the equation in the title (referred to hereafter as the equation (E)) a, b are positive constants and h, p are continuous functions depending only on the arguments shown, with p ω-periodic in t, that is p(t, x, y, z) = p(t + ω, x, y, z) for some constant ω > 0 and for arbitrary t, x, y and z. We shall be concerned specifically with the existence of ω-periodic solutions of equations (E), in which the function h is such that |h(x)| → ∞ as |x| → ∞.
A multiplicative property of R-sequences and H1-sets
Authors: Rhodes C. P. L.
Published: November 2008

Let R be a commutative ring which may not contain a multiplicative identity. A set of elements a1,…,ak in R will be called an H1-set (this notation is explained in section 1) if for each relation r1a1 + … +rkak = 0 (ri ∈ R) there exist elements sij ∈ R such that

where Xl,…,Xk are indeterminates. Any R-sequence is an H1-set, but there do exist H1-sets which are not R-sequences (see section 1). Throughout this note we consider an H1-set a1,…,ak which we suppose to be partitioned into two non-empty sets bl…, br and cl,…, cs. Our main purpose is to show that the ideals B = Rb1 + … + Rbr and C = Rc1 + … + Rcs satisfy Bm ∩ Cn = BmCn for all positive integers m and n (Corollary 1). This generalizes Lemma 2 of Caruth(2) where the result is proved when a1,…, ak is a permutable R-sequence. Our proof involves more detail than is necessary just for this, and we obtain various other properties of H1-sets. In particular we extend the main results of Corsini(3) concerning the symmetric and Rees algebras of a power of the ideal Ra1 +… + Rak (Corollary 3).
A new combinatorial approach to M24
Authors: Curtis R. T.
Published: November 2008

In this paper, we define M24 from scratch as the subgroup of S24 preserving a Steiner system S(5, 8, 24). The Steiner system is produced and proved to be unique and the group emerges naturally with many of its properties apparent.
A higher catastrophe machine
Authors: Woodcock A. E. R. , Poston Tim
Published: November 2008

An elaborated version of Zeeman's Catastrophe Machine (1), which physically illustrates the butterfly catastrophe, is described and analyzed.
A remark on the asymptotic theory of sums with random size
Authors: Galambos Janos
Published: November 2008

The aim of the present note is to point to a strange fact, through an example, in connexion with the asymptotic distribution of sums of a random number of independent and identically distributed (i.i.d.) random variables. We shall formulate our conclusion after the introduction of some notations.
Syntopogenous preordered spaces
Authors: Burgess D. C. J. , Fitzpatrick M.
Published: November 2008

The investigation of a topological, uniform or proximity space endowed with a preordering is by now well established. A topological space along with a preordering related to the topology in varying degrees has been studied, for example, in (3), (4), (5) and (6). ‘Uniform preordered spaces’ and ‘Proximity preordered spaces’ have been considered in (6) and (8) respectively. In each of these cases generalizations of varying proportions of the classical theories have been obtained.
On Killing vectors and invariance transformations of the Einstein–Maxwell equations
Authors: Woolley M. L.
Published: November 2008

It is shown that, in a simply connected four dimensional Riemannian space, an arbitrary divergence-free vector generates a one-parameter group of point transformations which leaves Maxwell's equations unchanged. This result is used to show that, if the metric tensor of a simply connected vacuum Einstein–Maxwell space-time admits a group of motions which is also an invariance group of the electromagnetic field tensor, then there exists a one-parameter family of metric tensors all of which satisfy the Einstein–Maxwell equations with the invariant electromagnetic field as source.
Cliques in random graphs
Authors: Bollobas B. , Erdös P.
Published: November 2008

Let 0 < p < 1 be fixed and denote by G a random graph with point set , the set of natural numbers, such that each edge occurs with probability p, independently of all other edges. In other words the random variables eij, 1 ≤ i < j, defined by

are independent r.v.'s with P(eij = 1) = p, P(eij = 0) = 1 − p. Denote by Gn the subgraph of G spanned by the points 1, 2, …, n. These random graphs G, Gn will be investigated throughout the note. As in (1), denote by Kr a complete graph with r points and denote by kr(H) the number of Kr's in a graph H. A maximal complete subgraph is called a clique. In (1) one of us estimated the minimum of kr(H) provided H has n points and m edges. In this note we shall look at the random variables

the number of Kr's in Gn, and

the maximal size of a clique in Gn.
On finite groups containing no element of order six
Authors: Tyrer A. P.
Published: November 2008

We prove the following theorem:

Theorem A. Let G be a finite simple group of order divisible by 3. Suppose

(i) G contains no element of order 6;

(ii) G has cyclic Sylow 3-subgroups;

(iii) G does not involve A4;

(iv) Every 3′-simple section of G is a Suzuki group.

