bluuebunny/crossref_metadata_2025 huggingface dataset:
The Poincare Polynomial of a Symmetric Product
Authors: Macdonald I. G.
Published: November 2008

Let X be a compact polyhedron, Xn the topological product of n factors equal to X. The symmetric group Sn operates on Xn by permuting the factors, and hence if G is any subgroup of Sn we have an orbit space Xn/G obtained by identifying each point of Xn with its images under G. In particular Xn/Sn is the nth symmetric product of X, and if G is a cyclic subgroup of order n then Xn/G is the nth cyclic product of X.
On stability of parallel flow of an incompressible fluid of variable density and viscosity
Authors: Drazin P. G.
Published: November 2008

Some aspects of generation of water waves by wind and of turbulence in a heterogeneous fluid may be described by the theory of hydrodynamic stability. The technical difficulties of these problems of instability have led to obscurities in the literature, some of which are elucidated in this paper. The stability equation for a basic steady parallel horizontal flow under the influence of gravity is derived carefully, the undisturbed fluid having vertical variations of density and viscosity. Methods of solution of the equation for large Reynolds numbers and for long-wave disturbances are described. These methods are applied to simple models of wind blowing over water and of fresh water flowing over salt water.
A property of certain even integral functions
Authors: Monticelli F.
Published: November 2008

In some problems where sine, cosine or Bessel functions are involved (e.g. (l), p. 35 et seq.), one has to satisfy certain initial conditions so that one obtains relations of the type

where t is a real parameter such that |t| < 1, and ±αn(n = 1, 2, …) are real simple zeros of an even or odd integral function f(z), being z = x + iy. An's are coefficients to be determined and, of course, ±ω is a real number that is not a zero of f(z).
Stable operations in mod p K-theory
Authors: Maunder C. R. F.
Published: November 2008

There comes a time in the development of a cohomology theory when a discussion of cohomology operations becomes necessary. In the case of complex K-theory, the subject of the present paper, such operations have of course already been investigated by Adams (see (2)), so that any further discussion might appear superfluous. Powerful as Adams's results are, however, the situation still leaves something to be desired: it is not known just what other operations can be defined in K-theory, and it is an inconvenience from the standpoint of stable homotopy theory that Adams's operations are not themselves stable.
Lipschitz classes on 0-dimensional groups
Authors: Walker P. L.
Published: November 2008

1. Let G be a compact metric 0-dimensional Abelian group. Its dual or character group Γ is a discrete countable torsion group. We denote elements of G by x, of Γ by y, the value of the character y at x by (x, y), and the Fourier transform of f by
Gaps and steps for the sequence nθ mod 1
Authors: Slater Noel B.
Published: November 2008

For a given θ(0 < θ < 1) and r = 0, 1, 2, 3,… there are two related problems on the fractional parts {rθ}, namely:

the gap problem: for any φ (0 < φ < 1) to determine the gaps between the successive r for which

the step problem: for the set

rearranged in ascending order, to determine the steps into which the interval [0,1] is thereby partitioned. There are subsidiary problems of the numbers of gaps or steps of the various lengths, and also of the permutation of 1,…, N when (2) is arranged in order.
On the differential equation governing the rear stagnation point in magnetohydrodynamics and Goldstein's ‘backward’ boundary layers
Authors: Leibovich S.
Published: November 2008

Existence and uniqueness proofs for a boundary-value problem associated with a magnetohydrodynamic Falkner–Skan equation are presented. Relevant special cases of the problem herein considered include the magnetohydrodynamic rear stagnation point flow, and the non-magnetic ‘backward boundary layers’ of Goldstein(2).
On a generalized Stieltjes transform
Authors: Kapoor V. K.
Published: November 2008

The Stieltjes transform is a consequence of the iteration of two Laplace transforms, namely if

then is referred to as the Stieltjes transform of g(t).
A note on the real zeros of the basic confluent hypergeometric function
Authors: Mishra Ramadhar
Published: November 2008

Some years back, Slater (4) discussed the approximations, based on the expansion in series, for the cases 1F1(a; b; x) = 0, when either of b and x or a and x are fixed. These approximations were based essentially on the well-known Newton's method of approximation and were helpful in the numerical evaluation of the small real zeros of the confluent hypergeometric function 1F1(a; b; x;). In this note, we deal with the corresponding problem for the basic confluent hypergeometric function 1Φ1(a; b; x;).
An explicit determination of the empty space-times on which the wave equation satisfies Huygens' principle
Authors: McLenaghan R. G.
Published: November 2008

The validity of Huygens' principle in the sense of Hadamard's ‘minor premise’ is investigated for scalar wave equations on curved space-time. A new necessary condition for its validity in empty space-time is derived from Hadamard's necessary and sufficient condition using a covariant Taylor expansion in normal coordinates. A two component spinor calculus is then employed to show that this necessary condition implies that the plane wave space-times and Minkowski space are the only empty space-times on which the scalar wave equation satisfies Huygens' principle.
Whittaker's method for systems of non-linear equations
Authors: Nicolaides R. A.
Published: November 2008

This paper is concerned with a generalization to n dimensions of a method for obtaining an approximation to a root of an algebraic or transcendental equation. We prove a convergence theorem for the method, and show how the method becomes of considerable practical utility in the case when the Jacobian matrix of the functions constituting the system is strictly diagonally dominant in a neighbourhood.
Longitudinal wave propagation in an infinite isotropic elastic plate with a penny-shaped crack
Authors: Paul H. S.
Published: November 2008

