For a wide class of one-parameter families of Thue equations of arbitrary degree n[ges ]3 all solutions are determined if the parameter is sufficiently large. The result is based on the Lang–Waldschmidt conjecture, on the primitivity of the associated number fields and on an index bound, which does not depend on the coefficients. By applying the theory of Hilbertian fields and results on thin sets, primitivity is proved for almost all choices (in the sense of density) of the parameters.

Let E2(T) denote the error term in the asymptotic formula for ∫T0[mid ]ζ(½+it)[mid ]4dt. It is proved that there exist constants A>0, B>1 such that for T[ges ]T0>0 every interval [T, BT] contains points T1, T2 for which

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and that ∫T0[mid ]E2(t) [mid ]adt[Gt ]T1+(a/2) for any fixed a[ges ]1. These results complement earlier results of Motohashi and Ivić that ∫T0E2(t) dt[Lt ]T3/2 and that ∫T0E22(t) dt[Lt ]T2logCT for some C>0. Omega-results analogous to the above ones are obtained also for the error term in the asymptotic formula for the Laplace transform of [mid ]ζ(½+it)[mid ]4.

An integral version of a classical embedding theorem concerning quaternion algebras B over a number field k is proved. Assume that B satisfies the Eichler condition, that is, some infinite place of k is not ramified in B, and let Ω be an order in a quadratic extension of k. The maximal orders of B which admit an embedding of Ω are determined. Although most Ω embed into either all or none of the maximal orders of B, it turns out that some Ω are ‘selective’, in the sense that they embed into exactly half of the isomorphism types of maximal orders of B. As an application, the maximal arithmetic subgroups of B*/k* which contain a given element of B*/k* are determined.

In this paper we shall prove the main part ((3) implies (1)) of the following theorem. The proof of the theorem will be completed in a forthcoming paper [6].

The determination of the modular character tables of the sporadic O'Nan group, its automorphism group and its covering group is completed by the calculation of the 7-modular decomposition numbers. The results are obtained with the assistance of the systems GAP, MOC, and MeatAxe, and by applying new condensation methods.

A central issue in finite group modular representation theory is the relationship between the p-local structure and the p-modular representation theory of a given finite group. In [5], Broué poses some startling conjectures. For example, he conjectures that if e is a p-block of a finite group G with abelian defect group D and if f is the Brauer correspondent block of e of the normalizer, NG(D), of D then e and f have equivalent derived categories over a complete discrete valuation ring with residue field of characteristic p. Some evidence for this conjecture has been obtained using an important Morita analog for derived categories of Rickard [11]. This result states that the existence of a tilting complex is a necessary and sufficient condition for the equivalence of two derived categories. In [5], Broué also defines an equivalence on the character level between p-blocks e and f of finite groups G and H that he calls a ‘perfect isometry’ and he demonstrates that it is a consequence of a derived category equivalence between e and f. In [5], Broué also poses a corresponding perfect isometry conjecture between a p-block e of a finite group G with an abelian defect group D and its Brauer correspondent p-block f of NG(D) and presents several examples of this phenomena. Subsequent research has provided much more evidence for this character-level conjecture.

In many known examples of a perfect isometry between p-blocks e, f of finite groups G, H there are also perfect isometries between p-blocks of p-local subgroups corresponding to e and f and these isometries are compatible in a precise sense. In [5], Broué calls such a family of compatible perfect isometries an ‘isotypy’.

In [11], Rickard addresses the analogous question of defining a p-locally compatible family of derived equivalences. In this important paper, he defines a ‘splendid tilting complex’ for p-blocks e and f of finite groups G and H with a common p-subgroup P. Then he demonstrates that if X is such a splendid tilting complex, if P is a Sylow p-subgroup of G and H and if G and H have the same ‘p- local structure’, then p-local splendid tilting complexes are obtained from X via the Brauer functor and ‘lifting’. Consequently, in this situation, we obtain an isotypy when e and f are the principal blocks of G and H.

Linckelmann [9] and Puig [10] have also obtained important results in this area.

In this paper, we refine the methods and program of [11] to obtain variants of some of the results of [11] that have wider applicability. Indeed, suppose that the blocks e and f of G and H have a common defect group D. Suppose also that X is a splendid tilting complex for e and f and that the p-local structure of (say) H with respect to D is contained in that of G, then the Brauer functor, lifting and ‘cutting’ by block indempotents applied to X yield local block tilting complexes and consequently an isotypy on the character level. Since the p-local structure containment hypothesis is satisfied, for example, when H is a subgroup of G (as is the case in Broué's conjectures) our results extend the applicability of these ideas and methods.

In [2] Bieri and Strebel introduced a geometric invariant for finitely generated abstract metabelian groups that determines which groups are finitely presented. For a valuable survey of their results, see [6]; we recall the definition briefly in Section 4. We shall introduce a similar invariant for pro-p groups.

