bluuebunny/crossref_metadata_2025 huggingface dataset:
Adding propositional connectives to countable infinitary logic
Authors: Friedman Harvey
Published: November 2008

For countable admissible α, one can add a new infinitary propositional connective to so that the extended language obeys the Barwise compactness theorem, and the set of valid sentences is complete α-r.e.

Aside from obeying the compactness theorem and a completeness theorem, ordinary finitary predicate calculus is also truth-functionally complete.

In (1), Barwise shows that for countable admissible A, provides a fragment of which obeys a compactness theorem and a completeness theorem. However, we of course lose truth-functional completeness, with respect to infinitary propositional connectives that operate on infinite sequences of propositional variables. This raises the question of studying extensions of the language obtained by adding infinitary propositional connectives, in connexion with the Barwise compactness and completeness theorems, and other metatheorems, proved for Some aspects of this project are proposed in (3). It is the purpose of this paper to answer a few of the more basic questions which arise in this connexion.

We have not attempted to study the preservation of interpolation or implicit definability. This could be quite interesting if done systematically.
On a theory of laminar flow in channels of a certain class. II
Authors: Fraenkel L. E. , Eagles P. M.
Published: November 2008

This paper continues (and concludes) the mathematical analysis begun in (8) of a formal theory of viscous flow in channels with slowly curving walls. In that paper, the theory was shown to yield strict asymptotic expansions, in powers of the small curvature parameter, of exact solutions of the Navier-Stokes equations, but the proofs were restricted to a set of Reynolds numbers and wall divergence angles that is distinctly smaller than the set on which the formal approximation is defined. In the present paper, we study in more detail a certain linear, partial differential operator TN, the invertibility of which is essential to the proofs. This operator is shown to be invertible (and the formal theory is thereby justified) on a parameter domain that is much larger than and may well be the whole of . A key step is to associate with TN a family of operators that approximate TN locally and have much simpler coefficients.
A new formulation of the Liénard–Chipart stability criterion
Authors: Barnett S.
Published: November 2008

The Liénard–Chipart criterion for determining whether all the zeros of a real nth degree polynomial a(λ) have negative real parts involves calculation of only about half the Hurwitz determinants, the order of the largest being n − 1. It is shown, by using the companion matrix of a polynomial formed from a(λ), that the required determinants are equal to minors of a matrix whose order is ½ n or ½(n−1) according as n is even or odd. In either case this matrix is very easy to obtain A simple application of the result provides a criterion for a(λ) to be stable and aperiodic.
A convolution-integral representation for a class of linear operators
Authors: Rowlands K.
Published: November 2008

Let be the complex vector space consisting of all complex-valued functions of a non-negative real variable t. For each positive number u, the shift operator Iu is the mapping of into itself defined by the formula

A linear operator T which maps a subspace of into itself is said to be a V-operator (13) if:

(a) for each x in , the complex-conjugate function x* is in ;

(b) both and \ are invariant under the shift operators;

(c) every shift operator commutes with T.

(Property (a) ensures that every function x in can be uniquely expressed as x1 + ix2, where x1 and x2 are real functions in .)
A finite evaluation of a special exponential sum
Authors: Mordell L. J.
Published: November 2008

Let f(x) and g(x) be rational functions of x with integer coefficients, p an odd prime, e(x) = e(2πix/p), e(u/v) = e(w), where u = vw (mod p)† and χ(x) is a character mod p. It is well known that very abstruse methods are required to find an estimate for sums such as

where in the summation, ‡ the values of x are omitted for which eitherf(x) or g(x) is not defined.
On the biquadratic Gauss sum
Authors: McGettrick Andrew D.
Published: November 2008

1. Professor J. W. S. Cassels (in (l)) recently enunciated a conjecture concerning Kummer sums. His conjecture suggests that there is a close connexion between the Kummer sum and a certain product of Weierstrass elliptic functions.

In this paper we consider the biquadratic Gauss sum. A theory similar to that used in (1) and in (4) suggests that there is a corresponding connexion between the biquadratic sum and a product of Jacobian elliptic functions. The enunciation of this conjecture appears in section 6. But first of all we require some preliminary results.
The product of the distributions and
Authors: Fisher B.
Published: November 2008

In the following a distribution or a generalized function f is denned, as by Gelfand and Shilov(2) or by Temple (3), as a continuous linear functional on the space K of infinitely differentiable test functions ø having compact support. The value of f at a test function ø will be denoted by (f, ø).

A sequence of test functions {øn} is said to be a null sequence if

(1) the support of each øn is contained in some fixed domain D independent of n,

(2) the sequence converges uniformly to zero in D, as n tends to infinity, for all m.
Reflexion properties of an isotropic gyrational medium with Epstein stratifications
Authors: Westcott B. S.
Published: November 2008

A recent search for analytical profiles in plane-stratified isotropic gyrational media has revealed particular Epstein models for permittivity and magneto-electric properties of the media when subjected to normally incident electromagnetic waves. The reflexion and transmission coefficients of such a model are derived and possible transparencies are discussed.
Basis normalizers and Carter subgroups in a class of locally finite groups
Authors: Graddon C. J. , Hartley B.
Published: November 2008

We shall be working throughout this paper in the class of locally finite groups introduced in (3) and further discussed in (5) and (6), and all groups appearing will be assumed to belong to this class. By definition, is the largest subgroupclosed class of locally finite groups satisfying the conditions:

U1. If G ε then G has a finite series

with locally nilpotent factors.
The graph-theoretic analogue of Tietze's characterization of a convex set and its generalization
Authors: Guay M. D.
Published: November 2008

Introduction. One of the most satisfying theorems in the theory of convex sets states that a closed connected subset of a topological linear space which is locally convex is convex. This was first established in En by Tietze and was later extended by other authors (see (3)) to a topological linear space. A generalization of Tietze's theorem which appears in (2) shows if S is a closed subset of a topological linear space such that the set Q of points of local non-convexity of S is of cardinality n < ∞ and S ~ Q is connected, then S is the union of n + 1 or fewer convex sets. (The case n = 0 is Tietze's theorem.)
Motion of a classical particle with spin
Authors: Ellis J. R.
Published: November 2008

The helical solutions of the Frenkel-Thomas equations for a free spinning particle are discussed following manifestly covariant lines. For the purposes of expressing the equations in Lagrangian and Hamiltonian form, the definition of spin by H. C. Corben is not entirely satisfactory being frame-dependent. The use of a spin ‘four-vector’ is discussed which makes the solution of the equations shorter and more elegant than that of Corben. Such a derivation necessitates the use of the Frenet-Serret formulae. By basing a Lagrangian formalism on this definition of spin we show that the covariant Euler-Lagrange equations (with multipliers) lead directly to the Frenkel-Thomas equations. Such a derivation is thus an improvement on those of other authors and suggests a more suitable canonical formalism for these equations.
Desargues configurations and their collineation groups
Authors: Coxeter H. S. M.
Published: November 2008

When two triangles, ABC and A′B′C′, are perspective from a point 0, their pairs of corresponding sides meet on a line 0, the axis of perspective. The line OA passes through A′ and some point on 0. These four points have a certain cross ratio which is the same if B or C is used instead of A. The reciprocal cross ratio arises if A′B′C′ is regarded as the first triangle and ABC the second. The complete figure contains ten pairs of perspective triangles, yielding twenty cross ratios. In section 6 a technique, suggested by D. W. Babbage and John Rigby, is used to express these cross ratios in the form
Applications of variational principles to classical perturbation theory in general relativity
Authors: Robinson D. C.
Published: November 2008

The equations satisfied by perturbations of solutions of the Euler-Lagrange equations corresponding to a Lagrangian density L are derived from the first and second variations of L and these new Lagrangian densities are used to construct Noether and Euler identities. It is shown how these identities can be used in general relativity to construct conserved quantities and local uniqueness theorems.
Measures of the non-convexity of sets and the Shapley–Folkman–Starr theorem
Authors: Cassels J. W. S.
Published: November 2008

The object of this note is to show that elementary probability considerations suggest a very natural way of measuring the non-convexity of a set in euclidean space or, more generally, in a real Hilbert space . In particular they give a proof, much simpler and under less restrictive conditions, of results due to Shapley, Folkman and Starr which are of importance in Mathematical Economics ((1),(2)).
Structure of a foliated neighbourhood
Authors: Harrison Jenny
Published: November 2008

C. Ehresmann (2) has shown that if a leaf L of a smooth foliation has a foliated neighbourhood, then there exists a fibre bundle over L, normal to the leaves, with discrete structure group. Using the concept of a microbundle and the n-isotopy extension theorem, we find a similar result for both PL and TOP categories, and, in addition, show that the structure group can be chosen to be the holonomy group of L. As for applications we show that holonomy characterizes the foliated neighbourhood of a leaf (proved by Haefliger in the differentiable case (3)). In particular, if the holonomy group of a compact leaf L is trivial then the leaf has a trivial foliated neighbourhood, and if it is finite it has a neighbourhood of compact leaves which are covering spaces of L. Another corollary is the known result that a proper submersion with compact fibre is a fibration. Finally we use the fact that the constructed normal microbundle can be chosen to have its fibres contained in the leaves of a transverse foliation to demonstrate isotopy uniqueness of normal microbundles.
Unbounded operators and random Fourier series
Authors: Sanders J. W.
Published: November 2008

0·0. Let G be an infinite compact abelian group with dual X. Parseval's identity shows that if f ∈ C(G) and ω ∈ l∞(X) then . Edwards has shown in (2) that L2(G) here cannot, in general, be replaced by any smaller Lp(G) space. Precisely: there exist f ∈ C(G) and ω: X → {± 1} such that . We strengthen this result by showing much more can be said about the summability of the Fourier series of f than . For example, when G is the circle group, f can be chosen to also satisfy

The functions introduced here and called darts, generalize this type of series condition.
Regularity of the Segre stratification
Authors: Gibson C. G.
Published: November 2008

Arnol'd (1) remarks that the partition by Segre symbol of the endomorphisms of the complex vector space is a finite constructible stratification: in this paper we verify this statement, and show moreover that it is a Whitney stratification. Some of the arguments are by no means special to the case in hand; however, there seems little point in attempting a general formulation. It is a pleasure to acknowledge helpful conversations with Professor C. T. C. Wall and Dr E. J. N. Looijenga.
Geometric versions of Whitney regularity
Authors: Trotman D. J. A.
Published: November 2008

Let Xm and Yn be C1 manifolds embedded in , m < n < p, and let . In (4) C. T. C. Wall considered the following conditions:

(as) For any local C1 retraction at x, π: , x has a neighbourhood U such that is a submersion.
Maximal one-sided ideals in Banach algebras
Authors: Green Michael D.
Published: November 2008

In (1) Akemann and Rosenfeld introduced a property for Banach algebras which they called (*). A Banach algebra satisfies (*) if every maximal one-sided ideal in is closed. They proved that certain classes of Banach algebra with satisfy (*), and they mentioned a conjecture that if is a Banach algebra with , then satisfies (*). In this paper we show that, if is a Banach algebra with a bounded right (left) approximate identity, then maximal left (right) ideals in are closed, and we give a counter-example to the above conjecture. We also give an independent proof that C*-algebras satisfy (*).
Inclusion conditions for Hurwitzian and Schur sets in
Authors: Smith Russell A.
Published: November 2008

A condition of frequency domain type on the vector polynomial G(z) is obtained which is both necessary and sufficient for the Hurwitzian set of G(z) to include the special Hurwitzian set , where E(z) = col (1, z, …, zn). This result is extended to Hurwitzian sets arising from scalar delay-differential equations. Analogous conditions are also given for each of the inclusions where denotes the Schur set of G(z).
Continuous singular measures equivalent to their convolution squares
Authors: Brown Gavin , Hewitt Edwin
Published: November 2008

Throughout this paper, G will denote a locally compact, non-discrete, Abelian group (subjected to various conditions) and X wi11 denote the character group of G. All terminology and notation are as in (7). The measure algebra M(G), as is known, is a very complicated entity. We address ourselves here to some novel peculiarities of the subspace Ms(G) of continuous measures in M(G) that are singular with respect to Haar measure λ.
Non-singular models of specialized Weddle surfaces
Authors: Edge W. L.
Published: November 2008

There are, in projective space Σ of three dimensions, two famous quartic surfaces: W, the Weddle surface with six nodes ((13), p. 69, footnote) and K, the Kummer surface with sixteen ((s), p. 246, (7) passim). They are in birational correspondence and have the same non-singular model: the octavic base surface F of the net of quadrics in [5], which contain a given line λ and for which a given simplex S is self-polar ((5); (6)). One naturally takes S, with vertices X0, X1, X2, X3, X4, X5 as simplex of reference for homogeneous coordinates x0, x1, x2, x3, x4, x5; F is invariant under the harmonic inversions hj in the vertices Xj and opposite bounding primes xj = 0 of S. These six hj, mutually commutative and having identity for their product, generate an elementary abelian group of order 32. This representation of throws into prominence what may, in this context, be called its positive subgroup , of order 16, consisting of identity and the 15 products hjhk = hkhj; these are harmonic inversions in the edges XjXk and opposite bounding solids xj = xk = 0 of S. The coset of consists of the six hj and their ten products in threes, complementary products being the same (h0h1h2 ≡ h3h4h5) because of the product of all six hj being identity. These ten products are harmonic inversions in the ten pairs of opposite plane faces of S.
On the growth of recurrence sequences
Authors: Loxton J. H. , van der Poorten A. J.
Published: November 2008

In this note, we discuss some questions on the arithmetic properties of recurrence sequences. Our primary purpose is to mention some methods which originated as lemmas in certain transcendence studies, but in order to give context to these ideas, we briefly review other techniques applicable to the problems considered. These problems concern the distribution of zeros in a recurrence, the rate of growth of its terms and the size of the greatest prime factor of the terms.
Injective W*-algebras
Authors: Wassermann Simon
Published: November 2008

Let M be a W*-algebra. If π is a non-degenerate faithful normal representation of M on a Hilbert space H, M is said to be injective [(8), § 5] if there is a projection of norm one from L(H) onto π(M). Tomiyama [(13), theorem 7.2] has shown that this definition is independent of the choice of representation π, and, moreover, that M is injective if and only if the commutant π(M)′ is injective for some such π. M is said to be semidiscrete [(8), § 3] if the identity map on M can be approximated in the topology of simple-weak* convergence by completely positive normal linear maps in L(M) of finite rank which preserve the identity. If π is a non-degenerate faithful normal representation of M on a Hilbert space H, a *-homomorphism η of the algebraic tensor product M ⊙ π (M)′ into L(H) is defined by

If η has a bounded extension to M ⊗ π(M)′ (the completion relative to the spatial C*-cross norm) we shall say that M is amenable.
Singular infinitely divisible distributions whose characteristic functions vanish at infinity
Authors: Brown Gavin
Published: November 2008

In the course of discussing dynamical systems which enjoy strong mixing but have singular spectrum, E. Hewitt and the author, (2), recently constructed families of symmetric random variables which satisfy inter alia the following properties:

(i) Zt is purely singular and has full support,

(ii) χ(t)(x) → 0 as ± x → ∞, where χ(t) is the characteristic function of Zt,

(iii)′ Zt+s, Zt + Zs have the same null events,

(iv) whenever s ≠ t, Zt and Zs + a are mutually singular for every (possibly zero) constant a.
On the coefficients of certain automorphic functions
Authors: Hayman W. K. , Patterson S. J. , Pommerenke Ch.
Published: November 2008

Suppose that

is regular in the unit disk D = {|z| < 1}, and assumes there values ω lying in a domain Δ. It is natural to ask what effect this restriction has on the coefficients aν. The strongest order result we can hope to prove is that
The 2-primary J-homomorphism
Authors: Snaith Victor
Published: November 2008