The stress distribution, subject to a constant pressure over the entire surface of a penny-shaped crack is discussed by Sneddon(4). Recently, Robertson (3) has considered the diffraction of a plane longitudinal wave by a penny-shaped crack on a semi-infinite elastic solid. In the present analysis, the propagation of longitudinal wave in an infinite isotropic elastic plate with a penny-shaped crack in the middle has been investigated. The plane longitudinal wave is moving in the positive direction of z-azis and is impinging on the surface of the penny-shaped crack. The dual integral equation technique of Noble(l) is utilized to solve the mixed boundary-value problem. The analysis closely follows the method used in the author's previous paper (2). The vertical displacement is analysed numerically.
Determination of phase velocities of a circular helix using the method of internal constraints
Authors: Lincoln J. W. , Nikkhah M. , Volterra E.
Published: November 2008

A procedure is described to determine phase velocities of a circular helix. Computations are made for three helix configurations using the method of internal constraints. The results so obtained are compared with the results obtained by using the elementary theory proposed by J. H. Michell, and the extended elementary theory proposed by A. B. Basset.
The combinatorial structure of the Hughes plane
Authors: Room T. G.
Published: November 2008

The main purpose of this paper is to describe the construction of an incidence table for the Hughes plane of order q2. To do this it is necessary first to construct a table of the same pattern for the Galois plane, and this requires the expression in terms of explicit matrices of the Singer cyclic group of the plane as the composition of cyclic groups of orders q2 + q + 1 and q2 − q + 1. The two tables constructed have some combinatorial properties of possible interest.

In the Galois plane of order q2 there are two types of polarities, one with q2 + 1 singular points on a conic, and the other with q3 + 1 singular points on the Hermitian analogue of a conic. Correspondingly in the Hughes plane there are polarities with ½(q3 + q + 2) and with ½(q3 + 2q2 − q + 2) singular points.

The paper concludes with the complete incidence table for the Hughes plane of order 25, together with the data necessary for its construction.

The Hughes plane and some of its properties are described in Hughes(1), Zappa(2), Rosati(3) and Ostrom(4). The property which enables the Hughes plane to be most easily constructed from the corresponding Galois plane, and which will be used in this paper, is to be found in Room (5).
The calculation of matrix elements of three particle operators in α-particle nuclei
Authors: Nash G. F.
Published: November 2008

A method is given for evaluating matrix elements of three particle operators in many nucleon states of [44…4] symmetry.
The union of sets of interpolation for general algebras
Authors: Körner T. W.
Published: November 2008

The union of 2 disjoint sets of interpolation need not be of interpolation even if every finite subset is of interpolation with bounded constant.
A note on the normalizer condition
Authors: Hartley B.
Published: November 2008

The first example of a non-trivial group satisfying the normalizer condition but having trivial centre was obtained by Heineken and Mohamed (1). In fact, the group they constructed satisfies the stronger condition that all its proper subgroups are subnormal and nilpotent. In this note we use a construction suggested by their approach to obtain such a group G as a subgroup of the restricted wreath product Cp ≀ Cp∞. The fact that G is given as a subgroup of a well known group makes it rather easier to investigate its properties, and in particular to see that its centre Z(G) is trivial.
A note on (k, n)-arcs
Authors: Wilson B. J.
Published: November 2008

In a finite projective plane S2, q of order q, a (k, n)-arc is a set of k points, of which some n, but no n + 1 are collinear. Barlotti (1) proved that if (n, q) = 1 then
Approximating sequences and Hausdorff measure
Authors: Gardner R. J.
Published: November 2008

Approximating sequences have been extensively studied in many branches of mathematics, for example, in number theory (approximating real numbers by rationals) and in numerical analysis (approximations to functions by polynomials). In (1), A. Hyllengren introduced a type of approximating sequence ‘majorizing sequences’ which he used in solving a problem in complex analysis. In this note we study a very similar concept, which is general enough to be applicable to any separable metric space, and which turns out to have strong connexions with the theory of Hausdorff measures (as did Hyllengren's majorizing sequences).
The summation of Fourier series by logarithmic means
Authors: Kwee B.
Published: November 2008

Let {sn} be the sequence of partial sums of the series and let

If , we say that the series is summable (l) to s, and if , the series is said to be summable (l2) to s. It is clear that .
Complex bundles with two sections
Authors: Rees Elmer
Published: November 2008

Atiyah(2) defined the geometrical dimension of an element to be less than k + 1 (g dim (x) ≤ k) if there is a k-dimensional bundle over X whose stable equivalence class is x. If ξ is an n-plane complex bundle over X, we say that it has r sections if there is an (n − r)-plane bundle η such that ξ is isomorphic to η ⊕ εr where εr is the trivial r-plane bundle over X. If X has dimension 2n or less, then ξ has r sections if and only if g dim (ξ − n) ≤ n − r.
High frequency diffraction by a soft circular disc
Authors: Newby J. C.
Published: November 2008

The problem is governed by a Jones integral equation and the solution is shown to depend upon a single function which occurs naturally after a contour deformation has produced extensive cancellation in the work. The far-scattered field off the axis of symmetry is found in detail, yielding terms which are believed to be new.
Convex series
Authors: Jameson G. J. O.
Published: November 2008

1. Introduction. Let A be a subset of a Hausdorff topological linear space. By a convex series of elements of A we mean a series of the form where an∈A and λn ≥ 0 for each n, and . We say that A is:

(i) CS-closed if it contains the sum of every convergent convex series of its elements;

(ii) CS-compact if every convex series of its elements converges to a point of A (this bold terminology is chosen because sets satisfying this condition turn out to have properties analogous to those of compact sets).
Corner regions in the asymptotic solution of ε∇2u = ∂u/∂y with reference to MHD duct flow
Authors: Cook L. Pamela , Ludford G. S. S. , Walker J. S.
Published: November 2008

In the asymptotic solution of the title equation for simple boundary data on the rectangle 0 ≤ x ≤ l, 0 ≤ y ≤ 1 the bottom corners are not passive, as has previously been supposed, but in fact generate the parabolic side layers at x = o, l. The top corners, though passive, lead to unconventional elliptic quarter-plane problems. Implications for the MHD duct problem are indicated.
Handle-theory and engulfing
Authors: Buoncristiano S.
Published: November 2008

This paper is concerned with the engulfing of a polyhedron X from one end δ_ W of a cobordism (W, δ_ W, δ+W). In the case where the pair (W, δ_ W) is highly connected this version of Engulfing is dealt with in Rourke and Sanderson (2) by combining the method used in the proof of the h-cobordism theorem (eliminating handles) with a simple procedure involving handle-moves. A handle-move is an ambient isotopy which shrinks a handle H onto a small regular neighbourhood of its fibre D. Thus, if a closed polyhedron misses D, a handle-move can be applied to cause X to slip off H.
On a conjecture of Littlewood
Authors: Hall R. R.
Published: November 2008

Let K be a hyper-cube of side 2 in n dimensions where n is any positive integer, and H an arbitrary, fixed (n − 1)-dimensional hyper-plane through the centre of K.
On a quantum mechanical model for a maser I
Authors: Hasler Martin
Published: November 2008

The model is closely connected with a model by Lamb and Scully (10). Atoms described as two-level systems, initially in an incoherent superposition of the two levels, interact successively during a time T with an electromagnetic field of which only one mode is taken into consideration. In the limit as infinitely many atoms have interacted, it is shown that the field either approaches a thermal distribution or is excited to arbitrarily high Photon numbers according to whether or not the lower level of the atoms is initially more probable than the upper level. It is also shown that in any case the correlations between pure Photon number states converge to 0. If the atoms are initially in the upper level it is proved that the Photon number grows roughly as the square root of the number of atoms that have interacted. Throughout the discussion number-theoretical properties of T play a disturbing role. The last mentioned result in fact depends on a sharp (but arbitrary) value for T and is therefore disqualified for physical interpretation.
Operations on K-theory of torsion-free spaces
Authors: Adams J. F. , Hoffman P.
Published: November 2008

The object of this paper is to study stable operations on the K-cohomology of torsion-free spaces and spectra. Such a study was begun by Novikov ((4), § 8). We hope the present work may among other things serve to clarify some of the obscure points in this part of his work.
Infinite loop maps in geometric topology
Authors: Madsen I. , Snaith V. , Tornehave J.
Published: November 2008

0. Introduction. Let and be connected cohomology theories on the category of all CW spaces. In this paper we examine the relationship between natural homomorphisms (stable operations) of degree zero from to and natural homomorphisms from to in a number of cases of particular interest in topology – foremost the K-theories and the bundle theories classifying ‘surgery problems’. There are two parts to this. On the one hand we ask under what conditions a homomorphism from to extends to a stable operation and on the other hand when is a stable operation determined by its restriction to .
The associated graded ring of an integral group ring
Authors: Passi Inder Bir S. , Vermani Lekh Raj
Published: November 2008

Let G be an Abelian group, the symmetric algebra of G and the associated graded ring of the integral group ring ZG, where (AG denotes the augmentation ideal of ZG). Then there is a natural epimorphism (4)

which is given on the nth component by

In general θ is not an isomorphism. In fact Bachmann and Grünenfelder(1) have shown that for finite Abelian G, θ is an isomorphism if and only if G is cyclic. Thus it is of interest to investigate ker θn for finite Abelian groups. In view of proposition 3.25 of (3) it is enough to consider finite Abelian p-groups.
Syntopogenous preordered spaces (II)
Authors: Burgess D. C. J. , Fitzpatrick M.
Published: November 2008

In (1) the authors introduced the idea of a syntopogenous preordered space (SPS), thereby generalizing simultaneously the concepts of ‘uniform preordered space’ and ‘proximity preordered space’. In the case of a classical topology the SPS condition was seen to be equivalent to the condition, ‘convex plus T1-preordered’. In the present paper we show that, in the general case, the SPS condition may be characterized as the sum of two generalizations of the logically independent conditions of convexity and T1-preorderedness.
On the generation of short internal waves by cylinders oscillating at the surface separating two infinite liquids
Authors: Gorgui M. A. , Kassem S. E.
Published: November 2008

This paper is an investigation of the short-wave asymptotic motion due to a cylinder heaving at the surface separating two infinite liquids. The cylinder is assumed to have a smooth cross-section of an arbitrary shape that intersects the surface of separation at right angles. A non-rigorous argument is used to get the asymptotic expansion of the velocity potentials of the motion in the two liquids in the far field. The asymptotic evaluation of the coefficients describing wave-making, virtual-mass and damping are also obtained in terms of the limit potentials of the motions in the two liquids.
Cushions, cigars and diamonds: an area-perimeter problem for symmetric ovals
Authors: Croft H. T.
Published: November 2008