Let [ ] be the algebraic closure of [ ]p and U be the formal power series algebra [ ][lobrk ]T[robrk ], with group of units U×. Let Q be a finitely generated abelian pro-p group. We write ℤp[lobrk ]Q[robrk ] for the completed group algebra of Q over ℤp. Let T(Q) be the abelian group Hom(Q, U×) of continuous homomorphisms from Q to U×. We write 1 for the trivial homomorphism. Each v∈T(Q) extends to a unique continuous algebra homomorphism vmacr; from ℤp[lobrk ]Q[robrk ] to U.

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.

Consider a standard braid diagram as a three-dimensional figure viewed from the top; what happens when this figure is looked at from the side? Then a new braid can be obtained, and studying the connection between the initial braid and the derived braid so obtained provides both a new simple proof for the existence of the right greedy normal form of positive braids and a geometrical interpretation for the automatic structure of the braid groups.

There has been interest recently concerning when a left ordered group is locally indicable. Chiswell and Kropholler proved that every left ordered solvable-by-finite group is locally indicable, while Bergman gave examples of left ordered groups which are not locally indicable. This paper proves that every left ordered elementary amenable group is locally indicable. Every solvable-by-finite group is elementary amenable, and every elementary amenable group is amenable. The author leaves it as an open problem as to whether every left ordered amenable group is locally indicable.

Let P be a fixed p-group for an odd prime. We are interested in the localized classifying spaces BG(p) (hereafter denoted simply by BG) for various groups G sharing the same Sylow p-subgroup P. If such groups G become bigger then BG becomes smaller, because a transfer argument shows that BG is a stable summand of BP and of intermediate subgroups. For this reason, C. B. Thomas [16] first found that, in many cases, the odd component of the cohomology of sporadic simple groups should be simple even if the cohomology of P is quite complicated. In particular, when G is the biggest Janko group J4, D. J. Green [6] showed that Heven(G)/3 is the Dickson algebra of rank two. The Janko group has a Sylow 3-subgroup 31+2+ = E, the extra special 3-group of order 33 and of exponent 3. Tezuka and Yagita [14] studied the even-dimensional cohomology of all sporadic simple groups with [mid ]P[mid ] = p3; indeed, in these cases P is isomorphic to the extra-special group p1+2+ = E.

In this paper, we study BG for simple groups G with a Sylow p-subgroup E. We note the importance of G-conjugacy classes of p-pure elementary abelian p-subgroups of rank two. Here ‘p-pure’ means that all nonzero elements in the subgroup are G-conjugate. In Section 1, we see that BG is expressed as a homotopy pushout of B(NG(E)) and of B(p-pure subgroup), when NG(E) = NG(Z(E)). The last condition is always satisfied if p>3. In Section 2, the case p = 3 is studied; for example, we see the homotopy equivalence BJ4 ≅ BRu. In Section 3, we show that stable homotopy splitting is given from the dominant summands of E and of non-p-pure subgroups. When p = 3 the splitting is given explicitly; for example, BJ4 is the summand induced from the trivial representation in F3[Out(E)]. In the last section, we add the list of sporadic simple groups (due to Yoshiara) and cohomology H*(G)/√0 for p = 3 from the paper [14] for the reader's convenience. The author thanks Satoshi Yoshiara, who pointed out to him the paper of Benson and taught him the importance of p-pure subgroups. He also thanks Michishige Tezuka; indeed, most of the results in this paper are natural consequences of results in the joint paper [14] with Tezuka.

A coherent system ([Escr ], V) consists of a holomorphic bundle plus a linear subspace of its space of holomorphic sections. Motivated by the usual notion in geometric invariant theory, a notion of slope stability can be defined for such objects. In the paper it is shown that stability in this sense is equivalent to the existence of solutions to a certain set of gauge theoretic equations. One of the equations is essentially the vortex equation (that is, the Hermitian–Einstein equation with an additional zeroth order term), and the other is an orthonormality condition on a frame for the subspace V⊂H0([Escr ]).

The paper considers nonlinear, probability measure-valued dynamical systems that generalise those classical Lotka–Volterra equations in which, to use ecological terminology, the total size of a finite number of populations of interacting species is conserved. In this generalisation there is a ‘different species’ at each point of an arbitrary measurable space.

Such infinitely-many-species analogues of the classical Lotka–Volterra equations appear as hydrodynamic-type limits of stochastic systems of randomly coalescing masses related to those that have been used to model physical and chemical processes of agglomeration, coagulation and condensation.

One natural instance of this generalisation has closed-form solutions, including a family of solutions that exhibit soliton-like behaviour. The large time asymptotics of other classes of examples can be completely described using analogues of Lyapunov function techniques. Moreover, there are conserved quantities in the form of relative entropies that generalise those found by Volterra in the classical case. Finally, each solution has a series expansion as a time-varying, geometric mixture of a fixed sequence of probability measures. The existence of this expansion is related to the fact that the system is in martingale problem duality with a function-valued Markov process.