Then G ≅ SL(2, 2n) for some odd n ≥ 3.
Best approximation in von Neumann algebras
Authors: Robertson A. Guyan
Published: November 2008

We investigate here the question of uniqueness of best approximation to operators in von Neumann algebras by elements of certain linear subspaces. Recall that a linear subspace V of a Banach space X is called a Chebyshev subspace if each vector in X has a unique best approximation by vectors in V. Our first main result characterizes the one-dimensional Chebyshev subspaces of a von Neumann algebra. This may be regarded as a generalization of a result of Stampfli [(4), theorem 2, corollary] which states that the scalar multiples of the identity operator form a Chebyshev subspace. Alternatively it may be regarded as a generalization of the commutative situation in which a continuous complex-valued function f on a compact Hausdorff space X spans a Chebyshev subspace of C(X) if and only if f does not vanish on X [(3), p. 215]. Our second main result is that a finite dimensional * subalgebra, of dimension > 1, of an infinite dimensional von Neumann algebra cannot be a Chebyshev subspace. This imposes limits to further generalization of Stampfli's result.
Inequalities associated with the inversion of elastic stress-deformation relations and their implications
Authors: Ogden R. W.
Published: November 2008

For an elastic solid the constitutive law can be written in terms of the deformation gradient α and its conjugate nominal stress s ≡ s(α), and also in terms of the right stretch u and its conjugate stress τ ≡ τ(u). It is shown that for isotropic elastic solids s(α) is invertible, in the local sense, for all u in the domain of u-space where τ(u) is locally invertible, with the exception of certain configurations which correspond to planes in τ-space.In the global sense a given s corresponds to four distinct τ's, and s is invertible to give four distinct α's when the corresponding τ's are uniquely invertible. That there are four branches of the inversion α(s) is of fundamental importance in that it clarifiesthe extent to which non-uniqueness of solution of boundary-value problems can be expected.

The implications of these results in respect of the complementary variational principle are discussed, and the controversy surrounding the use of nominal stress in this principle resolved.

Consequences of the required restrictions on τ(u) are examined and discussed in relation to inequalities which may be regarded as entailing physically reasonable response. It is intimated that τ(u) is invertible in the domain of elastic response.
Zero divisors and idempotents in group rings
Authors: Linnell P. A.
Published: November 2008

1. Introduction. Let kG denote the group algebra of a group G over a field k. In this paper we are first concerned with the zero divisor conjecture: that if G is torsion free, then kG is a domain. Recently, K. A. Brown made a remarkable breakthrough when he settled the conjecture for char k = 0 and G abelian by finite (2). In a beautiful paper written shortly afterwards, D. Farkas and R. Snider extended this result to give an affirmative answer to the conjecture for char k = 0 and G polycyclic by finite. Their methods, however, were less successful when k had prime characteristic. We use the techniques of (4) and (7) to prove the following:

Theorem A. If G is a torsion free abelian by locally finite by supersoluble group and k is any field, then kG is a domain.
Phase-integral treatment of transmission through an inverted Morse potential
Authors: Lundborg Bengt
Published: November 2008

The transmission of a quantal particle through a barrier obtained by inverting a Morse potential is studied by means of the ‘symmetric’ phase-integral approximations introduced by N. Fröman and further developed by N. Fröman and P. O. Fröman. The calculations are performed up to the fifth-order approximation. Approximate expressions for important quantities such as transmission and reflexion coefficients, phases of the wave function and also for quantities related to the phenomenon of resonances, which may occur when the wave function is given as a standing wave, are compared to the corresponding exact quantities, and this comparison shows the great accuracy of these phase-integral approximations.
On separation axioms for certain types of ordered topological space
Authors: Burgess D. C. J. , Fitzpatrick M.
Published: November 2008

An ordered topological space (E, τ, ≥) is a set E endowed with a topology τ and a partial order ≥. For such a space order separation axioms have been studied by Nachbin (6) and McCartan (3). In this paper we discuss the consequences for these axioms of the imposition, in turn, of four conditions on (E, τ, ≥), namely convexity (6), continuity, anticontinuity and bicontinuity (5).
The evaluation in terms of Γ-functions of the periods of elliptic curves admitting complex multiplication
Authors: Zucker I. J.
Published: November 2008

Let

and K′ = K(k′), where k2 + k′2 = 1

with
A dual approach to Černikov modules
Authors: Hartley B.
Published: November 2008