In this paper every space will be 2-local, for example BO will mean the 2-localization of the space usually denoted BO.
Properties of two-way transmission in generalized wave propagation problems
Authors: Heading J.
Published: November 2008

A deeper mathematical insight into physical problems is gained when a relatively elementary problem is generalized mathematically beyond the bounds of physical reality, particularly when the results are independent of the generalization. See, for example, Heading (3).
On the edge-reconstruction of planar graphs
Authors: Fiorini S.
Published: November 2008

The object of this paper is to prove that the following classes of graphs are edge-reconstructible: (i) 4-connected planar graphs with minimum valency 5; (ii) maximal planar graphs with minimum valency at least 4.
On words of minimal length under endomorphisms of a free group
Authors: Edmunds Charles C.
Published: November 2008

In (1) it is remarked that theorem 5·10 is an analogue, for endomorphisms of a free group, of a result of A. Shenitzer ((3), theorem 2) about automorphisms. With this in mind, D. Solitar asked the author if there might also be an analogue for the corollary on page 276 of(3). There is such an analogue and, after a preliminary discussion of definitions and notation, we will describe it.
On coverings of a finite group by abelian subgroups
Authors: Mason David R.
Published: November 2008

In (2), Erdös and Straus pose the following two conjectures.

If G is a finite group of order n and X is a subset of G consisting of pairwise non-commuting elements, then |X| ≤ n/2 + 1.
Almost-primes in arithmetic progressions and short intervals
Authors: Heath-Brown D. R.
Published: November 2008

In this paper we shall investigate the occurrence of almost-primes in arithmetic progressions and in short intervals. These problems correspond to two well-known conjectures concerning prime numbers. The first conjecture is that, if (l, k) = 1, there exists a prime p satisfying
A lattice of extension rings for a commutative ring
Authors: Kirby D. , Adranghi M. R.
Published: November 2008

The work of this note was motivated in the first place by North-cott's theory of dilatations for one-dimensional local rings (see, for example (4) and (5)). This produces a tree of local rings as in (4) which corresponds, in the abstract case, to the branching sequence of infinitely-near multiple points on an algebroid curve. From the algebraic point of view it seems more natural to characterize such one-dimensional local rings R by means of the set of rings which arise by blowing up all ideals Q which are primary for the maximal ideals M of R. This set of rings forms a lattice (R), ordered by inclusion, each ring S of which is a finite R-module. Moreover the length of the R-module S/R is just the reduction number of the corresponding ideal Q (cf. theorem 1 of Northcott (6)). Thus the lattice (R) provides a finer classification of the rings R than does the set of reduction numbers (cf. Kirby (1)).
Homomorphs and formations of given derived class
Authors: Lafuente J.
Published: November 2008

A homomorph H is a normal Schunck class if and only if there exists a derived class χ such that H = χ*; moreover, in this case one has H′ = χ (for the definitions, see below). These results give to the derived classes a decisive significance on the study of the normal Schunck classes (see (5)). The aim of this paper is to study the homomorphs H such that H′ is a fixed derived class: we prove that these homomorphs compose a complete and distributive lattice for the inclusion relation (the maximum of this lattice being a normal Schunck class). We construct the greatest and the smallest formations whose derived class is given. We prove finally that, except in trivial cases, a normal Schunck class is not a formation.
Non-linear optimized fibre reinforced plates in bending
Authors: Lowe P. G. , Yasseri S. F.
Published: November 2008

1. Introduction. In recent years a good deal of progress has been made in obtaining solutions to practical problems concerned with reinforcement optimization of what, in practice, might be reinforced concrete or other fibre reinforced composites in bending (1). Almost all the solutions have been obtained using a linear expression relating bending moment to the amount of fibre required. In what follows a non-linear expression will be adopted; the expression is obtained by including the next term in a parameter series expansion.
Generic bifurcations of forced oscillations of integrable mechanical systems
Authors: Cleave J. P.
Published: November 2008

Although integrable Hamiltonian systems are non-generic (Robinson (13)) they have some importance in classical mechanics, e.g. the two-body problem, a free rigid body not subject to a gravitational field, the Toda lattice (Moser(12)). Arnol'd (cf. (2) and (1), appendix 26) proved that under quite general conditions action-angle coordinates can be introduced. Accordingly we consider a fixed system with n degrees of freedom in the standard form

where H0 is a smooth function of the action variables Li only and I ∈ I and I is an open set of n-tuples of positive reals. now subjected to a periodic impressed force, the resulting system, , being determined by a small, periodic perturbation of the energy function

where e is a small positive real and K(c, I; α0, α1, …, an) is a smooth function of parameters c ranging over a smooth r-manifold E, I ∈ I, and K has period 2π in each of the angles αi, i.e. (α0, …, αn) ∈ Tn (n-torus). The purpose of this paper is to define the forms of bifurcation of oscillations of the perturbed system (K):
Footnote to a paper of Baker and Mines
Authors: Pym John
Published: November 2008