Inequalities involving classical operators of harmonic analysis, such as maximal functions, fractional integrals and singular integrals of convolutive type have been extensively investigated in various function spaces. Results on weak and strong type inequalities for operators of this kind in Lebesgue spaces are classical and can be found for example in [4, 20, 24]. Generalizations of these results to Zygmund spaces are presented in [4]. An exhaustive treatment of the problem of boundedness of such operators in Lorentz and Lorentz–Zygmund spaces is given in [3]. See also [8, 9] for further extensions in the framework of generalized Lorentz–Zygmund spaces.

As far as Orlicz spaces are concerned, a characterization of Young functions A having the property that the Hardy–Littlewood maximal operator or the Hilbert and Riesz transforms are of weak or strong type from the Orlicz space LA into itself is known (see for example [13]). In [17, 23] conditions on Young functions A and B are given for the fractional integral operator to be bounded from LA into LB under some restrictions involving the growths and certain monotonicity properties of A and B.

The main purpose of this paper is to find necessary and sufficient conditions on general Young functions A and B ensuring that the above-mentioned operators are of weak or strong type from LA into LB. Our results for (fractional) maximal operators are presented in Section 2, while Section 3 deals with fractional and singular integrals. In particular, we re-cover a result concerning the standard Hardy–Littlewood maximal operator which has recently been proved in [2, 11, 12]. Finally, in Section 4, the resolvent operator of some differential problems is taken into account and a priori bounds for Orlicz norms of solutions to elliptic boundary value problems in terms of Orlicz norms of the data are established. Let us mention that part of the results of the present paper were announced in [6].

It is proved that, for a Banach space X, the following properties are equivalent: (a) (X, weak) is fragmentable by a metric d(., .) that majorizes the norm topology (that is, the topology generated by d contains the norm topology), (b) (X, weak) is fragmentable by a metric d(., .) that majorizes the weak topology, and (c) (X, weak) is sigma-fragmentable by the norm. The paper gives a game characterization of these equivalent properties and uses it to show that a large class of Banach spaces (including the spaces that are Čech-analytic in their weak topology) enjoy the properties (a)–(c). In particular, if the unit ball B of X is a Borel subset of the second dual ball (B**, weak*) then X possesses the properties (a)–(c). This provides an alternative approach to the proofs of some of the results of J. E. Jayne, I. Namioka and C. A. Rogers on sigma-fragmentability of Banach spaces. It is also proved that C(T), with the topology p of pointwise convergence, is sigma-fragmentable by the norm whenever T=∪i[ges ]1Ti and (C(Ti), p), i[ges ]1, is sigma-fragmentable by the norm. This answers a question of R. Haydon. Finally, a simple proof is given that l∞ does not have any of the properties (a)–(c) and that (l∞(Γ), weak) is not sigma-fragmentable by any metric, provided that the cardinality of Γ is bigger than the continuum.

Let Γ be an anisotropic fractal. The aim of the paper is to investigate the distribution of the eigenvalues of the fractal differential operator

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acting in the classical Sobolev space W˚12(Ω) where Ω is a bounded C∞ domain in the plane ℝ2 with Γ⊂Ω. Here −Δ is the Dirichlet Laplacian in Ω and trΓ is closely related to the trace operator trΓ.

The restrictions Bspq(Ω) and Fspq(Ω) of the Besov and Triebel–Lizorkin spaces of tempered distributions Bspq(ℝn) and Fspq(ℝn) to Lipschitz domains Ω⊂ℝn are studied. For general values of parameters (s∈ℝ, p>0, q>0) a ‘universal’ linear bounded extension operator from Bspq(Ω) and Fspq(Ω) into the corresponding spaces on ℝn is constructed. The construction is based on a new variant of the Calderón reproducing formula with kernels supported in a fixed cone. Explicit characterizations of the elements of Bspq(Ω) and Fspq(Ω) in terms of their values in Ω are also obtained.

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.

In the years 1979–80 Sacks and Uhlenbeck [19] and Schoen and Yau [21] proved independently the following beautiful theorem.

Suppose [Cscr ] is a category with a symmetric monoidal structure, which we will refer to as the smash product. Then the Picard category is the full subcategory of objects which have an inverse under the smash product in [Cscr ], and the Picard group Pic([Cscr ]) is the collection of isomorphism classes of such invertible objects. The Picard group need not be a set in general, but if it is then it is an abelian group canonically associated with [Cscr ].

There are many examples of symmetric monoidal categories in stable homotopy theory. In particular, one could take the whole stable homotopy category [Sscr ]. In this case, it was proved by Hopkins that the Picard group is just Z, where a representative for n can be taken to be simply the n-sphere Sn [8, 19]. It is more interesting to consider Picard groups of the E-local category, for various spectra E (all of which will be p-local for some fixed prime p in this paper). Here the smash product of two E-local spectra need not be E-local, so one must relocalize the result by applying the Bousfield localization functor LE. The best-known case is E=K(n), the nth Morava K-theory, considered in [8].

In this paper we study the case E=E(n), where E(n) is the Johnson–Wilson spectrum. In this case the E-localization functor is universally denoted Ln, and we denote the category of E-local spectra by [Lscr ]. Our main theorem is the following result.