Let G be a group and A a right G-module. If the additive group A+ of A is a Černikov group, that is, a direct sum of finitely many cyclic and quasi-cyclic groups, we shall call A a Černikov module over G or over the integral group ring . Suppose that A+ is, furthermore, a divisible p-group, where p is a prime. Since the endomorphism ring of a quasi-cyclic p-group is isomorphic to the ring of p-adic integers, we find that is a free -module of finite rank. We can make A* into a right G-module in the usual way, and since A* is actually just the Pontrjagin dual of A, Pontrjagin duality shows that A → A* gives rise to a contravariant equivalence between the categories of divisible Černikov p-torsion modules over and G-modules which are -free of finite rank. Since the latter category is to some extent familiar, at least when G is finite – for its objects determine representations of G over the field of p-adic numbers, a field of characteristic zero – we may hope to exploit this correspondence systematically to study divisible Černikov p-modules. This is our main theme.
The asymptotic covariance of two counters
Authors: Kingman J. F. C.
Published: November 2008

A conjecture of Cox and Isham about the asymptotic form of the covariance of counts in two counters, linked by a partially shared input, is established by means of Smith's theory of cumulative processes and a variant of Wald's identity.
Some remarks on the existence of spacelike hypersurfaces of constant mean curvature
Authors: Goddard A. J.
Published: November 2008

Bernstein's theorem states that the only complete minimal graphs in R3 are the hyperplanes. We shall produce evidence in favour of some conjectural generalizations of this theorem for the cases of spacelike hypersurfaces of constant mean curvature in Minkowski space and in de Sitter space. The results suggest that the class of asymptotically simple space-times admitting a complete spacelike hypersurface of constant mean curvature may well be considerably smaller than the general class of asymptotically simple space-times.
A new type of convergence
Authors: Maddox I. J.
Published: November 2008

In his important paper (1), Lorentz defined the space f of almost convergent sequences, using the idea of Banach limits. If x ∈ l∞(R), the space of bounded real sequences, and

where the inf is taken over all sets n(1), n(2), …, n(r) of natural numbers, then a Banach limit L may be defined as a linear functional on l∞(R) which satisfies
Formulae for high derivatives of composite functions
Authors: Fraenkel L. E.
Published: November 2008

This note concerns a question of elementary calculus. Given a smooth composite function u = g o f [with values u(x) = g(f(x))], we write explicit formulae for its derivatives, of arbitrary order, in terms of derivatives of f and g. We consider (A) the general case,

in which E, F and G are Banach spaces, and U, V are open sets; (B) the finite-dimensional case E = ℝM and F = ℝN, where ℝM denotes real M-dimensional Euclidean space; and (C) the particular case of (B) (due to restricting g to part of an M-dimensional surface in ℝM + 1) in which N = M + 1 and u(x)= g(x, φ (x)), so that φ denotes a real-valued (scalar-valued) function of x = (x1, …, xM).
On interpolation by analytic maps in infinite dimensions
Authors: Globevnik J.
Published: November 2008

Let A be the complex Banach algebra of all bounded continuous complex-valued functions on the closed unit ball of a complex Banach space X, analytic on the open unit ball, with sup norm. For a class of spaces X which contains all infinite dimensional complex reflexive spaces we prove the existence of non-compact peak interpolation sets for A. We prove some related interpolation theorems for vector-valued functions and present some applications to the ranges of analytic maps between Banach spaces. We also show that in general peak interpolation sets for A do not exist.
Waves created against a vertical cliff – a uniform asymptotic solution
Authors: Pramanik A. K.
Published: November 2008

The initial-value problem of waves generated by a moving oscillatory surface pressure against a vertical cliff is solved and a uniform asymptotic analysis of the unsteady state is given. The same problem with no cliff is solved by Kaplan, and by Debnath and Rosenblat but their solutions are not uniform.
Inequalities between means of positive operators
Authors: Bhagwat K. V. , Subramanian R.
Published: November 2008

One of the most fruitful – and natural – ways of introducing a partial order in the set of bounded self-adjoint operators in a Hilbert space is through the concept of a positive operator. A bounded self-adjoint operator A denned on is called positive – and one writes A ≥ 0 - if the inner product (ψ, Aψ) ≥ 0 for every ψ ∈ . If, in addition, (ψ, Aψ) = 0 only if ψ = 0, then A is called positive-definite and one writes A > 0. Further, if there exists a real number γ > 0 such that A — γI ≥ 0, I being the unit operator, then A is called strictly positive (in symbols, A ≫ 0). In a finite dimensional space, a positive-definite operator is also strictly positive.
An upper bound for the multidimensional dimer problem
Authors: Minc Henryk
Published: November 2008

