For a non-zero integer n, let d(n) denote the number of positive divisors of n. Let a, b and c be integers with a>0, and set Δ=b2−4ac. If the quadratic polynomial ax2+bx+c is irreducible over the rational numbers Q (that is, if Δ is not the square of an integer), then one has

formula here

as X→∞, for some λ depending on a, b and c (see [7]). In this paper we discuss the way in which λ depends on a, b and c, giving a precise, compact expression in terms of class numbers. This extends previous work for the case a=1, Δ<0 (see [4]).

For the case a=1, b=0, a much better description of the error is given in [2], with the following expression for λ:

formula here

Here ρ is a multiplicative function, defined below, and (p/q) is the Legendre/Jacobi symbol.

When a system of differential equations is derivable from a variational principle, an outstanding consequence is the connection which ensues between symmetries and conservation laws. The idea behind the present work is to use this consequence as a new characterization of the variational principle itself. For general systems of equations there is a rather weak form of the connection, as will be demonstrated; but it has highly arbitrary and non-invariant features, and, for instance, there is not a formula associating a particular infinitesimal invariance with a particular constant of the motion. In terms of the modern technical concept which will be employed, the transformation is not natural. However, if the system is derivable from a variational principle, there follows a preferred transformation; and it has certain invariance properties which accord with the concept of naturality.

In this paper we consider algebraic integers, θ, such that θ and θ¯ (θ ╪ θ¯) are the only algebraic conjugates of θ that lie outside the unit circle.

Let S1 be the set of algebraic integers, ρ, necessarily real, all of whose conjugates, except ρ, satisfy | z < 1.

An infinitely long canal with uniform cross-section is filled with inviscid fluid. It is required first to show that any small two-dimensional motion of the fluid can be represented as the superposition of normal mode disturbances. A suitable generalized Green's function G(x, y; ξ) is constructed and is used to set up an integral equation (2·9) for the velocity potential on the free surface. It is shown that the eigenfunctions are complete and so are their (possibly time-dependent) extensions to the whole canal, in the sense that an arbitrary disturbance possesses a unique representation. In section 5, it is required to find asymptotic approximations to the large eigenvalues of (2·9). For this purpose a different integral equation (5·5) is set up on the canal, the kernel of which is the sum of a degenerate kernel and a small kernel. The solutions of this equation can therefore be obtained by iteration. The form of the mth eigenvalue is shown to be

for sufficiently large m.

This paper treats an extension of the problem considered by the authors in a recent paper (1). The minimum energy principle of the classical theory of elasticity was used in the above paper for evaluating the elastic field when an elliptic region (the inclusion, which could be of a material different from the rest) undergoes spontaneous dimensional change in an otherwise unstrained infinite medium (the matrix). By modification of this method, it has been possible to deal with the case when the inclusion is spherical or circular and the matrix is under uniform tension at infinity (2). The present paper deals with the much more general case when the matrix is under tension, at infinity, inclined at any angle to the major axis of the elliptic inclusion. The solution has been possible by the combination of the complex variable method coupled with minimum energy principle and superposition methods of linear elasticity theory. As a consequence we immediately derive almost without further calculation many particular cases, viz. (i) the inclusion problem in a matrix under axial tension parallel to either of the axes, (ii) under all round uniform tension (or pressure) etc. It is obvious that the results for the respective cases of a circular inclusion can be deduced from these results.

It also solves the problem of composite sections under external forces at infinity because of the complete freedom in choosing the elastic constant of the inclusion which can be different from that of the matrix. As a corollary, it solves the problem of a cavity under stress at infinity.

A lattice ring is a lattice group ((l), page 214) and a ring in which ab ≥ 0 whenever a ∧ b ≥ 0.

In any lattice group (commutative or not) we define a+ = a ∨ 0, a− = (−a) ∨ 0 and |a| = a+ + a−. Itisknown((1). pages 219,220) that a+ ∧ a− = 0, a = a+ − a−, |a| = a+ ∨ a−, and that a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c), and a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c). For a non-empty subset M of a lattice group we define

Let G be a locally compact group and let Ĝ be its character group. Among other results, the minimal left ideals of L∞(G)* and LUC(G)* are all found when G is abelian. The main tool used for this study is the set of (topological) χ-invariant functionals (χ∈Ĝ), defined in this paper.

In [5] Kalf obtained a characterization of the Friedrichs extension TF of a general semi-bounded Sturm–Liouville operator T, the only assumptions made on the coefficients being those necessary for T to be defined. The domain D(TF) of TF was described in terms of ‘weighted’ Dirichiet integrals involving the principal and non-principal solutions of an associated non-oscillatory Sturm–Liouville equation. Conditions which ensure that members of D(TF) have a finite Dirichlet integral were subsequently given by Rosenberger in [7].

We study a (p-adic) geometric analogue for abelian varieties over a function field of characteristic p of the cyclotomic Iwasawa theory and the non-commutative Iwasawa theory for abelian varieties over a number field initiated by Mazur and Coates respectively. We will prove some analogue of the principal results obtained in the case over a number field and we study new phenomena which did not happen in the case of number field case. We also propose a conjecture (Conjecture 1.6) which might be considered as a counterpart of the principal conjecture in the case over a number field.

