It is proved that the number of countable models of a countable supersimple theory is either 1 or infinite. This result is an extension of Lachlan‘s theorem on a superstable theory.

One of the most famous theorems in number theory states that there are infinitely many positive prime numbers (namely p = 2 and the primes p ≡ 1 mod4) that can be represented in the form x21+x22, where x1 and x2 are positive integers. In a recent paper, Fouvry and Iwaniec [2] have shown that this statement remains valid even if one of the variables, say x2, is restricted to prime values only. In the sequel, the letter p, possibly with an index, is reserved to denote a positive prime number. As p21+p22 = p is even for p1, p2 > 2, it is reasonable to conjecture that the equation p21+p22 = 2p has an infinity of solutions. However, a proof of this statement currently seems far beyond reach. As an intermediate step in this direction, one may quantify the problem by asking what can be said about lower bounds for the greatest prime divisor, say P(N), of the numbers p21+p22, where p1, p2 [les ] N, as a function of the real parameter N [ges ] 1. The well-known Chebychev–Hooley method combined with the Barban–Davenport–Halberstam theorem almost immediately leads to the bound P(N) [ges ] N1−ε, if N [ges ] No(ε); here, ε denotes some arbitrarily small fixed positive real number. The first estimate going beyond the exponent 1 has been achieved recently by Dartyge [1, Théorème 1], who showed that P(N) [ges ] N10/9−ε. Note that Dartyge's proof provides the more general result that for any irreducible binary form f of degree d [ges ] 2 with integer coefficients the greatest prime divisor of the numbers [mid ]f(p1, p2)[mid ], p1, p2 [les ] N, exceeds Nγd−ε, where γd = 2 − 8/(d + 7). We in particular want to point out that Dartyge does not make use of the specific features provided by the form x21+x22. By taking advantage of some special properties of this binary form, we are able to improve upon the exponent γ2 = 10/9 considerably.

Let F1, …, Ft be diagonal forms of degree k with real coefficients in s variables, and let τ be a positive real number. The solubility of the system of inequalities

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in integers x1, …, xs has been considered by a number of authors over the last quarter-century, starting with the work of Cook [9] and Pitman [13] on the case t = 2. More recently, Brüdern and Cook [8] have shown that the above system is soluble provided that s is sufficiently large in terms of k and t and that the forms F1, …, Ft satisfy certain additional conditions. What has not yet been considered is the possibility of allowing the forms F1, …, Ft to have different degrees. However, with the recent work of Wooley [18,20] on the corresponding problem for equations, the study of such systems has become a feasible prospect. In this paper we take a first step in that direction by studying the analogue of the system considered in [18] and [20]. Let λ1, …, λs and μ1, …, μs be real numbers such that for each i either λi or μi is nonzero. We define the forms

formula here

and consider the solubility of the system of inequalities

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in rational integers x1, …, xs. Although the methods developed by Wooley [19] hold some promise for studying more general systems, we do not pursue this in the present paper. We devote most of our effort to proving the following theorem.

Let G be a finite group and let K be a hilbertian field. Many finite groups have been shown to satisfy the arithmetic lifting property over K, that is, every G-Galois extension of K arises as a specialization of a geometric branched covering of the projective line defined over K. The paper explores the situation when a semidirect product of two groups has this property. In particular, it shows that if H is a group that satisfies the arithmetic lifting property over K and A is a finite cyclic group then G = A [rtimes ] H also satisfies the arithmetic lifting property assuming the orders of H and A are relatively prime and the characteristic of K does not divide the order of A. In this case, an arithmetic lifting for any A[wreath ]H-Galois extension of K is explicitly constructed and the existence of the arithmetic lifting for any G-Galois extension is deduced. It is also shown that if A is any abelian group, and H is the group with the arithmetic lifting property then A[wreath ]H satisfies the property as well, with some assumptions on the ground field K. In the construction properties of Hilbert sets in hilbertian fields and spectral sequences in étale cohomology are used.

The moduli theory of Calabi–Yau threefolds is investigated, and using Griffiths' work on the period map, some finiteness results are derived. In particular, a prediction of Morrison's cone conjecture is confirmed.

Cellular algebras have recently been introduced by Graham and Lehrer [5, 6] as a convenient axiomatization of all of the following algebras, each of them containing information on certain classical algebraic or finite groups: group algebras of symmetric groups in any characteristic, Hecke algebras of type A or B (or more generally, Ariki Koike algebras), Brauer algebras, Temperley–Lieb algebras, (q-)Schur algebras, and so on. The problem of determining a parameter set for, or even constructing bases of simple modules, is in this way reduced (but of course not solved in general) to questions of linear algebra.