A recently proved upper bound for the permanents of (0,1) matrices is used to improve the Fowler-Rushbrooke upper bound for the constant λd occurring in the d-dimensional dimer problem, d ≥ 3.
On fibre spaces and nilpotency
Authors: James I. M.
Published: November 2008

Recall that a categorical covering of a space B is a covering by closed sets each of which is contractible in B. Suppose that B admits a finite categorical covering, and hence one where the number of sets is minimal. The category of B is then defined to be one less than that minimum number. Category is generally associated with nilpotency, in homotopy theory. In this note we describe a further illustration of this, from the theory of fibre spaces.
On the duality between the behaviour of sums of independent random variables and the sums of their squares
Authors: Hall Peter
Published: November 2008

LetXnj, 1 ≤j≤kn, be independent, asymptotically negligible random variables for eachn≥ 1. In certain cases there exists a duality between the behaviour of ΣjXnjand. We extend one of the known forms of this duality, and show that, under mild conditions on the truncated moments of theXnj, the convergence ofto 1 in the mean of orderp(p≥ 1) is equivalent to the convergence of ΣjXnjto the standard normal law, together with the convergence of its 2pth absolute moment to that of a standard normal variable. A similar result holds in the case of convergence to a Poisson law.
Algebras of measures on C-distinguished topological semigroups
Authors: Dzinotyiweyi H. A. M.
Published: November 2008

Let S be a (jointly continuous) topological semigroup, C(S) the set of all bounded complex-valued continuous functions on S and M (S) the set of all bounded complex-valued Radon measures on S. Let (S) (or (S)) be the set of all µ ∈ M (S) such that x → │µ│ (x-1C) (or x → │µ│(Cx-1), respectively) is a continuous mapping of S into ℝ, for every compact set C ⊆ S, and . (Here │µ│ denotes the measure arising from the total variation of µ and the sets x-1C and Cx-1 are as defined in Section 2.) When S is locally compact the set Ma(S) was studied by A. C. and J. W. Baker in (1) and (2), by Sleijpen in (14), (15) and (16) and by us in (3). In this paper we show that some of the results of (1), (2), (14) and (15) remain valid for certain non-locally compact S and raise some new problems for such S.
Finite Bol loops
Authors: Burn R. P.
Published: November 2008

In this paper, we prove that, for any prime p, a Bol loop of order 2p or of order p2 is necessarily a group, and we show that there exist exactly six non-associative Bol loops of order 8.
On G-spaces, Serre classes and G-nilpotency
Authors: Hilton Peter , Roitberg Joseph , Singer David
Published: November 2008

0. Introduction. The concept of a nilpotent action on a nilpotent group was studied in (2), and was crucial to the definition of a nilpotent map (or nilpotent fibration) in (4). These ideas had, of course, already been treated systematically in (1)†. The concept found a new, and related, application to homotopy theory in (3), where the following situation was studied. Let X be a nilpotent space and let G be a group which ‘acts’on X. Here it is not strictly speaking necessary to specify the nature of the action, provided that there is an induced action of G on the homotopy and homology groups of X. Thus G might be a group of base-point preserving homeomorphisms of X, or a group of (based) homotopy classes of self-homotopy-equivalences of X. We then discussed in (3) questions related to the nilpotency of the induced actions of G on the homotopy and homology groups of X and also relativized the theory to a discussion of fibrations on which the group G acts; to obtain interesting results related to nilpotency, it turned out to be at least necessary to suppose the fibration F → E → B quasinilpotent (7), meaning that all spaces are connected and π1B acts nilpotently on the homology groups of F.
Euler operators and conservation laws of the BBM equation
Authors: Olver Peter J.
Published: November 2008

The BBM or Regularized Long Wave Equation is shown to possess only three non-trivial independent conservation laws. In order to prove this result, a new theory of Euler-type operators in the formal calculus of variations will be developed in detail.
Constants relating a Hermitian operator and its square
Authors: Partington J. R.
Published: November 2008

Let T be a linear operator on a complex normed space X. Its spatial numerical range V(T) is denned as
On strong almost convergence
Authors: Maddox I. J.
Published: November 2008

The concept of strong almost convergence was introduced in (2), where the matrices summing every strongly almost convergent sequence, leaving the limit invariant, were characterized.
The determination of a function from its projections
Authors: Falconer K. J.
Published: November 2008