It is well known that the so-called Beauville number B = 25.33.52.7.11.13.17.19 is a universal bound of the global indices of ℚ–Calabi–Yau 3-folds, but it has been unknown whether this number is best possible or not.

Let a finite group G act topologically on a closed smooth manifold Mn. One of the most natural and basic questions is whether such an action can be smoothed. More precisely, let γ:G × Mn → Mn be a topological action of G on Mn. The action γ can be smoothed if there exists a smooth action and an equivariant homeomorphism It is well known that for n ≤ 2 every finite topological group action on Mn is smoothable. However already for n = 3 there are examples of topological actions on 3-manifolds which cannot be smoothed (see [1, 2] and references there). All these actions fail to be smoothable because of bad local behaviour.

The intuition behind the notion of a cycle-free partial order (CFPO) is that it should be a partial ordering (X, ≤ ) in which for any sequence of points (x0, x1;…, xn–1) with n ≤ 4 such that xi is comparable with xi+1 for each i (indices taken modulo n) there are i and j with j ╪ i, i + 1 such that xj lies between xi and xi+1. As its turn out however this fails to capture the intended class, and a more involved definition, in terms of the ‘Dedekind–MacNeille completion’ of X was given by Warren[5]. An alternative definition involving the idea of a betweenness relation was proposed by P. M. Neumann [1]. It is the purpose of this paper to clarify the connections between these definitions, and indeed between the ideas of semi-linear order (or ‘tree’), CFPO, and the betweenness relations described in [1]. In addition I shall tackle the issue of the axiomatizability of the class of CFPOs.

The main focus in this paper is the algebraic K-theory and higher Chow groups of linear varieties and schemes. We provide Kunneth spectral sequences for the higher algebraic K-theory of linear schemes flat over a base scheme and for the motivic cohomology of linear varieties defined over a field. The latter provides a Kunneth formula for the usual Chow groups of linear varieties originally obtained by different means by Totaro. We also obtain a general condition under which the higher cycle maps of Bloch from mod-lv higher Chow groups to mod-lv étale cohomology are isomorphisms for projective nonsingular varieties defined over an algebraically closed field of arbitrary characteristic p [ges ] 0 with l ≠ p. It is observed that the Kunneth formula for the Chow groups implies this condition for linear varieties and we compute the mod-lv motivic cohomology and mod-lv algebraic K-theory of projective nonsingular linear varieties to be free ℤ/lv-modules.

1. Let f(x) be a complex function belonging to LP (−∞, ∞); i.e. let f(x) be measurable, and |f(x)|p integrable, over (−∞, ∞). The function

is called the Fourier transform of f(x), if the integral on the right exists, in some sense, for almost every value of y. It is well known that, if 1 ≤ p ≤ 2, the integral (1) converges in mean, with index p′ = p/(p – l)† i.e. that

where

In an earlier paper [10], we constructed a ‘locally finite approximation away from a given prime p’ of the classifying space BG of a Lie group with finite component group. Such an approximation consists of a locally finite group g and a homotopy class of maps which in particular induces an isomorphism in cohomology with finite coefficients of order prime to p. The usefulness of such a construction is that it reduces various homotopy-theoretic questions concerning the space BG to the corresponding questions concerning Bπ for finite subgroups π. For example, we demonstrated in [10] how H. Miller's proof of the Sullivan conjecture concerning maps from , where π is a finite group and X is a finite-dimensional complex, can be extended to maps BG→X for G a Lie group with finite component group.

Waves in a rotating, stratified fluid of variable depth are considered. The perturbation pressure is used throughout as the dependent variable. This proves to have some advantages over the use of the vertical velocity. Some previous three-dimensional solutions for internal waves in a wedge are shown to be incorrect and the correct solutions presented. A WKB analysis is then performed for the general problem and the results compared with the exact solutions for a wedge. The WKB solution is also applied to long surface waves on a rotating ocean.

Within the Tate–Shafarevich group of an elliptic curve E defined over a number field K, there is a canonical subgroup defined by imposing stronger conditions at the places above a given prime p. This group appears naturally in the Iwasawa theory for E. We propose a study of what one can say about the relation to the full Tate–Shafarevich group. Some numerical examples are included, as well as a few conjectures.

We determine those pseudovarieties of groups ${\bf H}$ for which the power monoids $P(G)$, ranging over all groups $G$ in ${\bf H}$, satisfy the same profinite identities (i.e. pseudoidentities) as all semidirect products of ${\cal J}$-trivial monoids by groups in ${\bf H}$. That is, in the language of finite monoid theory, we characterize all solutions to the pseudovariety equation ${\bf PH}\,{=}\,{\bf J}\ast {\bf H}$. The characterization is in terms of the geometry of the Cayley graphs of the free pro-${\bf H}$ groups as well as in terms of the pro-${\bf H}$ topology of a finitely generated free group.

It is well known that the rotational specific heat of a diatomic gas is given by

where R is the gas constant, σ = h2/8π2AKT, h is Planck's constant, T is the absolute temperature, K is Boltzmann's constant, and A is the moment of inertia of the molecule.