P. R. Scott (1) has asked which two-dimensional closed convex set E, centro-symmetric in the origin O, and containing no other Cartesian lattice-point in its interior, maximizes the ratio A/P, where A, P are the area, perimeter of E; he conjectured that the answer is the ‘rounded square’ (‘cushion’ in what follows), described below. We shall prove this, indeed in a more general setting, by seeking to maximize

where κ is a parameter (0 < κ < 2); the set of admissible E is those E centro-symmetric in 0 that do not contain in their interior certain fixed lattice-points. There are two problems, the unrestricted one , where there is no given upper bound on A (it will become apparent that this problem only has a finite answer when κ ≥ 1) and the restricted one , when one is given a bound B and we must have A ≤ B. Special interest attaches to the case B = 4, both because of Minkowski's theorem: any E symmetric in O and containing no other lattice-point has area at most 4; and because it turns out that it is a ‘natural’ condition: the algebraic expressions simplify to a remarkable extent. Hence in what follows, the ‘restricted case ’ shall mean A ≤ 4.
A note on homology surgery and the Casson–Gordon invariant
Authors: Akbulut Selman
Published: November 2008

We give a ‘picture’ proof to a theorem of M. Freedman (2) which shows the failure of the 4-dimensional homology surgery theory and the homology splitting theorem. Our proof employs the language of the framed links (3) and it involves calculating Casson-Gordon invariant of a certain algebraically slice knot. We use framed links to represent 1-connected 4-manifolds with boundary by attaching 2-handles along them onto B4 via the framings. We adapt the notation ≈ for diffeomorphisms, and for diffeomorphisms between the boundaries of manifolds. Here manifolds refer to smooth manifolds.
A continuum model of many-body interactions in a perfect crystal
Authors: Parry G. P.
Published: November 2008

Theoretical and numerical calculations of the mechanical properties of single crystals usually presuppose pairwise interactions between the atoms of the lattice. It follows from this assumption that the Cauchy relations hold in respect of the Green measure of strain. Here we account for many-body interactions between the atoms of the lattice. The main result is that a continuum model of n-body interactions must possess completely symmetric (n − 1)th order Green strain moduli. Existing strain energy functions, used to model crystal elasticity, are thereby given some physical significance.
On fibre spaces and nilpotency. II
Authors: James I. M.
Published: November 2008

1. Introduction. Let X be a space and let E be a fibre space over E. A fibre-preserving map f: E → E determines, for each point x ∈ X, a map fx: Ex → Ex of the fibre over x. In a previous note (3) the situation was considered where fx is null-homotopic, for all x. In the present note we turn our attention to the situation where fx is homotopic to the identity on Ex, for all x ∈ X. If X admits a numerable categorical covering (as when X is an ANR) then such a fibre-preserving map f is a fibre homotopy equivalence, by the well-known theorem of Dold(1). Then the set Φ1(E) of fibre homotopy classes of such maps forms a normal subgroup of the group Φ*(E) of fibre homotopy classes of fibre homotopy equivalences. The purpose of this note is to prove

Theorem 1.1. Let X be a paracompact space of finite category. Let E be a fibre bundle over X of which the fibres are compact and path-connected ANR's. Then the Φ*(E)-growp Φ1(E) is Φ*(E)-nilpotent of class ≤ cat X.
A note on the uniqueness of small-amplitude water waves travelling in a region of varying depth
Authors: Fitz-Gerald G. F. , Grimshaw R. H. J.
Published: November 2008

The two-dimensional, irrotational, linear theory used in the investigation of the propagation of monochromatic surface gravity waves in a region of varying depth is considered. Uniqueness of the velocity potential is established for bottom profiles satisfying certain convexity conditions. These include the majority of profiles of physical interest.
Stable range in C*-algebras
Authors: Robertson A. Guyan
Published: November 2008

A unital C*-algebra A is said to have unitary 1-stable range (8) if for all pairs a, b of elements of A satisfying aA + bA = A there exists a unitary u in A such that a + bu is invertible. This concept is somewhat stronger than the usual stable range condition of algebraic K-theory ((3), chapter V). Handelman(8) shows among other things that finite AW*-algebras have unitary 1-stable range and uses this fact to study the algebraic K1 of a finite AW*-algebra. We prove below that a unital C*-algebra has unitary 1-stable range if and only if its group of invertible elements is dense. In addition we give some consequences of this fact and consider the related question of (unitary) polar decomposition in C*-algebras.
Some properties of the continued fraction expansion of (m/n) e1/q
Authors: Matthews K. R. , Walters R. F. C.
Published: November 2008

Introduction. Continued fractions of the form are called Hurwitzian if b1, …, bh, are positive integers, ƒ1(x), …, ƒk(x) are polynomials with rational coefficients which take positive integral values for x = 0, 1, 2, …, and at least one of the polynomials is not constant. f1(x), …, fk(x) are said to form a quasi-period.
Stable proofs of stable splittings
Authors: Cohen Ralph L.
Published: November 2008

V. P. Snaith's theorem giving splittings of the suspension spectrum of ΩnΣnX for a connected space X (8) has been exploited in several areas of homotopy theory. (See (2), (5) and (6), for instance.) Although this is a theorem about spectra, Snaith's proof (as well as a subsequent proof of Cohen, May, and Taylor (4)) proceeds unstably, on the space level. While the methods employed in these proofs are useful, they are somewhat complicated. The purpose of this note is to give a simple proof of this theorem, using only spectrum level arguments.
Subspaces with property (b) in locally convex spaces of quasi-barrelled type
Authors: Tsirulnikov Bella
Published: November 2008