Let H(t, θ) be the hyperplane in Rn (n ≥ 2) which is perpendicular to the unit vector θ and perpendicular distance t from the origin, that is H(t, θ) = {x ε Rn: x.θ = t} (Note that H(t, θ) and H(−t, −θ) are the same hyperplane.) If f(x)ε L1(Rn) we will denote by F(t, θ) the projection of f perpendicular to θ, that is the integral of f(x) over H(t, θ) with respect to (n − 1)-dimensional Lebesgue measure. By Fubini's Theorem, if f(x) ε L1 (Rn), F(t, θ) exists for almost all t for every θ.
The probability that a point of a tree is fixed
Authors: Harary Frank , Palmer Edgar M.
Published: November 2008

Using arguments involving combinatorial enumeration and asymptotics we compute the probability that a point of a random tree is fixed. The method is also applied to homeomorphically irreducible trees to illustrate how it works for other species of trees. To the nearest per cent, the limiting probability of a fixed point in a randomtree is 70%, and for homeomorphically irreducible trees it is 20%.
Strong and absolute summabilities of functions based on (D, α) summability methods
Authors: Mishra Babban Prasad , Srivastava Anand Prakash
Published: November 2008

1. Introduction. In 1966, Kuttner(4) constructed (D, α) summability methods for functions. In this paper, strong and absolute summabilities based on (D, α) summability methods are defined and some of their properties investigated.
The Smale invariants of an immersed projective space
Authors: Berrick A. J.
Published: November 2008

This note approaches the immersion/embedding problem for real projective n-space Pn in Rn+k by viewing the Smale invariant of the induced immersion Sn → Pn → Rn + k as an obstruction to extending to an immersion of Pn + 1 in Rn + k. First steps in this direction were taken in (8), by studying the elementary problem of extending axial maps.
Classical approximations to Gibbs states
Authors: Davies E. B.
Published: November 2008

We obtain two approximate representations of a one-particle Gibbs state, both of which become asymptotically exact in trace norm as m → ∞. The second representation is an integral of pure coherent states over phase space, and can therefore be regarded as a classical approximation to the Gibbs state. We also obtain a version of the second representation applicable to the microcanonical ensemble.
A characterization of the sine function
Authors: Roe J.
Published: November 2008

If a function and all its derivatives and integrals are absolutely uniformly bounded, then the function is a sine function with period 2π.
Periodic groups generated by finite amalgams
Authors: Hickin Kenneth K. , Phillips Richard E. , Plotkin J. M.
Published: November 2008

The following embedding theorem was proved in the paper (7). Let Π be a set of primes and P, H and K periodic Π-groups satisfying

Then there is a period Π-group J generated by isomorphic copies of H and K such that P ≤ J.
Continuity properties of k-plane integrals and Besicovitch sets
Authors: Falconer K. J.
Published: November 2008

Let Π be a k-dimensional subspace of Rn(n ≥ 2) and let Π┴ denote its orthogonal complement. If x ∈ Rn we shall write x = x0 + x′ with x0 ∈ Πand x′ ∈ Π ┴. If f(x) is a real measurable function on Rn, the k-plane integral F(Π,x′ )is defined as the integral of f over the affine subspace Π + x′ with respect to k-dimensional Lebesgue measure (assuming that the integral exists). If k = 1 we get the x-ray transform that arises in the problem of radiographic reconstruction, and if k = n − 1, the k-plane integral is the usual projection or Radon transform. The paper by Smith, Solmon and Wagner (4) contains a survey of results on k-plane integrals. Here we shall be interested in the behaviour of the F (Π, x′ ) regarded as a function of x′ for fixed Π for various classes of function f. We shall obtain some surprisingly strong results on the continuity and differentiability of F(Π,x′) with respect to x′ for almost all Π (in the sense of the appropriate Haar measure). As will be seen the dimensions n and k have a crucial effect on what may be said, and most of our results will be confined to the cases where k > ½n.
Generalized Nörlund means and consistency theorems
Authors: Maddox I. J.
Published: November 2008

It was recently shown in (1), corollary 1, that if (I) = (N, q) then I and (N, q) were consistent. By I we denote the unit infinite matrix, and (N, q) denotes a general complex Nörlund mean. The summability field (A) of any infinite matrix A = (ank) is defined as

where c is the space of convergent sequences. The summability field of (N, q) is, for simplicity, written as (N, q) instead of ((N, q)).
On homomorphisms between Weyl modules and Specht modules
Authors: Carter R. W. , Payne M. T. J.
Published: November 2008

The aim of this paper is to generalize a result of Carter and Lusztig(1) proving the existence of certain non-zero homomorphisms between Weyl modules for the special linear group SLn(K) over an infinite field of characteristic p, and between Specht modules for the symmetric group Sr over K.
On elastic waves from a finite line-source
Authors: Viswanathan K.
Published: November 2008

