Motivated by Cremona and Mazur's notion of visibility of elements in Shafarevich–Tate groups [6, 27], there have been a number of recent works which test its compatibility with the Birch and Swinnerton–Dyer conjecture and the Bloch–Kato conjecture. These conjectures provide formulas for the orders of Shafarevich–Tate groups in terms of values of $L$-functions. For example, one may see recent work of Agashe, Dummigan, Stein and Watkins [1, 2, 10, 11]. In their examples, they find that the presence of visible elements agrees with the expected divisibility properties of the relevant $L$-values.

It is proved that if $F$ is an infinite field with characteristic different from $2$, whose theory is supersimple, and $C$ is an elliptic or hyperelliptic curve over $F$ with generic ‘modulus’, then $C$ has a generic $F$-rational point. The notion of generity here is in the sense of the supersimple field $F$.

The Volterra convolution operator [Vscr ]f(x) = ∫x0ϕ (x−y)f(y)dy, where ϕ(·) is a non-negative non-decreasing integrable kernel on [0, 1], is considered. Under certain conditions on the kernel ϕ, the maximal Banach function space on [0, 1] on which the Volterra operator is a continuous linear operator with values in a given rearrangement invariant function space on [0, 1] is identified in terms of interpolation spaces. The compactness of the operator on this space is studied.

A method is given to solve any Thue–Mahler equation when the coefficients and variables come from the ring of integers of a number field. The method involves using an algorithm for solving an S-unit equation and a method of reducing the number of exponential variables. The method is used to solve the equation

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First, two comparison theorems are established between the first order delay differential equation

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and two related second order ordinary differential equations in the case when

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Next, by using these comparison theorems some new oscillation and non-oscillation criteria are given for (*) under condition (**). As a consequence, an open problem by Elbert and Stavroulakis is completely solved.

All references unless otherwise stated are to [1]. The lemma in Subsection 2.3 of [1] is incorrect as stated. The statements and proofs of the lemmas in Subsections 3.2 and 3.4 are in need of revision. These revisions do not affect any other results in [1].

The authors thank H.-J. Schneider for pointing out the errors.

The most powerful geometric tools are those of differential geometry, but to apply such techniques to finitely generated groups seems hopeless at first glance since the natural metric on a finitely generated group is discrete. However Gromov recognized that a group can metrically resemble a manifold in such a way that geometric results about that manifold carry over to the group [18, 20]. This resemblance is formalized in the concept of a ‘quasi-isometry’. This paper contributes to an ongoing program to understand which groups are quasi-isometric to which simply connected, homogeneous, Riemannian manifolds [15, 18, 20] by proving that any group quasi-isometric to H2×R is a finite extension of a cocompact lattice in Isom(H2×R) or Isom(SL˜(2, R)).

A class of harmonic function spaces is introduced and studied, namely the spaces $H^p_\sigma(G)$ of $\sigma$-harmonic $L^p$ functions on a locally compact group $G$, for $1\leq p \leq \infty$ and a given complex measure $\sigma$ on $G$ of unit norm. It is shown that there is a contractive projection from $L^p(G)$ onto $H^p_\sigma(G)$, for $1\lt p\leq \infty$, and structural results for $H^p_\sigma (G)$ are deduced. Given an adapted probability measure $\sigma$ on $G$, a uniqueness result is proved, that the space $H^p_\sigma(G)$ contains only constant functions, for $1\leq p\,{\lt}\,\infty$. For any $\sigma$, a result on the dimension of $H^1_\sigma (G)$ is proved.

Let f(x) be an integer-valued polynomial with no fixed integer divisor [ges ] 2, that is, for no integer d [ges ] 2 does d[mid ]f(x) for all integers x. One generalization of the famous Waring problem is to determine whether, for large enough s, the equation

formula here

is solvable in positive integers x1, …, xs for sufficiently large integers n. The existence of such s for every f was established by Kamke [5] in 1921. Subsequent authors (Pillai, Hua [2–4], Vinogradov, Načaev [7], and others) have studied the problem of bounding G(f), the least s for which (1.1) is solvable for all large n.

Questions of local solubility of (1.1), that is, solubility of the congruence

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play a more important and complicated role in this problem than in the classical Waring problem. Let Γ0(f) denote the least number s so that (1.2) is solvable for every pair n, q. It is well known that Γ0(xk) [les ] 4k for every k, but Hua [4] found that for every k, the polynomial

formula here

satisfies Γ0(fk) [ges ] 2k − 1 (take s = 2k−2, q = 2k and n = (−1)k in (1.2)). Clearly G(f) [ges ] Γ0(f), but one can say more by restricting the values of n under consideration, as has been done by several authors in the case f(x) = x4 (for example [1, 6]).

