Let D=π1..., πn, where π1, ... πn are distinct Gaussian primes ≡ (mod 4) and n is any positive integer. In this paper, we prove that the value of the Hecke L-function attached to the elliptic curve ED2: y2=x3−D2x at s=1, divided by the period ω defined below, is always divisible by 2n−1. Moreover, we give a simple combinatorial criterion for this value to be exactly divisible by 2n−1. Our results are in accord with the predictions of the conjecture of Birch and Swinnerton-Dyer, and, when D is rational, enable us to prove the conjecture of Birch and Swinnerton-Dyer for ED2 when the value at s=1 is exactly divisible by 2n−1.

In this paper we present results which relate the linearization of a G-equivariant vector field at an equilibrium to the linearization of the vector field obtained by projection on the orbit space of the group action. This allows us to propose a notion of hyperbolicity for an equilibrium of the projected vector field which is compatible with the orbit space structure. It provides an alternative to the notion of normal hyperbolicity for relative equilibria (flow invariant group orbits) developed by Field [9, 10].

1. Introduction

A class [Xscr] of groups is said to be countably recognizable, if every group all of whose countable subgroups are contained in countable [Xscr]-subgroups is itself an [Xscr]-group. Many examples of such classes are discussed in section 8·3 of [20]. In the present work we are concerned with the question of how far countable recognizability can be obtained for classes of finitary linear groups. Recall that a group is said to be finitary []-linear if it is isomorphic to a subgroup of FGL[](V), the group of all invertible []-linear transformations α of the []-vector space V with the property that the image of the endomorphism α−idV has finite []-dimension. This generalizes the notion of linearity. A survey about features of finitary linear groups is given in [18].

Upper bounds are established on the shifts in a minimal resolution of a multigraded module. Similar bounds are given on the coefficients in the numerator of the Backelin–Lescot rational expression for multigraded Poincaré series.

Recently, Hitoshi Murakami has shown that the quantum SU(2)- and SO(3)-invariants of 3-manifolds at roots of unity of prime order are algebraic integers. Unfortunately, his proof is by a very complicated computation. Here, a quite different and very simple proof is presented, based on the second author's method to show that the Turaev–Viro invariant is the square of the modulus of the Reshetikhin–Turaev invariant.

1. Introduction

It has been known for some time that contact structures show a high degree of topological flexibility in the sense that many topological operations can be performed on contact manifolds while preserving the contact property. For instance, Martinet [14] used a surgery description of 3-manifolds to show that every closed, oriented 3-manifold admits a contact structure, and alternative proofs of this result were given later by Thurston and Winkelnkemper [18], who based their proof on an open book decomposition, and Gonzalo [8], who used branched covers. These, however, are all strictly 3-dimensional constructions.

0. Introduction

Morin [M] gave a local normal form for singular maps having almost maximal rank, where almost maximal means maximal minus 1. The aim of the present paper is to give a global version of his normal form. We concentrate here on the case of [sum ]1, 1, 0-singular maps. (For the definition see [Bo], [A–G–V], [G–G] and also here below.) The case of [sum ]1, 0 singular maps was considered by Haefliger in [Ha], see also [Sz1] and [Sz2]. For the motivation in finding such a global normal form see [Sz1], [Sz2], and the final remarks in this paper.

Let G be a connected locally finite simplicial graph with rk(π1(G))[ges ]2 and let T be the universal cover of G. Consider a π1(G)-invariant conformal density μ of dimension d on ∂T. The total mass function ϕμ of μ is defined on the set of vertices of G. Let |ϕμ| be its l2-norm. Let Ω be the geodesic flow space of G and mμ the invariant measure on Ω associated to μ. The main results of this paper are the following:

(i) Assume that there exists an integer k[ges ]3 such that the degree at each vertex of G is [les ]k. Then, if d>½log(k−1) and mμ(Ω)<∞, we have ϕμ∈l2(V) and

|ϕμ|2[les ]Cmμ(Ω),

with C=(e2d−1)/(e2d−k+1).

(ii) Assume that there exists an integer k[ges ]3 such that the degree at each vertex of G is [ges ]k. Then, if ϕμ∈l2(V), we have d>½log(k−1) and

Cmμ(Ω)[les ]|ϕμ|2.

1. Introduction

In a recent paper [3], an extended Liouville–Green formula

formula here

was developed for solutions of the second-order differential equation

formula here

Here γM(x)∼Q−¼(x), QM(x)∼Q−½(x) and εM(x)→0 as x→∞, while M([ges ]2) is an integer and γM and QM can be defined in terms of Q and its derivatives up to order M−1. The general form of (1·1) had been obtained previously by Cassell [5], [6], [7] and Eastham [10], [11, section 2·4]. In particular, the proof of (1·1) in [10] and [11] depended on the formulation of (1·2) as a first-order system and then on a process of M repeated diagonalization of the coefficient matrices in a sequence of related differential systems.

The proportion ρk of gaps with length k between square-free numbers is shown to satisfy logρk=−(1+o(1))(6/π2)klogk as k→∞. Such asymptotics are consistent with Erdős's challenge to prove that the gap following the square-free number t is smaller than clogt/log logt, for all t and some constant c satisfying c>π2/12. The results of this paper are achieved by studying the probabilities of large deviations in a certain ‘random sieve’, for which the proportions ρk have representations as probabilities. The asymptotic form of ρk may be obtained in situations of greater generality, when the squared primes are replaced by an arbitrary sequence (sr) of relatively prime integers satisfying [sum ]r1/sr<∞, subject to two further conditions of regularity on this sequence.

It has been shown that univalent circle packings filling the complex plane C are unique up to similarities of C. Here we prove that bounded degree branched circle packings properly covering C are uniquely determined, up to similarities of C, by their branch sets. In particular, when branch sets of the packings considered are empty we obtain the earlier result.

We also establish a circle packing analogue of Liouville's theorem: if f is a circle packing map whose domain packing is infinite, univalent, and has recurrent tangency graph, then the ratio map associated with f is either unbounded or constant.

1. Introduction

Along with the prevalence of probabilistic methods in approximation theory the study of approximation of operators by other operator sequences, or the study of limiting properties of operator sequences, has attracted more and more attention. It was shown by Stancu [15] in the 1960s that Szász operators and Baskakov operators are limiting operators of Stancu–Mühlbach operator sequences in an appropriate sense. Later, Butzer and Hahn [4, 5, 11] established rates of convergence of some probabilistic type operators towards their limiting operators as applications of their general approximation theorems in probability theory. In 1988, Khan [13] proved that Szász operators are limiting operators of Bleimann–Butzer–Hahn operator sequences. Recently de la Cal, Luquin, Adell and San Miguel investigated the limiting properties of a series of probabilistic type operator sequences and exhibited that their limiting operators might be Bernstein, Szász, Baskakov, gamma or beta operators [1, 2, 6, 7, 8].

We consider the diffraction of the dominant acoustic wave mode which propagates out of the mouth of a semi-infinite waveguide made of a soft and hard half plane. This semi-infinite waveguide is symmetrically located inside an infinite waveguide whose infinite plates are soft and hard. The whole system constitutes a trifurcated waveguide. Another trifurcated waveguide is obtained by interchanging the infinite plates. A closed form solution of the resulting matrix Wiener–Hopf equation is obtained for each configuration. Thus we present exact closed-form solutions to two new waveguide trifurcation problems.

The proof of Lemma 2·4 in the paper above is not correct because of mistakes in computations (4)–(8) using Araki operations.