Hazebroek's problem of a finite line-source in an infinite elastic medium is reconsidered. Here the object is to present a simpler method of solution by treating bhe line-source as equivalent to an appropriate distribution of point sources over the surface of a prolate spheroid which degenerates into this line in the limit when its minor axis tends to zero. This is achieved by using the property of Green's functions. The solution thus obtained is verified to be identical with that derived by Hazebroek.
Some elastostatic and potential problems for an elliptic disc
Authors: Walpole L. J.
Published: November 2008

We consider some elastostatic and related potential problems in which an unbounded uniform medium contains a plane elliptic disc as an inhomogeneity. Various appropriate conditions on the disc and at infinity are specified. These problems have many physical applications. In the elastostatic case the disc can be a crack or rigid inclusion, for example. The potential problems include the classical one of finding the electrostatic field around an elliptic disc charged to a given potential.
Norm and spectral radius for algebraic elements of a Banach algebra
Authors: Young N. J.
Published: November 2008

The purpose of this note is to show that, for any algebraic element a of a Banach algebra and certain analytic functions f, one can give an upper bound for ‖f(a)‖ in terms of ‖a‖ and the spectral radius ρ(a) of a. To illustrate the nature of the result, consider the norms of powers of an element a of unit norm. In general, the spectral radius formula

contains all that can be said (that is, the limit ρ(a) can be approached arbitrarily slowly). If we have the additional information that a is algebraic of degree n we can say a good deal more. In the case of a C*-algebra we have the neat result that, if ‖a‖ ≤ 1,

(see Theorem 2), while for a general Banach algebra we have at least
Resolving acyclic images of three-manifolds
Authors: Bryant J. L. , Lacher R. C.
Published: November 2008

Our main result is that a locally contractible Z2-acyclic image of a 3-manifold without boundary is the cell-like image of a 3-manifold without boundary (all mappings being proper). Consequently, such images are generalized 3-manifolds (that is, finite-dimensional and Z-homology 3-manifolds). A refinement of the proof allows the omission of the Z2-acyclicity hypothesis over a zero-dimensional set; an application is the result of D. R. McMillan, Jr., to the effect that a generalized 3-manifold with compact zero-dimensional singular set admits a resolution if and only if a deleted neighbourhood of the singular set embeds in a compact, orientable 3-manifold.
Convex bodies with almost all k-dimensional sections polytopes
Authors: Dalla Leoni , Larman D. G.
Published: November 2008

It is a well-known result of V. L. Klee (2) that if a convex body K in En has all its k-dimensional sections as polytopes (k ≥ 2) then K is a polytope.
Constitutive surfaces in fluid mechanics
Authors: Sewell M. J. , Porter D.
Published: November 2008

The new concept of a constitutive surface is introduced into inviscid fluid mechanics, with special reference to compressible gas dynamics and to shallow water theory. The detailed shape of such surfaces is calculated, including the manner of their transition across singularities where the Mach or Froude number passes through unity. The calculation draws upon recent work describing the transition of a Legendre transformation through its singularity. For example, mass flow Q, total energy h and flow stress P are always related by part of a ‘swallowtail’ surface, regardless of the particular motion.

The addition of dynamical conditions defines particle history tracks which always lie on constitutive surfaces even for unsteady flow, except that they may jump from one part to another of such a surface when shock waves or hydraulic jumps are passed through.

Illustrations given include the steady flow of a general gas through a standing normal shock, general shallow water theory, and flow along a sloping-sided channel. Connections with existing literature are described.
How to solve a quadratic equation in integers
Authors: Grunewald Fritz J. , Segal Daniel
Published: November 2008

In answer to a question posed by J. L Britton in (2), we sketch in this note an effective procedure to decide whether an arbitrary quadratic equation

with rational coefficients, has a solution in integers. A similar procedure will in fact decide whether such an equation over a (suitably specified) algebraic number field k has a solution in any (suitably specified) order in k; but we shall not burden the exposition by giving chapter and verse for this claim.
Finite Bol loops II
Authors: Burn R. P.
Published: November 2008

In this paper, we prove that for any odd prime p, there are exactly two non-associative, non-Moufang, (right) Bol loops of order 4p.
Characterizing the rate of convergence in the central limit theorem. II
Authors: Hall Peter
Published: November 2008

We obtain upper and lower bounds of the same order of magnitude for the error between the distribution of a sum of independent and identically distributed random variables, and a normal approximation by a portion of a Chebychev-Cramér series. Our results are sufficiently general to contain the familiar characterizations by Ibragimov(4), Heyde and Leslie (3) and Lifshits(5), and complement some of those obtained earlier by the author (2).
3-primary exponents
Authors: Neisendorfer Joseph A.
Published: November 2008

The purpose of this paper is to show that 3n annihilates the 3-primary component of the homotopy groups of the 2n + 1-dimensional sphere. In the terminology of (2) and (3), S2n+1 has exponent 3n at 3.