The singular series

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where e(z) = e2πiz, encapsulates the local solubility information. In particular, [Sfr ]s, f(n) [ges ] 0 for every n and [Sfr ]s, f(n) > 0 if and only if (1.2) is soluble for every q. Define G(f) to be the least number s so that for every δ >0 and every n>n0(δ) with [Sfr ]s, f(n) [ges ] δ, (1.1) is soluble. The reason for taking [Sfr ]s, f(n) [ges ] δ instead of [Sfr ]s, f(n) > 0 is that we wish to exclude from consideration certain n lying in sparse sequences for which (1.1) is insoluble but [Sfr ]s, f(n) > 0. For example, taking f(x) = x4, s = 15 and nj = 79·16j (j = 0, 1, …), it can be shown that (1.1) is not soluble for n = nj, that [Sfr ]s, f(nj) > 0 for all j, and that [Sfr ]s, f(nj) → 0 as j → ∞. It is known that G(x4) = 16 (see [1]) and that G(x4) [ges ] 11 almost holds (see [6]).

This paper presents a rigidity theorem for infinite-dimensional Bergman spaces of hyperbolic Riemann surfaces, which states that the Bergman space $A^{1}(M)$, for such a Riemann surface $M$, is isomorphic to the Banach space of summable sequence, $l^{1}$. This implies that whenever $M$ and $N$ are Riemann surfaces that are not analytically finite, and in particular are not necessarily homeomorphic, then $A^{1}(M)$ is isomorphic to $A^{1}(N)$. It is known from V. Markovic that if there is a linear isometry between $A^{1}(M)$ and $A^{1}(N)$, for two Riemann surfaces $M$ and $N$ of non-exceptional type, then this isometry is induced by a conformal mapping between $M$ and $N$. As a corollary to this rigidity theorem presented here, taking the Banach duals of $A^{1}(M)$ and $l^{1}$ shows that the space of holomorphic quadratic differentials on $M,\ Q(M)$, is isomorphic to the Banach space of bounded sequences, $l^{\infty }$. As a consequence of this theorem and the Bers embedding, the Teichmüller spaces of such Riemann surfaces are locally bi-Lipschitz equivalent.

In this paper we study the L1 mean

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of the exponential sum M(α)= [sum ]n[les ]Xμ(n)e(nα), where μ(n) is the Möbius function and e(x)=e2πix. From the Cauchy–Schwarz inequality and Parseval's identity, we have

formula here

and it is an interesting problem to investigate whether (2) reflects the true order of magnitude of (1).

In this paper we describe our discovery that the sporadic simple groups HS and M22 are contained in the simple Chevalley group E7(5).

The work of [9] produces a short list of the possibilities for a sporadic simple subgroup of an exceptional group of Lie type. Apart from possible embeddings of M22 and HS in groups of type E7 in characteristic 5, all of the embeddings of [9] are already known to occur. Thus our paper completes the classification of sporadic simple subgroups of exceptional groups of Lie type.

We give two proofs of the embedding HS<E7(5). The first of these proofs is entirely computer free, while the second proof makes some use of machine calculations. As a step in our hand proof of HS<E7(5) we establish the embedding M22<E7(5): of course, since M22 is a subgroup of HS, this result also follows as a consequence of our computer proof of HS<E7(5).

We were led to conjecture the inclusion HS<E7(5) for the following reasons (similar arguments are presented in [9]). The double cover 2.HS has a faithful 56- dimensional character, whose values are compatible with the character values of groups of type E7 acting on their natural 56-dimensional module. Now HS and 2.HS contain a subgroup 52[ratio ]20, an elementary abelian group of order 25 extended by a cyclic group of order 20 acting faithfully on the 52. Since 20 is not the order of an element in the Weyl group W(E7) = 2×S6(2), it can be shown that 52[ratio ]20 does not embed in groups of type E7(K), where K is a field of characteristic prime to 5. Thus HS embeds in E7(q) only if 5[mid ]q. On the other hand, all local subgroups of 2.HS embed in 2.E7(5), where our conjecture HS<E7(5).

Throughout, G denotes the double cover 2.E7(5), G¯ denotes the simple group E7(5), and V denotes the natural 56-dimensional module for G over GF(5). Most of our notation follows that of the atlas [4]. The two sections of our paper are completely independent.

The interaction between harmonic analysis and operator theory has been fruitful. In [1] Arveson related results in the theory of operator algebras to spectral synthesis. He defined synthesis for subspace lattices and proved that certain classes of lattices are synthetic. The main result in this paper is a generalization of a result of Arveson for the case of subspace maps. In order to describe in more detail the content of the present work, we need to introduce some definitions and facts from [2] and [9].

