bluuebunny/crossref_metadata_2025_split huggingface dataset:
An application of the Thue–Siegel–Roth theorem to elliptic functions
Authors: Coates J. 
Published: November 2008

Let α1, …, αn be n ≥ 2 algebraic numbers such that log α1,…, log αn and 2πi are linearly independent over the field of rational numbers Q. It is well known (see (6), Ch. 1) that the Thue–Siegel–Roth theorem implies that, for each positive number δ, there are only finitely many integers b1,…, bn satisfyingwhere H denotes the maximum of the absolute values of b1, …, bn. However, such an argument cannot provide an explicit upper bound for the solutions of (1), because of the non-effective nature of the theorem of Thue–Siegel–Roth. An effective proof that (1) has only a finite number of solutions was given by Gelfond (6) in the case n = 2, and by Baker(1) for arbitrary n. The work of both these authors is based on arguments from the theory of transcendental numbers. Baker's effective proof of (1) has important applications to other problems in number theory; in particular, it provides an algorithm for solving a wide class of diophantine equations in two variables (2).
Linear theory of Cosserat surface and elastic plates of variable thickness
Authors: Green A. E. , Naghdi P. M. , Wenner M. L. 
Published: November 2008

AbstractWithin the scope of the linear isothermal theory of an elastic Cosserat surface, constitutive equations are derived for an initially flat Cosserat surface in which the initial director (along the normal to the initial surface) is allowed to depend on the surface coordinates. These constitutive equations correspond to those for bending and stretching of a transversely isotropic three-dimensional plate. Special attention is given to the relevance and applicability of the results to bending of (three-dimensional) plates of variable thickness and comparison is made with a set of equations for elastic plates of variable thickness obtained, by an approximation procedure, from the three-dimensional equations.
Classification of crystal lattices
Authors: Schwarzenberger R. L. E. 
Published: November 2008

There are two natural ways in which to attempt a classification of the lattices which occur in crystallography or, more generally, of n-dimensional lattices. The first method is algebraic and proceeds by classifying the symmetry groups of the lattices. Thus an n-dimensional lattice is a discrete subgroupof Rn and determines a symmetry groupwhich is a subgroup of the orthogonal group On. Two lattices T1 and T2 are said to determine the same crystal system if the symmetry groups G(T1) and G(T2) are conjugate: that is there exists a linear isomorphism ø: Rn → Rn such that
A convolution involving Bell polynomials
Authors: Heselden G. P. M. 
Published: November 2008

AbstractA convolution formula is established for Bell polynomials. This is expressed in seven equivalent ways and used to derive further properties of these polynomials. The application of these results to some twenty-seven special polynomial sets is shown and illustrated in the case of binomial, Hermite, Gegenbauer and generalized Bernoulli sets.
Quadratic Langrangians and static gravitational fields
Authors: Buchdahl H. A. 
Published: November 2008

AbstractIt is shown that the solutions of the field equations generated by the most general effective quadratic LagrangianaR2+bRklRklwhich are static, singularity-free, of classC4and asymptotically flat (of rankA3) are (i) conformally flat when 3a+b= 0, (ii) flat when 3a+b≠ 0 (b≠ 0), and (iii) have zero scalar curvature whenb= 0.
The combinatorial structure of the Hughes–Zassenhaus plane of order 25
Authors: Room T. G. 
Published: November 2008

AbstractThe Hughes–Zassenhaus plane, , of order 25 is the simplest of the exceptional Hughes planes (see, e.g. (1), p. 391).In this paper, an incidence table is constructed analogous to tables for the field plane and the regular Hughes plane, and the following rather surprising properties of the plane emerge:(i) The table has much greater internal symmetry than either of the analogous tables,(ii) any polarity with regard to a conic in the central subplane Δ0 of extends to two pairs of polarities in one pair with 30 and the other with 60 singular points each in addition to the six in Δ0.
Adding propositional connectives to countable infinitary logic
Authors: Friedman Harvey 
Published: November 2008

AbstractFor 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.
A new formulation of the Liénard–Chipart stability criterion
Authors: Barnett S. 
Published: November 2008

AbstractThe 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 formulaA 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 aswhere 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

AbstractA 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 serieswith locally nilpotent factors.
Motion of a classical particle with spin
Authors: Ellis J. R. 
Published: November 2008

AbstractThe 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.
Applications of variational principles to classical perturbation theory in general relativity
Authors: Robinson D. C. 
Published: November 2008

AbstractThe 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.
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 satisfyThe functions introduced here and called darts, generalize this type of series condition.
Inclusion conditions for Hurwitzian and Schur sets in
Authors: Smith Russell A. 
Published: November 2008

AbstractA 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).
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 byIf 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 thatis 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
Homomorphs and formations of given derived class
Authors: Lafuente J. 
Published: November 2008

AbstractA 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.
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 formwhere 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 functionwhere 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):
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.
On Hurwitz extensions by PSL2(7)
Authors: Cohen Jeffrey M. 
Published: November 2008

AbstractIn 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.
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 matricessatisfyingand such thatThough 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.
On the Hamiltonian structure of evolution equations
Authors: Olver Peter J. 
Published: November 2008

AbstractThe 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.
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 bywhere k ∈ ℕ = {1,2,3,…} andThus, 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.
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 inequalityhas a solution in integers x1, x2, …, xr.
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.
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

AbstractIn 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 readwhere 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.