Let S be a semigroup with a compact topology in which multiplication is continuous on the left (i.e. xi→x implies xiy→xy for each y in S). Then S has a minimal left ideal L which is compact; each idempotent e in L is a right identity for L (xe = xfor each x ∈ L)and L = Se; Ge = eL is a group and L is the union of all such groups; and if f is a second idempotent in L, the canonical map x ↦ fx of Ge to Gf is an algebraic isomorphism (see Ruppert(2) for these facts). Baker and Milnes(1), §4(A), have observed that, in the case in which S is the Stone–Cech compactification of a discrete abelian group, the canonical map from Ge to Gf may not be a homeomorphism. (This contrasts with the situation in compact semigroups with separately continuous multiplication.) We present a simple proof of a more definitive result.
Affine cubic functions
Authors: Wall C. T. C.
Published: November 2008

Although the classification of affine cubic curves was undertaken by Newton(4), in one of the first major exercises ever in coordinate geometry (see Cayley(2) for a fuller account), a parallel study of cubic functions seems not to have been contemplated till recently. The essential difference is that a function f defines a pencil of curves – its level curves – which have to be considered simultaneously. The author's interest in the subject arose from problems in singularity theory (concerning canonical stratifications), and a later paper in the series will have applications of this kind. Here we study the simplest case as an introduction. Our techniques are entirely classical, but the results are hard to find elsewhere. In later papers we intend to study cubic functions on ℝ2, and on ℂ3.
Orientation-reversing involutions on homology 3-spheres
Authors: Galewski David E. , Stern Ronald J.
Published: November 2008

An important problem in geometric topology is to find a homology 3-sphere M with Rohlin invariant μ(M) = ½ such that M # M bounds an acyclic 4-manifold. If such an M exists then, for instance, all closed topological manifolds of dimension ≥ 5 support a polyhedral structure(3, 4). One way of producing such examples is to find a homology 3-sphere M with μ(M) = ½ such that M admits an orientation-reversing diffeomorphism h, for then M # M would be diffeomorphic to M # − M which bounds an acyclic manifold. In this paper we observe by elementary means that if h is further assumed to be an involution, then this is not possible, namely:

Theorem. Let M be a homology 3-sphere which admits an orientation reversing involution. Then M bounds a parallelizable rational 4-ball. In particular, μ(M) = 0.
Integer solutions of systems of quadratic equations
Authors: Britton J. L.
Published: November 2008

As is well known, there is an algorithm for deciding if a system of linear equations with coefficients from the set Z of integers has a solution in integers. The purpose of this paper is to answer the following question: does this remain true if ‘linear’ is replaced by ‘quadratic’?
On Hurwitz extensions by PSL2(7)
Authors: Cohen Jeffrey M.
Published: November 2008

In this paper, a new family of factors of (2, 3, 7) is obtained which contains groups found in three other papers. It is shown that for all n, there exists an m such that there are at least n isomorphism types of Hurwitz groups of order m. Finally, presentations for all groups considered are obtained.
Essential geometric morphisms between toposes over finite sets
Authors: Haigh John
Published: November 2008

We show that if {Gi}J ε I is a generating set for an (elementary) topos ℰ then {P(Gi)}iεI is a cogenerating set for x2130;. From this we show that if topos ℰ contains an object G whose subobjects generate ℰ, then ΩG is a cogenerator for ℰ. Let denote the topos of finite sets and functions. We also show that if ℰ1 is a topos and ℰ2 is a bounded -topos then every geometric morphism ℰ1 → ℰ2 is essential.
The embedding problem for Markov chains with three states
Authors: Frydman Halina
Published: November 2008

In this paper we consider the embedding problem for Markov chains with three states. A non-singular stochastic matrix P is called embeddable if there exists a two-parameter family of stochastic matrices

satisfying

and such that

Though extensive characterizations of embeddable n × n stochastic matrices have been given in (l), (2), (3), (6), and further characterizations of embeddable 3 × 3 stochastic matrices in (4), they do not provide, except in the case of 2 × 2 stochastic matrices, easily applicable necessary and sufficient conditions for embeddability.
Qualitative analysis of a model for boundary effects in the Taylor problem
Authors: Schaeffer David G.
Published: November 2008

Boundary effects in the classical bifurcation experiments of fluid mechanics have been the subject of much recent investigation, most notably for us the work of Benjamin(2) on bifurcating steady flows in the Taylor problem in a very short annulus. His paper broaches a fundamental conceptual issue: how does the number of cells in the primary flow, an integer, depend on the length L of the annulus, a real parameter? By the primary flow we mean the flow which develops at high Reynolds number R as the rotation speed is increased gradually from rest, L being held constant (say L = L*). The relevant part of Benjamin's data is presented in Fig. 0·1. For the very short annuli of his experiment the primary flow possessed either two of four cells only. (See Section 1 concerning the bias towards an even number of cells.) If L* > L2, L1 < L* < L2 or L* < L1 the primary flow has 4, 2 or 2 cells respectively; however when L1 < L* < L2 the evolution of the flow as R is increased is not smooth but suffers a jump as R passes the middle of the three intersections of the line {L = L*} with the cusped curve Γ in the figure. (It is now widely accepted that the development of cells in the flow occurs over a range of Reynolds numbers; this point is not the issue here.) The curve Γ divides the plane into two regions such that there are one or two stable, experimentally observable flows according as (R, L) lies in region I or region II, respectively. We may refer to the two stable states in region II as 2 cell or 4 cell, although their cellular structure is only partially developed at the moderate values of the Reynolds number occurring in the figure. One or the other of the two flows in region II loses stability across Γ, giving rise to the possibility of jumps when, as above, (R, L) crosses the boundary moving from region II to region I. In general if (R, L) is varied quasi-statically along a path which begins in region I, passes through region II, and re-enters region I, a jump will occur on re-entry if and only if the point H lies between the two points of Γ where the path crosses the boundary (between in the sense of distance along Γ).
Dedekind sums and Hecke operators
Authors: Parson L. Alayne
Published: November 2008