The present paper has two aims. First, we make explicit an inductive construction of cellular algebras which has as input data of linear algebra, and which in fact produces all cellular algebras (but no other ones). This is what we call ‘inflation’. This construction also exhibits close relations between several of the above algebras, as can be seen from the computations in [6]. Among the consequences of the construction is a natural way of generalizing Hochschild cohomology. Another consequence is the construction of certain idempotents which is used in the second part of the paper.

The second aim is to study Morita equivalences of cellular algebras. Since the input of many of the constructions of representation theory of finite-dimensional algebras is a basic algebra, it is useful to know whether any finite-dimensional cellular algebra is Morita equivalent to a basic one by a Morita equivalence that preserves the cellular structure. It turns out that the answer is ‘yes’ if the underlying field has characteristic other than 2. However, there are counterexamples in the case of characteristic 2, or more generally for any ring in which 2 is not invertible. This also tells us that the notion of ‘cellular’ cannot be defined only in terms of the module category. However, in any characteristic we find some useful Morita equivalences which are compatible with cellular structures.

Two rings A and B are said to be derived Morita equivalent if the derived categories Db(Mod A) and Db(Mod B) are equivalent. If A and B are derived Morita equivalent algebras over a field k, then there is a complex of bimodules T such that the functor T[otimes ]LA−[ratio ]Db(Mod A) → Db(Mod B) is an equivalence. The complex T is called a tilting complex.

When B = A the isomorphism classes of tilting complexes T form the derived Picard group DPic(A). This group acts naturally on the Grothendieck group Ko(A).

It is proved that when the algebra A is either local or commutative, then any derived Morita equivalent algebra B is actually Morita equivalent. This enables one to compute DPic(A) in these cases.

Assume that A is noetherian. Dualizing complexes over A are complexes of bimodules which generalize the commutative definition. It is proved that the group DPic(A) classifies the set of isomorphism classes of dualizing complexes. This classification is used to deduce properties of rigid dualizing complexes.

Finally finite k-algebras are considered. For the algebra A of upper triangular 2×2 matrices over k, it is proved that t3 = s, where t, s ∈ DPic(A) are the classes of A*:= Homk(A, k) and A[1] respectively. In the appendix, by Elena Kreines, this result is generalized to upper triangular n×n matrices, and it is shown that the relation tn+1 = sn−1 holds.

It is proved that the variety of all 4-Engel groups of exponent 4 is a maximal proper subvariety of the Burnside variety [Bfr ]4, and the consequences of this are discussed for the finite basis problem for varieties of groups of exponent 4. It is proved that, for r [ges ] 2, the 4-Engel verbal subgroup of the r-generator Burnside group B(r, 4) is irreducible as an [ ]2GL(r, 2)-module. It is observed that the variety of all 4-Engel groups of exponent 4 is insoluble, but not minimal insoluble.

The main theorem shows that whenever certain amalgamated products act geometrically on a CAT(0) space, the space has non-locally connected boundary. One can now easily construct a wide variety of examples of one-ended CAT(0) groups with non-locally connected boundary. Applications of this theorem to right-angled Coxeter and Artin groups are considered. In particular, it is shown that the natural CAT(0) space on which a right-angled Artin group acts has locally connected boundary if and only if the group is ℤn for some n.

Let k be an algebraically closed field of characteristic p > 0, and let G be a connected, reductive algebraic group over k. In [8] and [11], conditions on the dimension of rational G modules were seen to imply semisimplicity of these modules. In [8], certain of these conditions were extended to cover the finite groups of Lie type. In this paper, we extend some of the results of [11] to cover these finite Lie type groups. The main such extension is the following result.

The starting point of our investigation is the remarkable paper [2] in which Bestvina and Brady gave an example of an infinitely related group of type FP2. The result about right-angled Artin groups behind their example is best interpreted by means of the Bieri–Strebel–Neumann–Renz Σ-invariants.

For a group G the invariants Σn(G) and Σn(G, ℤ) are sets of non-trivial homomorphisms χ[ratio ]G→ℝ. They contain full information about finiteness properties of subgroups of G with abelian factor groups. The main result of [2] determines for the canonical homomorphism χ, taking each generator of the right-angled Artin group G to 1, the maximal n with χ ∈ Σn(G), respectively χ ∈ Σn(G, ℤ).

In [6] Meier, Meinert and VanWyk completed the picture by computing the full Σ-invariants of right-angled Artin groups using as well the result of Bestvina and Brady as algebraic techniques from Σ-theory. Here we offer a new account of their result which is totally geometric. In fact, we return to the Bestvina–Brady construction and simplify their argument considerably by bringing a more general notion of links into play. At the end of the first section we re-prove their main result. By re-computing the full Σ-invariants, we show in the second section that the simplification even adds some power to the method. The criterion we give provides new insight on the geometric nature of the ‘n-domination’ condition employed in [6].