In fact, a stronger result is proved. Localize at 3 and let Ω2nS2n + 1〈2n + 1〉 denote the 2n-fold loop space of the 2n + 1-connected cover of S2n+1. Then Ω2nS2n + 1〈2n + 1〉 has a null homotopic 3n-th power map.
Convergence of non-homogenous stochastic chains. II
Authors: Mukherjea Arunava , Chaudhuri Ranjan
Published: November 2008

Earlier study by one of the authors on the convergence of non-homogeneous stochastic chains is continued here. Several results more general than those considered earlier relating convergence of such chains to ergodicity of certain smaller chains are presented.
Constructing maximal geodesics in strongly causal space-times
Authors: Beem John K. , Ehrlich Paul E.
Published: November 2008

Let (M, g) be an arbitrary space-time of dimension ≥ 2 and let d = d(g): M × M → ℝ ∪ {∞} (where d(p, q) = 0 for q∉J+(p)) denote the Lorentzian distance function of (M, g). Also let C(M, g) denote the space of Lorentzian metrics for M globally con-formal to g. Here g1 is said to be globally conformal to g if there exists a smooth function Ω: M → (0, ∞) such that g1 = Ωg.
Some divisibility problems in branching processes
Authors: Biggins J. D. , Shanbhag D. N.
Published: November 2008

In this paper the major concern is in investigating various divisibility properties of the random variable W, the limit of the Heyde martingale in supercritical branching processes. Throughout the paper ø(s) will be used to denote the Laplace-Stieltjes transform of W.
On the Dixmier property of simple C*-algebras
Authors: Riedel Norbert
Published: November 2008

A unital C*-algebra is said to satisfy the Dixmier property if for each element x in the closed convex hull of all elements of the form u*xu, u being a unitary in , intersects the centre of ((2), 2·7). The von Neumann algebras and also some other classes of C*-algebras are known to satisfy the Dixmier property (cf. (2), (3), (4), (6)). If is a simple C*-algebra which satisfies the Dixmier property then has at most one tracial state. In (3) Archbold raised the question whether there exists a unital simple C*-algebra which has at most one tracial state without satisfying the Dixmier property. In the present note we characterize the unital simple C*-algebras with at most one tracial state in terms of a condition which is similar to the Dixmier property, but is in fact formally weaker in the framework of simple C*-algebras. This characterization relies on the method used by Pedersen in (5) in order to show that for a unital simple C*-algebra which has at most one tracial state and at least one non-trivial projection the linear span of all projections in is dense in As an application we characterize those unital simple C*-algebras with a unique tracial state which satisfy the Dixmier property.
On the maximum sizes of certain (k, n)-arcs in finite projective geometries
Authors: Mason J. R. M.
Published: November 2008

In this paper the values of m(6)2,9, m(5)2,8 and m(5)2,9 have been determined as well as improved bounds on other arcs, and in particular a general construction for certain large arcs in planes of square order has been given. Table 5 summarizes the known values of m(n)2, q for 2 ≤ n ≤ q and q ≤ 9.
The Variational Principle and natural transformations II. Time dependent theory
Authors: Schafir R. L.
Published: November 2008

In the previous Paper I, the Variational Principle has been presented as a principle of natural transformation between conserved quantities and infinitesimal in variances, which preserves the geometrical character of the various standard objects under point transformations. So as to establish the theory as easily as possible, Paper I (2) confined itself to the simple case of autonomous dynamical systems, and their 2n dimensional space. The aim of the present paper is to extend the work to the 2n + 1 dimensional theory, in which time is included as an extra variable, and time dependent systems can be handled. It will be seen that within a suitable framework the extension is straightforward, and there are some rather more satisfactory features of the 2n + 1 dimensional theory - even for autonomous systems when included in the new formalism - than in the previous 2n dimensional theory. The exposition will now be rather concise; readers are requested to refer to Paper I for fuller discussion, particularly of the underlying ideas.
On the converse of absolute Abel summability
Authors: Kwee B.
Published: November 2008