By considering the action of the Hecke operators on the logarithm of the Dedekind eta function together with the modular transformation formula for this function, Knopp (8) proved an extension of an identity of Dedekind for the classical Dedekind sums first mentioned by H. Petersson. By looking at the action of the Hecke operators on certain Lambert series studied by Apostol(l) together with the transformation formulae for these series, Parson and Rosen (9) established an analogous identity for a type of generalized Dedekind sum. A special case of this identity was initially proved by Carlitz(6). In this note an elementary proof of these identities is given. The Hecke operators are applied directly to the Dedekind sums without invoking the transformation formulae for the logarithm of the eta function or for the Lambert series. (Recently, L. Goldberg has given another elementary proof of Knopp's identity.)
On the Hamiltonian structure of evolution equations
Authors: Olver Peter J.
Published: November 2008

The theory of evolution equations in Hamiltonian form is developed by use of some differential complexes arising naturally in the formal theory of partial differential equations. The theory of integral invariants is extended to these infinite-dimensional systems, providing a natural generalization of the notion of a conservation law. A generalization of Noether's theorem is proved, giving a one-to-one correspondence between one-parameter (generalized) symmetries of a Hamiltonian system and absolute line integral invariants. Applications include a new solution to the inverse problem of the calculus of variations, an elementary proof and generalization of a theorem of Gel'fand and Dikiî on the equality of Lie and Poisson brackets for Hamiltonian systems, and a new hierarchy of conserved quantities for the Korteweg–de Vries equation.
Tripos theory
Authors: Hyland J. M. E. , Johnstone P. T. , Pitts A. M.
Published: November 2008

One of the most important constructions in topos theory ia that of the category Shv (A) of sheaves on a locale (= complete Heyting algebra) A. Normally, the objects of this category are described as ‘presheaves on A satisfying a gluing condition’; but, as Higgs(7) and Fourman and Scott(5) have observed, they may also be regarded as ‘sets structured with an A-valued equality predicate’ (briefly, ‘A-valued sets’). From the latter point of view, it is an inessential feature of the situation that every sheaf has a canonical representation as a ‘complete’ A-valued set. In this paper, our aim is to investigate those properties which A must have for us to be able to construct a topos of A-valued sets: we shall see that there is one important respect, concerning the relationship between the finitary (propositional) structure and the infinitary (quantifier) structure, in which the usual definition of a locale may be relaxed, and we shall give a number of examples (some of which will be explored more fully in a later paper (8)) to show that this relaxation is potentially useful.
A numerical sequence and a family of polynomials arising from a question of completeness
Authors: Fraenkel L. E.
Published: November 2008

This paper concerns, in the first instance, a sequence {t(1), t(2), t(3), …} of positive numbers denned successively by

where k ∈ ℕ = {1,2,3,…} and

Thus, when k is odd, A(k) consists of those divisors of k that do not exceed k/3; when k is even, A (k) consists of those even divisors n of k that make k/n odd and do not exceed k/3. The sets A(k) are listed in Table 1 for k ≤ 105; the numbers t(1) to t(10) are 1, 1, 4/3, 1, 6/5, 4/3, 8/7, 1, 14/9, 6/5.
Rotation sets are closed
Authors: Ito Ryuichi
Published: November 2008

The rotation number ρ of an orientation preserving homeomorphism of a circle was originally defined by Poincaré It measures the asymptotic rotation of the orbit of each point under iterations of the homeomorphism (see, for example, (2)).
The inhomogeneous minimum of quadratic forms of signature ± 1
Authors: Raka Madhu
Published: November 2008

Let Qr be a real indefinite quadratic form in r variables of determinant D ≠ 0 and of type (r1, r2), 0 < r1 < r, r = r1 + r2, S = r1 − r2 being the signature of Qr. It is known (e.g. Blaney (3)) that, given any real numbers c1, c2,…, cr, there exists a constant C depending only on r and s such that the inequality

has a solution in integers x1, x2, …, xr.
Local homology and cohomology
Authors: Mitchell W. J. R.
Published: November 2008

The idea of associating local homology and cohomology groups to each point of a space is now over fifty years old, and it has proved useful in the theory of transformation groups (as in the local Smith theorems), in dimension theory and elsewhere. Nevertheless there is no uniformity as to definition or notation, and a good deal of the theory is written in a notation now superseded and often impenetrable to the modern reader. In this paper all the important definitions are collected in a single framework, and the relations between the various groups are explored. Particular attention is paid to the case of a cohomologically locally connected space, for which relatively complete results are obtained.
Rings associated with ideals and analytic spread
Authors: Rees D.
Published: November 2008

Let A be a Noether ring and let = (a1,…, ar) be an ideal of A. There are a number of graded rings that we can associate with . In this paper we shall be concerned with the following.
Bifurcation and hysteresis varieties for the thermal-chainbranching model with a negative modal parameter
Authors: Stewart Ian N.
Published: November 2008

