An independence algebra is an algebra A in which the subalgebras satisfy the exchange axiom, and any map from a basis of A into A extends to an endomorphism. Independence algebras fall into two classes; the first is specified by a set X, a group G, and a G-space C. The second is much more restricted; it is shown that the subalgebra lattice is a projective or affine geometry, and a complete classification of the finite algebras is given.

The second Stiefel–Whitney class of the quadratic form trA/k(ax2) is computed, where A is a central simple algebra over a perfect field k of characteristic different from 2, a ∈ A is a fixed element, and trA/k is the reduced trace. This class is related on the one hand to the class of A in the Brauer group, and on the other hand to corestrictions of quaternion algebras over certain factors arising from E[otimes ]kE, where E is a commutative étale algebra over k that depends on the semisimple part of a.

In Merel's recent proof [7] of the uniform boundedness conjecture for the torsion of elliptic curves over number fields, a key step is to show that for sufficiently large primes N, the Hecke operators T1, T2, …, TD are linearly independent in their actions on the cycle e from 0 to i∞ in H1(X0(N) (C), Q). In particular, he shows independence when max(D8, 400D4) < N/(log N)4. In this paper we use analytic techniques to show that one can choose D considerably larger than this, provided that N is large.

In 1920 Chowla made the following conjecture. Let pn denote the nth prime; if q [ges ] 3, (q, a) = 1 then there are infinitely many pairs of consecutive primes pn and pn+1 such that

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By considering the sum

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where χ is the non-principal character modulo 4 or 6, it is possible to prove the conjecture for q = 4 and q = 6 (a = ±1). In this paper we prove Chowla's conjecture for all q and a with (q, a) = 1. Moreover, we shall show that for any k there exist ‘strings’ of congruent primes such that

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For each modulus q the method used applies best to the following two sets of residue classes:

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Larger values of k in terms of pn+1 can be found for residue classes belonging to these sets.

It is determined when a block for the reduced enveloping algebra of a classical Lie algebra corresponding to a character in Levi form has finite representation type. These results refine earlier work of Premet.

In [1] Brauer puts forward a series of questions on group representation theory in order to point out areas which were not well understood. One of these, which we denote by (B1), is the following: what information in addition to the character table determines a (finite) group? In previous papers [5, 7–13], the original work of Frobenius on group characters has been re-examined and has shed light on some of Brauer's questions, in particular an answer to (B1) has been given as follows.

Frobenius defined for each character χ of a group G functions χ(k)[ratio ]G(k) → [Copf ] for k = 1, …, degχ with χ(1) = χ. These functions are called the k-characters (see [10] or [11] for their definition). The 1-, 2- and 3-characters of the irreducible representations determine a group [7, 8] but the 1- and 2-characters do not [12]. Summaries of this work are given in [11] and [13].

Let G be a group and P be a property of groups. If every proper subgroup of G satisfies P but G itself does not satisfy it, then G is called a minimal non-P group. In this work we study locally nilpotent minimal non-P groups, where P stands for ‘hypercentral’ or ‘nilpotent-by-Chernikov’. In the first case we show that if G is a minimal non-hypercentral Fitting group in which every proper subgroup is solvable, then G is solvable (see Theorem 1.1 below). This result generalizes [3, Theorem 1]. In the second case we show that if every proper subgroup of G is nilpotent-by-Chernikov, then G is nilpotent-by-Chernikov (see Theorem 1.3 below). This settles a question which was considered in [1–3, 10]. Recently in [9], the non-periodic case of the above question has been settled but the same work contains an assertion without proof about the periodic case.

The main results of this paper are given below (see also [13]).

It is proved that the automorphism group of any non-abelian free group F is complete. The key technical step in the proof is that the set of all conjugations by powers of primitive elements is first-order parameter-free definable in the group Aut(F).

A Coxeter system is a pair (W, S) where W is a group and where S is a set of involutions in W such that W has a presentation of the form

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Here m(s, t) denotes the order of st in W and in the presentation for W, (s, t) ranges over all pairs in S × S such that m(s, t) ≠ ∞. We further require the set S to be finite. W is a Coxeter group and S is a fundamental set of generators for W.

Obviously, if S is a fundamental set of generators, then so is wSw−1, for any w∈W. Our main result is that, under certain circumstances, this is the only way in which two fundamental sets of generators can differ. In Section 3, we will prove the following result as Theorem 3.1.

For a sequence (cn) of complex numbers, the quadratic polynomials fcn(z) := z2 + cn and the sequence (Fn) of iterates Fn := fcn∘…∘fc1 are considered. The Fatou set [Fscr ](cn) is by definition the set of all z ∈ [Copf ]ˆ such that (Fn) is normal in some neighbourhood of z, while the complement of [Fscr ](cn) is called the Julia set [Jscr ](cn). The aim of this article is to study the connectedness and stability of the Julia set [Jscr ](cn) provided that the sequence (cn) is bounded.

It is known that, for a transcendental entire function f, the Hausdorff dimension of J(f) satisfies 1 [les ] dimJ(f) [les ] 2. For each d ∈ (1, 2), an example of a transcendental entire function f with dimJ(f) = d is given. It is then indicated how this function can be modified to produce a transcendental meromorphic function F with one pole with dimJ(F) = d. These appear to be the first examples of Julia sets with non-integer dimensions whose dimensions have been calculated exactly.