Let ${\cal H}_1$ and ${\cal H}_2$ be separable complex Hilbert spaces, and ${\cal P}_i$ be the lattice of all (orthogonal) projections on ${\cal H}_i, i = 1, 2$ . Following Erdos [1], we let ${\cal M}({\cal P}_1, {\cal P}_2)$ denote the set of all maps $\varphi : {\cal P}_1 \longrightarrow {\cal P}_2$ which are 0-preserving and $\vee$ -continuous (that is, they preserve arbitrary suprema). We will call such maps subspace maps. It was shown in [2] that each $\varphi \in {\cal M}({\cal P}_1, {\cal P}_2)$ uniquely defines semi-lattices ${\cal S}_{1\varphi} \subseteq {\cal P}_1$ and ${\cal S}_{2\varphi} = \varphi({\cal P}_1) \subseteq {\cal P}_{2}$ such that $\varphi$ is a bijection between ${\cal S}_{1\varphi}$ and ${\cal S}_{2\varphi}$ and is uniquely determined by its restriction to ${\cal S}_{1\varphi}$ . Moreover, ${\cal S}_{1\varphi}$ is meet-complete and contains the identity projection while ${\cal S}_{2\varphi}$ is join-complete and contains the zero projection. If ${\cal S}_{1\varphi}$ and ${\cal S}_{2\varphi}$ are commutative, we say that $\varphi$ is a commutative subspace map.

Let f[ratio ][Copf ]n, 0→[Copf ]p, 0 be a [Kscr ]-finite map germ, and let i=(i1, …, ik) be a Boardman symbol such that [sum ]i has codimension n in the corresponding jet space Jk(n, p). When its iterated successors have codimension larger than n, the paper gives a list of situations in which the number of [sum ]i points that appear in a generic deformation of f can be computed algebraically by means of Jacobian ideals of f. This list can be summarised in the following way: f must have rank n−i1 and, in addition, in the case p=6, f must be a singularity of type [sum ]i1,i2.

Periodic neutral functional differential equations are considered. Sufficient conditions for existence, uniqueness and global attractivity of periodic solutions are established by combining the theory of monotone semiflows generated by neutral functional differential equations and Krasnosel'skii's fixed-point theorem. These results are applied to a concrete neutral functional differential equation that can model single-species growth, the spread of epidemics, and the dynamics of capital stocks in a periodic environment.

Most of our notation is taken from James [7], where further details of the representation theory of the symmetric groups may be found; note, however, that we write functions on the left.

Let $n$ be a non-negative integer, and $\lambda$ a partition of $n$ . Say that two $\lambda$ -tableaux are row equivalent if one can be obtained from the other by permuting the entries within each row, and define column equivalence similarly. Let $\sim_{\rm row}$ and $\sim_{\rm col}$ denote these relations.

The work in this paper is part of an ongoing program to classify maximal orders on surfaces via their ramification data. Del Pezzo orders and ruled orders have already been classified by the authors and others. In this paper, we classify numerically Calabi–Yau orders which are the noncommutative analogues of minimal surfaces of Kodaira dimension zero.

We show that solutions of analytic elliptic partial differential equations of the form

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in a simply connected domain Ω ⊂ ℝn can be extended holomorphically into [Nscr ] (Ω) ⊂ [Copf ]n, the so-called kernel of the harmonicity hull of Ω. This extends results of Avanissian, Aronszajn etc., on polyharmonic functions and also results of Vekua, Khavinson and Shapiro in ℝ2. We also find the domain of influence for solutions of a certain subclass of these operators, in terms of their Cauchy data on analytic hypersurfaces in [Copf ]n (a complex Huygens' principle). As an application, we investigate reflection properties of these solutions and, in particular, solutions of the Helmholtz equation in ℝ3.

In this paper, we give a lower bound of the p-part of the reduced Euler characteristic of the order complex of the centric p-radical subgroups by studying vertices of indecomposable summands of the reduced Lefschetz module. This bound is in fact best possible for at least some groups, and also provides a uniform explanation of the observed phenomenon on the reduced Euler characteristic for some sporadic simple groups.

The paper calculates the average density of the normalized Hausdorff measure on the fractal set generated by a conformal iterated function system. It equals almost everywhere a positive constant given by a truncated generalized s-energy integral, where s is the corresponding Hausdorff dimension. As a main tool a conditional Gibbs measure is determined. The appendix proves an appropriate extension of Birkhoff's ergodic theorem which is also of independent interest.