A subspace G of a locally convex space E has property (b) if for every bounded set B of E the codimension of G in the linear hull of G ∪ B is finite, (5). Extending the results of (5) and (14), we prove that, if the strong dual of E is complete, then subspaces with property (b) inherit the following properties of E: σ-evaluability, evaluability, the property of being Mazur, semibornological and bornological. We also prove that a dense subspace with property (b) of a Mazur space is sequentially dense, and of a semibornological space – dense in the sense of Mackey (locally dense, following M. Valdivia).
NoncommutativeLp-spaces with 0 <p< 1
Authors: Saito Kichi-Suke
Published: November 2008

The noncommutative Lp-spaces (1 ≤p≤ ∞) of unbounded operators associated with a regular gauge space (a von Neumann algebra equipped with a faithful normal semifinite trace) are studied by many authors ((4), (5) and (7)). It is well-known that the noncommutativeLp-spaces (1 ≤P< ∞) are Banach spaces and the dual ofLpisLq(1 ≤p< ∞, 1/p+ 1/q= 1) by means of a Radon-Nikodym theorem.
A pencil of specialized canonical curves
Authors: Edge W. L.
Published: November 2008

1. The present paper continues the investigation begun in (5). This earlier paper concluded with the description of a pencil P of canonical curves of genus 6 on a del Pezzo quintic surface F, but had first to describe the relevant features of the mapping of prime sections of F on a plane p by cubic curves circumscribing a quadrangle in p. It remains now to consider F and P in their ambient [5] in relation to G, the group, isomorphic to the symmetric group of degree 5, of self-projectivities of F.
Hadamard–Landau inequalities in uniformly convex spaces
Authors: Partington J. R.
Published: November 2008

The inequality

for fεLp(− ∞, ∞)or Lp(0, ∞) (1≤p ≤ ∞), and its extension

for T an Hermitian or dissipative linear operator, in general unbounded, on a Banach space X, for xεX, have been considered by many authors. In particular, forms of inequality (1) have been given by Hadamard(7), Landau(15), and Hardy and Little-wood(8),(9). The second inequality has been discussed by Kallman and Rota(11), Bollobás (2) and Kato (12), and numerous further references may be found in the recent papers of Kwong and Zettl(i4) and Bollobás and Partington(3).
Some James numbers of Stiefel manifolds
Authors: Ōshima Hideaki
Published: November 2008

The purpose of this note is to determine some unstable James numbers of Stiefel manifolds. We denote the real numbers by R, the complex numbers by C, and the quaternions by H. Let F be one of these fields with the usual norm, and d = dimRF. Let On, k = On, k(F) be the Stiefel manifold of all orthonormal k–frames in Fn, and q: On, k → Sdn−1 the bundle projection which associates with each frame its last vector. Then the James number O{n, k} = OF{n, k} is defined as the index of q* πdn−1(On, k) in πdn−1(Sdn−1). We already know when O{n, k} is 1 (cf. (1), (2), (3), (13), (33)), and also the value of OK{n, k} (cf. (1), (13), (15), (34)). In this note we shall consider the complex and quaternionic cases. For earlier work see (11), (17), (23), (27), (29), (31) and (32). In (27) we defined the stable James number , which was a divisor of O{n, k}. Following James we shall use the notations X{n, k}, Xs{n, k}, W{n, k} and Ws{n, k} instead of OH{n, k}, , Oc{n, k} and respectively. In (27) we noticed that O{n, k} = Os{n, k} if n ≥ 2k– 1, and determined Xs{n, k} for 1 ≤ k ≤ 4, and also Ws{n, k} for 1 ≤ k ≤ 8. On the other hand Sigrist (31) calculated W{n, k} for 1 ≤ k ≤ 4. He informed the author that W{6,4} was not 4 but 8. Since Ws{6,4} = 4 (cf. § 5 below) this yields that the unstable James number does not equal the stable one in general.
Lacunary theorems for Cesàro means
Authors: Kuttner B. , Maddox I. J.
Published: November 2008

Suppose that (ni) = (n1, n2,…) and (mi) are infinite sequences of positive integers with ni < mi < ni+1. It is well-known and easily proved that, if a series σak is (C, 1) summable to s and has lacunae (ni, mi) such that ak = 0 (ni < k < mi) with

then where
Non-oscillation and asymptotic properties for a class of forced second-order nonlinear equations
Authors: Erbe Lynn H. , Rao V. Sree Hari , Rao K. V. V. Seshagiri
Published: November 2008

Consider the second order nonlinear equation

where, r, p, f ε C ([a, + ∞), R), R = (–∞, +∞) with r > 0 and λ > 0 is the quotient of odd integers. One of the prototypes of this class of equations is the generalized Thomas-Fermi of Emden-Fowler equation
Corrigenda
Authors: Friedlander F. G.
Published: November 2008

In this paper [1], two corrections must be made.

(1) In the second member of (27), m is to be replaced by m + 1; this agrees with (18), and gives the correct result.