It is shown that, if Q is a finitely generated abelian group, a finitely generated Q-group A is m-tame if and only if the mth tensor power of the augmentation ideal of ℤA is finitely generated over Am [rtimes ] Q, where Q acts diagonally on both Am and the tensor power. It is proved that quotients of metabelian groups of type FP3 are again of type FP3, and a necessary condition is found for a split extension of abelian-by-(nilpotent of class two) groups to be of type FP2. A conjecture is formulated that generalises the FPm-Conjecture for metabelian groups, and it is shown that one of the implications holds in the prime characteristic case.

The special linear group is the simply connected group and the projective linear group is the adjoint group of Lie type An. They are distinguished sections of the (reductive) general linear group which certainly is of this type as well (root system). We shall characterize the general linear group as the universal group of type An. Indeed we shall introduce corresponding algebraic groups and finite groups for each Lie type (to indecomposable root systems). Knowledge of the universal group implies knowledge of the related simply connected and adjoint groups; in certain respects the universal group even appears to be better behaved (automorphisms, Schur multiplier, character table).

The cohomology of [Mscr ](n, d), the moduli space of stable holomorphic bundles of coprime rank n and degree d and fixed determinant, over a Riemann surface Σ of genus g [ges ] 2, has been widely studied from a wide range of approaches. Narasimhan and Seshadri [17] originally showed that the topology of [Mscr ](n, d) depends only on the genus g rather than the complex structure of Σ. An inductive method to determine the Betti numbers of [Mscr ](n, d) was first given by Harder and Narasimhan [7] and subsequently by Atiyah and Bott [1]. The integral cohomology of [Mscr ](n, d) is known to have no torsion [1] and a set of generators was found by Newstead [19] for n = 2, and by Atiyah and Bott [1] for arbitrary n. Much progress has been made recently in determining the relations that hold amongst these generators, particularly in the rank two, odd degree case which is now largely understood. A set of relations due to Mumford in the rational cohomology ring of [Mscr ](2, 1) is now known to be complete [14]; recently several authors have found a minimal complete set of relations for the ‘invariant’ subring of the rational cohomology of [Mscr ](2, 1) [2, 13, 20, 25].

Unless otherwise stated all cohomology in this paper will have rational coefficients.

A family of transcendental meromorphic functions, fp(z), p ∈ N is considered. It is shown that, if p [ges ] 6, then the Hausdorff dimension of the Julia set of λfp satisfies dim J(λfp) [les ] 1/p, for 0 < λ < 1/6p, and dim J(λfp) [ges ] 1−(30 ln ln p/ln p), for p4p−1/105 ln p < λ < p4p−1/104 ln p. These results are used elsewhere to show that, for each d ∈ (0, 1), there exists a transcendental meromorphic function for which dim J(f) = d.

The paper establishes estimates for ideal measures of operators interpolated by the real method in terms of the measures of their restrictions to a sequence of spaces modelled on the intersection. It also shows that the estimates are optimal.

The paper presents diverse methods for estimating the covering number of a precompact subset of a Banach space when the entropy of the set of its extremal points is already known. In the case of a Hilbert space, the Gelfand diameters of the subset are also estimated.

Some constrained open mapping theorems are obtained via Ekeland's variational principle. The constraint need only be a closed subset when the mapping is assumed to be Lipschitz, or a closed convex cone when the mapping is assumed to be closed. Generalizations of some previous results of Welsh and others are obtained. Apart from the presence of a constraint and a different method, the differentiability assumptions made are weaker. As applications, two results on the constrained controllability of nonlinear systems are given.

A diffeomorphism f of the disk has a periodic orbit [Oscr ], and the paper considers the minimal topological entropy l(f, [Oscr ]) in the isotopy class of f with respect to this orbit. The structure of the periodic orbits with l(f, [Oscr ]) = 0 has been studied lately and is now well understood. For l(f, [Oscr ]) non-zero, the paper obtains a lower bound of l(f, [Oscr ]) that depends only on the period of [Oscr ] and which improves previously known estimates. This result is a consequence of a stronger bound in the case where the isotopy class is pseudo-Anosov. It is obtained by studying the induced dynamics on a graph.

For bounded Hilbert space operators X, An and Bn, n = 1, 2, …,

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for all 1 [les ] p < ∞ and

formula here

These inequalities involve some estimates for the norm of elementary operators with the range contained in the Schatten p-ideals.