Troels Jørgensen conjectured that the algebraic and geometric limits of an algebraically convergent sequence of isomorphic Kleinian groups agree if there are no new parabolics in the algebraic limit. We prove that this conjecture holds in ‘most’ cases. In particular, we show that it holds when the domain of discontinuity of the algebraic limit of such a sequence is non-empty (see Theorem 3.1). We further show, with the same assumptions, that the limit sets of the groups in the sequence converge to the limit set of the algebraic limit. As a corollary, we verify the conjecture for finitely generated Kleinian groups which are not (non-trivial) free products of surface groups and infinite cyclic groups (see Corollary 3.3). These results are extensions of similar results for purely loxodromic groups which can be found in [4]. Thurston [32] previously established these results in the case when the Kleinian groups are freely indecomposable (see also Ohshika [24, 25, 27]). Using different techniques from ours, Ohshika [26] has proven versions of these results for purely loxodromic function groups.

Asymptotics on the diagonal of the Green function and, as a consequence, asymptotic distribution of the spectrum for the semibounded Sturm–Liouville operator are obtained.

It is proved that the dual of a Banach space with the Mazur intersection property is almost weak* Asplund. Analogously, the predual of a dual space with the weak* Mazur intersection property is almost Asplund. Through the use of these arguments, it is found that, in particular, almost all (in the Baire sense) equivalent norms on [lscr ]1(Γ) and [lscr ]∞(Γ) are Fréchet differentiable on a dense Gδ subset. Necessary conditions for Mazur intersection properties in terms of convex sets satisfying a Krein–Milman type condition are also discussed. It is also shown that, if a Banach space has the Mazur intersection property, then every subspace of countable codimension can be equivalently renormed to satisfy this property.

A reflection principle is given for temperature functions on a rectangle in the plane with much weaker conditions than the classical continuity to zero at the boundary, which improves a continuous version of Widder.

As an application of this result a uniqueness theorem is given for solutions of the Cauchy problem of the heat equation on a semi-infinite rod.

Let M be a manifold with conical ends. (For precise definitions see the next section; we only mention here that the cross-section [Kscr ] can have a nonempty boundary.) We study the scattering for the Laplace operator on M. The first question that we are interested in is the structure of the absolute scattering matrix [Sscr ](s). If M is a compact perturbation of ℝn, then it is well-known that [Sscr ](s) is a smooth perturbation of the antipodal map on a sphere, that is,

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On the other hand, if M is a manifold with a scattering metric (see [8] for the exact definition), it has been proved in [9] that [Sscr ](s) is a Fourier integral operator on [Kscr ], of order 0, associated to the canonical diffeomorphism given by the geodesic flow at distance π. In our case it is possible to prove that [Sscr ](s) is in fact equal to the wave operator at a time t = π plus C∞ terms. See Theorem 3.1 for the precise formulation. This result is not too difficult and is obtained using only the separation of variables and the asymptotics of the Bessel functions.

Our second result is deeper and concerns the scattering phase p(s) (the logarithm of the determinant of the (relative) scattering matrix).

The Fourier series of the elements in the generalized Bergman spaces bp, q of harmonic functions over D and over [Copf ] (as well as those of holomorphic functions) is analysed. It is shown that the trigonometric system Ω = {r[mid ]k[mid ]eikϕ}k∈ℤ is never a basis of b1, 1 and b∞, 0 for any weighted L1-norm and L∞-norm over D. The same result holds in the special case of Bargmann–Fock space over [Copf ] (with respect to the weighted L1-norms and L∞-norms) which answers a question of Garling and Wojtaszczyk. On the other hand examples are given of weighted L1-norms and L∞-norms over [Copf ] where Ω is indeed a basis of b1, 1 and b∞, 0. Moreover, using similar methods, a weight is constructed on D where b∞, ∞ is not isomorphic to l∞ which shows that there are weighted spaces whose Banach space classifications differ completely from those which have been characterized so far.

The spectrum and essential spectrum of a self-adjoint operator in a real Hilbert space are characterized in terms of Palais–Smale conditions on its quadratic form and Rayleigh quotient respectively.

Let G be a discrete group and (G, G+) be a quasi-ordered group. Set G0+ = G+∩(G+)−1 and G1 = (G+[setmn ]G0+)∪{e}. Let [Fscr ]G1(G) and [Fscr ]G+(G) be the corresponding Toeplitz algebras. In the paper, a necessary and sufficient condition for a representation of [Fscr ]G+(G) to be faithful is given. It is proved that when G is abelian, there exists a natural C*-algebra morphism from [Fscr ]G1(G) to [Fscr ]G+(G). As an application, it is shown that when G = ℤ2 and G+ = ℤ+ × ℤ, the K-groups K0([Fscr ]G1(G)) ≅ ℤ2, K1([Fscr ]G1(G)) ≅ ℤ and all Fredholm operators in [Fscr ]G1(G) are of index zero.

Every subalgebra of [Lscr ]([Hscr ]) which is isometrically weak*-homeomorphic with H∞(Ω) is reflexive, when Ω is a finitely connected region in the complex plane.