(2) Theorem 2 is mis-stated.
On groups with real valued length function
Authors: Hoare A. H. M.
Published: November 2008

Groups with length functions were introduced by Lyndon [5], and those with integer valued length were shown by Chiswell [2] to be equivalent to groups acting on trees, investigated by Serre[6]. Watatani[7] showed that for a group with integer valued length function the edges of the tree form an orthonormal set in a Hilbert space on which the group acts as a group of isometries.
On the limit distributions of lightly trimmed sums
Authors: Mori Toshio
Published: November 2008

Let Xnn ≥ 1, be i.i.d.r.v.'s and Sn = X1+…+Xn. Let be Sn minus the r terms of largest absoluete value. Maller proved that if coverages in distribution to N(0, 1) then so does (Sn/bn)−an, assuming that Xn have a continuous symmetric distribution. We show that his resul;t is true without these extra assumptions. Some related results are also given.
On complex vector bundles on rational threefolds
Authors: BẮnicẮ Constantin , Putinar Mihai
Published: November 2008

It is known [14] that every topological complex vector bundle on a smooth rational surface admits an algebraic structure. In [10] one constructs algebraic vector bundles of rank 2 on with arbitrary Chern classes c1, c2 subject to the necessary topological condition c1 c2 = 0 (mod 2). However, in dimensions greater than 2 the Chern classes don't determine the topological type of a vector bundle. In [2] one classifies the topological complex vector bundles of rank 2 on and one proves that any such bundle admits an algebraic structure.
Riesz and Valiron means and fractional moments
Authors: Bingham N. H. , Tenenbaum G.
Published: November 2008

We shall be concerned here with two classical families of summability methods, and with links between them, together with applications in probability theory and elsewhere.
A note on trigonometric sums in several variables
Authors: Odoni R. W. K.
Published: November 2008

In [3, 4] we showed how the use of a random-walk analogue can be made to yield non-trivial information about the behaviour of certain trigonometric sums in one variable. Our aim here is to show how our method can be adapted to yield similar results for a broad class of trigonometric sums in several variables. Let

be a polynomial in v independent variables with integral coefficients. We choose integers n ≥ 0, d ≥ 1 and p ≥ 2 with p prime, and assume that f(x) has total degree ≤ d + 1. We shall consider the problem of obtaining non-trivial upper bounds for the absolute value of sums of the type

where P = {1, 2, …, p} and f is non-constant.
The packing measure of rectifiable subsets of the plane
Authors: Taylor S. James , Tricot Claude
Published: November 2008

It is usual to define Lebesgue outer measure in ℝ by using economical coverings by a sequence of open intervals. We start by outlining a different definition which gives the same answer for a bounded measurable E ⊂ ℝ. Put

Then λ0 defines a pre-measure, but is not an outer measure because it is not countably sub-additive. However it leads to an outer measure by defining

and it can be proved directly that λ is just another definition of Lebesgue outer measure. We do not give the details of this proof as it can also be deduced as a corollary of the main results in the present paper.
On a criterion for hyperplane sections
Authors: Bǎdescu Lucian
Published: November 2008

Throughout this paper we shall fix an algebraically closed field k. Consider the following:

Problem. Let (Y, L) be a normal polarized variety over k, i.e. a normal projective variety Y over k together with an ample line bundle L on Y. Then one may ask under which conditions the following statement holds:

(*) Every normal projective variety X containing Y as an ample Cartier divisor such that the normal bundle of Y in X is L, is isomorphic to the projective cone over Y.
De Bruijn enumeration applied to some genetical problems
Authors: Lloyd E. Keith
Published: November 2008

Several authors have considered relationships between individuals specified by the genes which they have in common, where two genes are regarded as the same if and only if they are identical by descent from some common ancestor. In particular Thompson [6] considered n pairs of genes, one pair from each of n individuals, and studied the number Nn of gene identity states, the number Mn, k of such states in which there are exactly k distinct genes and the number Dn of genetically distinct states (see Definitions 2 and 4 below). In this type of work the nature of the genes themselves is of no interest, but only which genes are identical to which, and precise definitions are best given in terms of orbits under group actions. Thompson did use some group theory in her work, but not the full machinery of Redfield-Pólya-de Bruijn enumeration.
Path sensitivity of material response at intrinsic eigenstates in classical plasticity
Authors: Hill R.
Published: November 2008

Categories of rigid/plastic response are investigated along arbitrary paths of homogeneous deformation. Successive yield surfaces are presumed to be convex and self-similar relative to certain canonical measures of stress and finite strain. After a general analysis attention is focussed on path-sensitive response in configurations where the canonical rate of hardening vanishes. The analysis is subsequently extended to other types of eigenstate, especially those associated with actual or prospective loading devices. The results are illustrated in the context of formability tests on sheet metal.
General function spaces, products and continuous lattices
Authors: Isbell John
Published: November 2008

The compact–open topology for function spaces is usually attributed to R. H. Fox in 1945 [16]; and indeed, there is no earlier publication to attribute it to. But it is clear from Fox's paper that the idea of the compact–open topology, and its notable success in locally compact spaces, were already familiar. The topology of course goes back to Riemann; and to generalize to locally compact spaces needs only a definition or two. The actual contributions of Fox were (1) to formulate the partial result, and the problem of extending it, clearly and categorically; (2) to show that in separable metric spaces there is no extension beyond locally compact spaces; (3) to anticipate, partially and somewhat awkwardly, the idea of changing the category so as to save the functorial equation. (Scholarly reservations: Fox attributes the question to Hurewicz, and doubtless Hurewicz had a share in (1). As for (2), when Fox's paper was published R. Arens was completing a dissertation which gave a more general result [1] – though worse formulated.)
A Tauberian theorem concerning weighted means and power series
Authors: Borwein David , Meir Amram
Published: November 2008