The theory of unfoldings of singularities, or ‘elementary catastrophe theory’ (Thom(10), Poston and Stewart(9), Golubitsky and Guillemin(3), Gibson(2)) has been generalized by Golubitsky and Schaeffer(5,6), providing a powerful method for analysing imperfect bifurcation. One recent application by Golubitsky, Keyfitz, and Schaeffer(4) concerns the ‘explosion peninsula’ in chemical reactions such as that between hydrogen and oxygen. They show that the ‘thermal-chainbranching model’ of Gray and Yang(7,8) is capable of reproducing the necessary qualitative features, a result for which only heuristic arguments had previously been available.
Multi-valued solutions of the wave equation
Authors: Friedlander F. G.
Published: November 2008

It was first pointed out by Sommerfeld, around the turn of the century, that certain multi-valued solutions of the wave equation in ℝ3 can be used to deal with the problem of scattering by a wedge, or reflection in a corner. The older literature on this subject is extensive; see ((4), chapter 5) for references up to 1958. The object of this Note is to give an explicit and elementary construction of a forward fundamental solution of the wave equation, of this type, in ℝn+1, where n ≥ 2; for n = 2 this includes Sommer-feld's original result.
On generic hypersurfaces in ℝn
Authors: Bruce J. W.
Published: November 2008

In this paper we consider certain questions concerning the differential geometry of generic hypersurfaces in ℝn. Our results prove, for example, that the curve of rib points of a generic surface in ℝ3 has transverse self-intersections.

In (4) Porteous discussed (amongst other things) the generic geometry of curves and surfaces in ℝ3. Subsequently Looijenga ((3) and see also (5)) gave a more precise definition of the term generic and showed that an open dense subset of smooth embeddings of manifolds in Euclidean space were indeed generic.
Totally knotted knots are prime
Authors: Gomez-Larran¯aga J. C.
Published: November 2008

Throughout, the word knot means a subspace of the 3-sphere S3 homeomorphic with the 1-sphere S1. Any knot can be expressed as a connected sum of a finite number of prime knots in a unique way (13), we consider the trivial knot a non-prime knot. (For higher dimensional knots, factorization and uniqueness have been studied in (1).) However given a knot it is difficult to determine if it is prime or not. We prove that totally knotted knots, see definition in §2, are prime in theorem 1, give a class of examples in theorem 2 and investigate how the last result can be applied to the conjecture that the family Y of unknotting number one knots are prime. (See problem 2 in (5).) At the end, prime tangles as defined by W. B. R. Lickerish are used to prove that in a certain family of knots, related somewhat to Y, there is just one non-prime knot: the square knot. The paper should be interpreted as being in the piecewise linear category. Standard definitions of 3-manifolds and knot theory may be found in (6) and (11) respectively.
Surface instabilities on an equibiaxially stretched elastic half-space
Authors: Reddy B. D.
Published: November 2008

The existence of non-homogeneous modes of deformation in an equibiaxially stretched elastic half-space is investigated. The surface of the half-space is assumed to be traction-free.

It is shown that three different eigenmodal deformations are possible, according as the cube of the critical stretch is greater than, less than, or equal to ¾.

For the class of strain-energy functions due to Ogden (9), necessary conditions for bifurcation are deduced-for equibiaxial tension. Results are given and discussed for single-term, as well as two- and three-term Ogden strain-energy functions, for equibiaxial tension and compression.
Constitutive branching in elastic materials
Authors: Hill R.
Published: November 2008

In finite elasticity a homogeneous strain is not always determined uniquely by the conjugate stress. The phenomenon is investigated for any constitutive work-function, and in both its global and local aspects. Uniqueness is lost in specific critical states. Basic properties of their aggregates in the stress and strain spaces are derived, and shown to be fundamental for the material response near individual critical states. The geometry of local branches is analyzed in detail. Further properties resulting from material isotropy are obtained. The general theory is illustrated by examples typifying the responses of metal crystals and rubberlike polymers.
On the K-theory of the extended power construction
Authors: McClure J. E. , Snaith V. P.
Published: November 2008

The construction of Dyer-Lashof operations in K-theory outlined in (6) and refined in (12) depends in an essential way on the descriptions of the mod-p K-theory of EZp, ×ZpXp and EΣ ×σ p Xp given there. Unfortunately, these descriptions are incorrect when p is odd except in the case where the Bockstein β is identically zero in K*(X; Zp), and even in this case the methods of proof used in (6) and (12) are not strong enough to show that the answer given there is correct. In this paper we repair this difficulty, obtaining a complete corrected description of K*(EZp ×ZpXp; Zp) and K*(EΣp) (theorem 3·1 below, which should be compared with ((12); theorems 3·8 and 3·9) and ((6); theorem 3)). Because of the error, the method used in (6) and (12) to construct Dyer-Lashof operations fails to go through for odd primes when non-zero Bocksteins occur, and it is not clear that this method can be repaired. We shall not deal with the construction of Dyer-Lashof operations in this paper. Instead, the first author will give a complete treatment of these operations in (5), using our present results and the theory of H∞-ring spectra to obtain strengthened versions of the results originally claimed in ((12); theorem 5·1). There is also a minor error in the mod-2 results of (12) (namely, the second formula in (12), theorem 3·8 (a) (ii)) should read

where B2 is the second mod-2 Bockstein, and a similar change is necessary in the second formula of ((12), theorem 3·8(b) (ii)). The correction of this error requires the methods of (5) and will not be dealt with here; fortunately, the mod-2 calculations of ((12), §6–9), (10) and (11) are unaffected and remain true as stated.
‘On projective planes of type (6, m)’
Authors: Lorimer P.
Published: November 2008

Alan Rahilly has pointed out to me that the proof of the Theorem in my paper (2) is incomplete. This correction will now complete it. I would also like to acknowledge here that the results which are actually established in (2) can also be found in the paper (5) by Praeger and Rahilly.