The series is said to be summable A to s if the power series converges to φ(x) for 0 ⋞ x < 1 and limx→1 φ(x) = s. If the function ⋞(x) is of bounded variation over the same interval, the series is said to be summable │A│. It is easy to prove │A│ ⊆ A, i.e. a series summable │A│ is necessarily summable A to the same sum.
Centralizers of reflections in crystallographic groups
Authors: Baskan T. , Macbeath A. M.
Published: November 2008

The study of hyperbolic 3-manifolds has recently been recognized as an increasingly important part of 3-manifold theory (see (9)) and for some time the presence of incompressible surfaces in a 3-manifold has been known to be important (see, for example, (4)). A particularly interesting case occurs when the incompressible surface unfolds in the universal covering space into a hyperbolic plane. The fundamental group of the surface is then contained in the stabilizer of the plane, or, what is the same thing, in the centralizer of the reflection defined by the plane. This is one motivation for studying centralizers of reflections in discrete groups of hyperbolic isometries, or, as we shall call them, hyperbolic crystallographic groups.
Homomorphs and wreath product extensions
Authors: Förster Peter
Published: November 2008

A homomorph is a class of (finite soluble) groups closed under the operation Q of taking epimorphic images. (All groups considered in this paper are finite and soluble.) Among those types of homomorphs that have found particular interest in the theory of finite soluble groups are formations and Schunck classes; the reader is referred to (2), § 2, for a definition of those classes. In the present paper we are interested in homomorphs satisfying the following additional closure property:

(W0) if A is abelian with elementary Sylow subgroups, then each wreath product A G (with respect to an arbitrary permutation representation of G) with G ∊ is contained in .
Asymptotic analysis of the Cooke-Noble integral equation
Authors: Rose L. R. F.
Published: November 2008

Those mixed boundary-value problems which can usefully be treated analytically often lead to the following mathematical problem. Two functions u(x), σ(x), defined over the interval ([0, ∞), take prescribed values over complementary portions of that interval; specifically, let

where p(x) is usually a simple function, for example a constant or a power of x. There exists a relation between u(x) and σ(x) which can be most simply expressed as a relation between their Hankel transforms. Using a circumflex to denote the Hankel transform, for example with

where Jv denotes as usual the Bessel function of the first kind of order v, we can state that relation between u and σ as follows:

where A(ξ) is a known function, determined at an earlier stage of the analysis. The problem is to derive u(x) for (xє [ 0, a), or σ(x) for x є (a, ∞).
Edth-a differential operator on the sphere
Authors: Eastwood Michael , Tod Paul
Published: November 2008

Introduction. In (9) Newman and Penrose introduced a differential operator which they denoted ð, the phonetic symboledth. This operator acts onspin weighted, orspinandconformally weightedfunctions on the two-sphere. It turns out to be very useful in the theory of relativity via the isomorphism of the conformal group of the sphere and the proper inhomogeneous Lorentz group (11, 4). In particular, it can be viewed (2) as anangular momentum lowering operatorfor a suitable representation of SO(3) and can be used to investigate the representations of the Lorentz group (4). More recently, edth has appeared in thegood cut equationdescribing Newman'sℋ-spacefor anasymptotically flatspace-time (10). This development is closely related to Penrose's theory oftwistorsand, in particular, toasymptotic twistors(14).
The singularities of H-space
Authors: Tod K. P.
Published: November 2008

The non-linear graviton construction of Penrose (9) and the ℋ-space construction of Newman (6) are two complementary techniques for constructing complex four dimensional space-times with quadratic metric and anti-self-dual curvature tensor.

In the former, the space-time is the space of holomorphic sections of a complex fibre space obtained by deforming part of flat twistor space. In the latter the space-time is the space of regular solutions of a differential equation, the good cut equation.

Pathologies arise in the non-linear graviton construction when the normal bundle of a holomorphic section changes. This is reflected in the ℋ-space construction by a change in the character of the solutions of the linearized good cut equation, the Newman equation.
On a theorem of Wielandt concerning simply primitive groups
Authors: Jones G. A. , Soomro K. D.
Published: November 2008

Let G be a simply primitive permutation group on a set Ω of order p2, where p is a prime (necessarily odd). In theorem 27·2 of (9), Wielandt states without proof:

Theorem A. (i) ¦G¦ is not divisible by p3;

(ii) if G has a pair of Sylow p-subgroups with nontrivial intersection, then G has an imprimitive subgroup of index 2 which is the direct product of two intransitive groups.
Vector bundles and complex polarizations
Authors: Woodhouse Nicholas
Published: November 2008

This paper discusses the local and global geometry of coisotropic foliations and complex polarizations on symplectic manifolds and draws attention to an analogy between coisotropic foliations and Hermitian vector bundles, in which connections and characteristic classes are modelled by objects in symplectic geometry.