Suppose throughout that {pn} is a sequence of non-negative numbers with p0 > 0, that

and that

Let {sn} be a sequence of real numbers.
The isomorphism class of a set of lattices
Authors: Wilson S. M. J.
Published: November 2008

Let R be a Dedekind domain with field of quotients K. Let A be a finite-dimensional K-algebra. We consider isomorphism classes and genera in a category whose objects are indexed sets of full R-lattices in some ambient A-module and whose morphisms are the A-homomorphisms of the ambient A-modules which map each lattice into its corresponding lattice. We find conditions under which the stable A-isomorphism class of one particular lattice in an indexed set will determine the stable class of the indexed set within its genus. We apply our methods to show that if L/K is a tame Galois extension of algebraic number fields then the stable isomorphism class of the set of ambiguous ideals in L considered as Galois modules over K is determined by the class of the ring of integers in L together with the inertia subgroups and their standard representations over the respective residue fields of R.
Schur indices and irreducible character degrees in finite solvable groups
Authors: Gow R.
Published: November 2008

Let G be a finite group and let Irr(G) denote the set of complex irreducible characters of G. Various authors have investigated the question of how information about the degrees of the characters in Irr (G) can provide information about the structure of G. Chapter 12 of [2] gives a survey of a number of results arising from such questions. Two well-known examples of theorems that relate character degrees and group structure are those due to Thompson (12·2 in [2]) and Itô (12.34 in [2]), which we recall here.
Bicompleteness of the fine quasi-uniformity
Authors: Künzi Hans-Peter A. , Ferrario Nathalie
Published: November 2008

A characterization of the topological spaces that possess a bicomplete fine quasi-uniformity is obtained. In particular we show that the fine quasi-uniformity of each sober space, of each first-countableT1-space and of each quasi-pseudo-metrizable space is bicomplete. Moreover we give examples ofT1-spaces that do not admit a bicomplete quasi-uniformity.

We obtain several conditions under which the semi-continuous quasi-uniformity of a topological space is bicomplete and observe that the well-monotone covering quasiuniformity of a topological space is bicomplete if and only if the space is quasi-sober.
Stability of periodic solutions near a collision of eigenvalues of opposite signature
Authors: Bridges Thomas J.
Published: November 2008

Some general observations about stability of periodic solutions of Hamiltonian systems are presented as well as stability results for the periodic solutions that exist near a collision of pure imaginary eigenvalues. Let I = ∮ p dq be the action functional for a periodic orbit. The stability theory is based on the surprising result that changes in stability are associated with changes in the sign of dI / dw, where w is the frequency of the periodic orbit. A stability index based on dI / dw is defined and rigorously justified using Floquet theory and complete results for the stability (and instability) of periodic solutions near a collision of pure imaginary eigenvalues of opposite signature (the 1: – 1 resonance) are obtained.
Extension of a theorem of J. H. Grace to transcendental entire functions
Authors: Clunie J. , Rahman Q. I.
Published: November 2008

The following result is due to J. H. Grace (see [4], p. 356, also see [9], §3).

Theorem A. if p is a polynomial such that p(–1) = p(1) then the derivative p' has a zero in each of the half-planes {Rez ≤0} and {Rez ≥0}.
On the nilpotence order of β1
Authors: Lee Chun-Nip , Ravenel Douglas C.
Published: November 2008

For p > 2, is the first positive even-dimensional element in the stable homotopy groups of spheres. A classical theorem of Nishida[1] states that all elements of positive dimension in the stable homotopy groups of spheres are nilpotent. In fact, Toda [4] proved . For p = 3 he showed that while . In [2] the second author computed the first thousand stems of the stable homotopy groups of spheres at the prime 5. One of the consequences of this computation is that while .
Permissible symmetries of coupled cell networks
Authors: Ashwin Peter , Stork Peter
Published: November 2008

We consider coupled sets of identical cells and address the problem of which symmetries are permissible in such networks. For example, n linearly coupled cells with one independent variable in each cell cannot be constructed with the symmetry group An, the alternating group on n symbols. Using a graphical technique, we show that it is possible to construct cell networks with any desired finite group of symmetries. In particular, we show that any subgroup of Sn can be realized as the symmetries of a group of n cells. Special forms of coupling (especially low order polynomial coupling) are shown to restrict the possible symmetries. We give some upper and lower bounds for the degree of polynomial required to realize several classes of subgroups of Sn.
Quartic curves in characteristic 2
Authors: Wall C. T. C.
Published: November 2008

Simple singularities in positive characteristic

Simple singularities in positive characteristic have been discussed by many authors, and the article [5] in particular establishes the subject on a firm footing. In it a simple, or ‘ADE’ singularity is defined by a list of normal forms and it is shown that the following conditions on a singularity are equivalent: (i) it is simple, (ii) it has finite deformation type, (iii) it has finite Cohen-Macaulay module type. Moreover, the normal forms for surface singularities coincide with the earlier list of Artin [1] and those for curves with the list of [9]: in those papers further characterizations were obtained.
Discrete coactions on Hilbert C*-modules
Authors: Ng Chi-Keung
Published: November 2008

In this paper, we will investigate discrete coactions on Hilbert C*-modules. In particular, we obtain a one-to-one correspondence between Hilbert C*-modules with discrete coactions and Hilbert C*-modules over the crossed products of the original C*-algebras which satisfies some nice properties (see 3·6 and 3·7). Then we will give some applications of this correspondence in the last three sections.