The trouble was in the proof of Proposition 2. Although the group G has a subgroup H1 which intersects each of its conjugates trivially, the same is not necessarily true of the image H1N/N of H1 in the 2-transitive representation G/N of G referred to in the paper. A theorem of M. E. O'Nan from (4) was used along with Proposition 2 to establish my theorem. What should have been done was to look at O'Nan's results more deeply and combine them with other results. Here is the way that it is done.

There is nothing actually wrong with the argument in my paper; it is just that it is not complete. The proof here will continue the argument of (2) using the notations established there.
Šarkovskii-minimal orbits
Authors: Coppel W. A.
Published: November 2008

My first paper (5), submitted for publication by my supervisor Philip Hall, dealt with iteration of a continuous map of an interval. This subject has recently become fashionable and many interesting results are discussed in the book of Collet and Eckmann (4). Several of these results hold for special types of map, for example those with negative Schwarzian derivative. However here, as in my original paper, we will be concerned with arbitrary continuous maps.
A note on the strong regularity of Nrlund means
Authors: Nurcombe J. R.
Published: November 2008

Let (pn), (qn) and (un) be sequences of real or complex numbers with

The sequence (sn) is strongly generalized Nrlund summable with index 0, to s, or s or snsN, p, Q if

and pnv=pnvpnv1, with p10. Strong Nrlund summability N, p was first studied by Borweing and Cass (1), and its generalization N, p, Q by Thorp (6). We shall say that (sn) is strongly generalized convergent of index 0, to s, and write snsC, 0, Q if sns and where sn=a0+a1++an. When qn all n, this definition reduces to strong convergence of index , introduced by Hyslop (4). If as n, the sequence (sn) is summable (, q) to s sns(, q).
Hyperbolic, fibred links and fibre-concordances
Authors: Soma Teruhiko
Published: November 2008

LetMbe a closed, connected, orientable 3-manifold. In Row [10], Jaco and Myers [3] and Myers [7], it was pointed out that the topological type of M is closely related to the knot theory inM. Therefore it is an interesting problem to find knots inMwith nice properties. Alexander provedMcontains a fibred link (see [9]). Myers proved, in [7],Mcontains a hyperbolic knot, and, in [8], every link inMis concordant to a hyperbolic link. In this paper we consider the fibred version of his results.
The norm of a Möbius transformation
Authors: Beardon A. F. , Wilker J. B.
Published: November 2008

A Möbius transformation z → (az + b)/(cz + d), ad — bc = 1, acts as a conformal transformation of the Riemann sphere , and its Poincaré extension acts as an isometry of hyperbolic 3-space modelled in the ball < 1. The size of this transformation can be measured by the matrix norm

or by the hyperbolic distance ρ through which its extension moves the point (0, 0, 0).
A generation theorem for operators commuting with group actions
Authors: Davies E. B.
Published: November 2008

We prove that an unbounded operator Z on a Banach space , which commutes with a representation of a Lie group G, is the generator of a contraction semigroup, under conditions on Z and G which have not previously been investigated. The case of an unbounded derivation Z on a C*-algebra is considered in particular detail.
3-transpositions in infinite groups
Authors: Weiss Richard
Published: November 2008

Let G be a group. A subset D will be called a set of 3-transpositions if |x| = 2 for xεD and |xy| = 3 whenever x, yεD do not commute. We will call the set D closed if xDx−1 = D for each xεD. For each xεD, let

For each subset X of D, we denote by [X] the graph with vertex set X where two elements x, yεX are joined by an edge whenever they commute. We denote by (X) the complement graph; thus two elements x, yεX are joined by an edge of (X) whenever they do not commute.
Generic 1-parameter families of caustics by reflexion in the plane
Authors: Fidal D. L. , Giblin P. J.
Published: November 2008

Let M (the mirror) be a plane oval (a smooth curve without inflexions), and let sεℝ2\M be the light source. Rays of light emanating from s are reflected by M, and the caustic by reflexion of M relative to s is the envelope of these reflected rays. In this article we suppose that M is generic (the precise assumption is stated later) and that s moves along a smooth curve in the plane; we are then able to describe how the local structure of the caustic changes. In order to state the result we recall a few facts from [3].
Mean values and classes of harmonic functions
Authors: Ramsey Thomas , Weit Yitzhak
Published: November 2008

Let μ be a finite complex Borel measure supported on the unit circle

.

In this paper, we are concerned with the characterization of the sets of functions satisfying the generalized mean value equation of the form

.

and for all ξ ∈ , | ξ | = R for some fixed R > 0.
Maximal abelian subgroups of free profinite groups
Authors: Haran Dan , Lubotzky Alexander
Published: November 2008

The aim of this note is to answer in the negative a question of W. -D. Geyer, asked at the 1983 Group Theory Meeting in Oberwolfach: Is a maximal abelian subgroup A of a free profinite group F necessarily isomorphic to , the profinite completion of