In Section 2, we give some results about discrete coactions on Hilbert C*-modules which mainly correspond to those about discrete coactions on C*-algebras (see [7]).
Dehn surgery and essential annuli
Authors: Hayashi Chuichiro
Published: November 2008

In this paper we consider Dehn surgery and essential annuli whose two boundary components are in distinct components of the boundary of a 3-manifold.

Let Nl be an orientable 3-manifold with boundary, Kl a knot in Nl, and N2 the 3-manifold obtained by performing γ-Dehn surgery Kl. In detail, let Vl be a regular neighbourhood Kl, X = Nl − int Vl the exterior of Kl, T the toral component ∂Vl of ∂X, and γ a slope on T. Then we obtain the 3-manifold N2 by attaching a solid torus V2 to X so that γ bounds a disc in V2. Let K2 be the core of V2. Let π be the slope of a meridian loop of Kl, and Δ the distance between the slopes π and γ, i.e. the minimal number of intersection points of the two slopes on T. Suppose for i = 1 and 2 that Ni contains a proper annulus Ai such that the two components of ∂Ai are essential loops on distinct incompressible components of ∂Ni. Then note that Ai is essential, i.e. incompressible and ∂-incompressible in Ni.
Some links with non-adequate minimal-crossing diagrams
Authors: Hara Masao , Yamamoto Makoto
Published: November 2008

To investigate invariants of links derived from their diagrams, the recent new polynomial invariants of links play important roles. Murasugi6, 7, Kauffman3 and Thistlethwaite 9 independently showed that the number of crossings in a proper connected alternating diagram of a link is the minimal-crossing number of the link and that the writhe of the diagram is invariant. Murasugi 8 also determined the minimal-crossing number of torus links. In 5, Lickorish and Thistlethwaite introduced the concept of an adequate link diagram and showed that the number of crossings in an adequate diagram of a semi-alternating link is the minimal-crossing number of the link. They also determined the minimal-crossing number of almost all Montesinos links. In this paper we show that for some links represented by plats and braids which are not adequate, the numbers of crossings in the diagrams are the minimal-crossing numbers of the links.
Compact open sets in duals and projections inL1-algebras of certain semi-direct product groups
Authors: Gröchenig Karlheinz , Kaniuth Eberhard , Taylor Keith F.
Published: November 2008

The main purpose of this paper is to study projections, that is, self-adjoint idempotents, inL1-algebras of semi-direct productsG= ℝ ⋉ ℝd,d≥ 2. We establish necessary and sufficient conditions for the existence of non-zero projections in terms of the action of ℝ on ℝd. In the cases where such projections exist, we describe minimal ones in detail.
Cluster sets of set-valued functions
Authors: Harbottle S. R.
Published: November 2008

In this paper we shall study the cluster sets of set-valued functions denned on the open unit disc Δ, and we shall, in particular, look at the relationship between the full cluster set and the cluster sets along radial and arbitrary monotonic paths.
A harmonic quadrature formula characterizing open strips
Authors: Armitage D. H. , Nelson C. S.
Published: November 2008

Let γn denote n-dimensional Lebesgue measure. It follows easily from the well-known volume mean value property of harmonic functions that if h is an integrable harmonic function on an open ball B of centre ξ0 in ℝn, where n ≥ 2, then

A converse of this result is due to Kuran [8]: if D is an open subset of ℝn such that γn(D) < + ∞ and if there exists a point ξo∈D such that

for every integrable harmonic function h on D, then D is a ball of centre ξ0. Armitage and Goldstein [2], theorem 1, showed that the same conclusion holds under the weaker hypothesis that (1·2) holds for all positive integrable harmonic functions h on D.
Two lemmas on primary cohomology operations
Authors: Hubbuck J. R.
Published: November 2008

Let X be a CW-complex with finite skeletons and with H*(X, Z) free of p-torsion, where p is some odd prime. We shall prove two theorems on the action of the Steenrod powers
A mathematical model for long waves generated by wavemakers in non-linear dispersive systems
Authors: Bona J. L. , Bryant P. J.
Published: November 2008

An initial-boundary-value problem for the equation

is considered for x, t ≥ 0. This system is a model for long water waves of small but finite amplitude, generated in a uniform open channel by a wavemaker at one end. It is shown that, in contrast to an alternative, more familiar model using the Korteweg–deVries equation, the solution of (a) has good mathematical properties: in particular, the problem is well set in Hadamard's classical sense that solutions corresponding to given initial data exist, are unique, and depend continuously on the specified data.
The jet from a horizontal slot at large Froude number
Authors: Keady G.
Published: November 2008

The two-dimensional flow in a jet, falling under gravity from a slot in a horizontal plane, is studied. The fluid is considered to be incompressible and inviscid; the flow is taken to be irrotational; and the reciprocal ε of the Froude number is considered to be small. By taking the complex potential as the independent variable we overcome the difficulty that the boundary geometry is not known in advance. The method of matched asymptotic expansions is applied. The first two terms of an inner asymptotic expansion and the first three of an outer one are found: the inner expansion is valid above and near the slot, but is inappropriate far downstream, while the outer expansion is valid far downstream, but fails to satisfy the conditions upstream. The two expansions are matched and ‘composite’ approximations, covering the whole flow field, are derived.