The author uses Padé approximation techniques and an elementary lemma on primes dividing binomial coefficients to sharpen a theorem of F. Beukers on fractional parts of powers of rationals. In particular, it is proven that ‖((N+ l)/N)k‖ > 3–k holds for all positive integers N and k satisfying 4 ≤ N ≤ k · 3k. Other results are described including an effective version of a theorem of K. Mahler for a restricted class of rationals.

All curves of the title are calculated up to an equivalence relation which is coarser than the relation of isogeny between the associated Jacobian varieties.

We show that a PD3-complex P such that π = π1(P) is infinite and has a non-trivial finite normal subgroup must be homotopy equivalent to RP2 × S1. Hence if A is an abelian normal subgroup of a 2-knot group πK which is not contained in the commutator subgroup πK′ and πK′ is infinite then A is torsion free.

In this note we present a new proof of a theorem of McClure on K*(Ω∞Σ∞X, Z/p) [11], in the special case when X is a finite complex with K1(X; Z/p) = 0. Although our method does not work in the full generality covered by his work, our argument requires neither a geometric interpretation of complex k-theory nor all the delicate coherence properties of its multiplication. Since BP-theory is not likely to possess such coherence properties [9], the possibility of generalizing his approach to the case of higher Morava K-theory does not seem feasible. On the contrary, the main ingredient of our approach is the rank formula for the Morava K-theory of the Borel construction [5], which works for any K(n); thus our approach is better adapted to the potential generalization [8]. Throughout the paper we assume that p > 2 so that mod p K-theory possesses a commutative multiplication, and denote by K*(−) the mod p K-theory. Since it is simpler to state our results in terms of CX, the combinatorial model for QX, rather than QX itself, we shall do so. This is sufficient, as when X is connected CX is homotopy equivalent to QX, and when not, K*(QX) can be easily recovered from K*(CX) (see e.g. [11]).

We define and study a class of groups which produce simple, unique trace, C*-algebras. This class strictly contains the class of weak Powers groups, as shown by the fact that it is closed under extensions. We provide answers to some open problems involving Powers and weak Powers groups.

We obtain normal forms for infinitesimally symplectic matrices (or linear Hamiltonian vector fields) that commute with the symplectic action of a compact Lie group of symmetries. In doing so we extend Williamson's theorem on normal forms when there is no symmetry present.

Using standard representation-theoretic results the symmetry can be factored out and we reduce to finding normal forms over a real division ring. There are three real division rings consisting of the real, complex and quaternionic numbers. Of these, only the real case is covered in Williamson's original work.

A set D of 3-transpositions in the group G is a normal set of elements of order 2 such that, for all d and e in D, the order of the product de is 1, 2, or 3. If G is generated by the conjugacy class D of 3-transpositions, we say that (G, D) is a 3-transposition group or (loosely) that G is a 3-transposition group. The study of 3-transposition groups was instituted by Bernd Fischer [6, 7, 8] who classified all 3-transposition groups which are finite and have no non-trivial normal solvable subgroups. Recently the present author and H. Cuypers[5] extended Fischer's result to include all 3-transposition groups with trivial centre. For this classification the present paper provides the extension of Fischer's paper [8] where he gave two basic reductions, the Normal Subgroup Theorem and the Transitivity Theorem stated below. Other results of help in the classification are also presented here.

Nielsen [14] gave the following commutator test for an endomorphism of the free group F = F2 = 〈x, y; Ø〉 to be an automorphism: an endomorphism ø: F → F is an automorphism if and only if the commutator [ø(x), ø(y)] is conjugate in F to [x, y]±1. He obtained this test as a corollary to his well-known result that every IA-automorphism of F (i.e. one which fixes F modulo its commutator subgroup) is an inner automorphism. Bachmuth et al. [4] have proved that IA-automorphisms of most two-generator groups of the type F/R′ are inner, and it becomes natural to ask if Nielsen's commutator test remains valid for those groups as well. Durnev[7] considered this question for the free metabelian group F/F″ and confirmed the validity of the commutator test in this case. Here we prove that Nielsen's test does not hold for a large class of F/R′ groups (Theorem 3·1) and, as a corollary, deduce that it does not hold for any non-metabelian solvable group of the form F/R″ (Corollary 3·2). In view of our Theorem 3·1, Nielsen's commutator test in these situations seems to have less appeal than his result that the IA-automorphisms of F are precisely the inner automorphisms of F. We explore some applications of this important result with respect to non-tameness of automorphisms of certain two- generator groups F/R (i.e. automorphisms of F/R which are not induced by those of the free group F). For instance, we show that a two-generator free polynilpotent group F/V, , has non-tame automorphisms except when V = γ2(F) or V = γ3(F), or when V is of the form [yn(U), γ(U)], n ≥ 2 (Theorem 4·2). This complements the results of [9] and [16] rather nicely, and is shown to follow from a more general result (Proposition 4·1). We also include an example of an endomorphism θ: x → xu, y→y of F which induces a non-tame automorphism of F/γ6(F) while the partial derivative ∂(u)/∂(x) is ‘balanced’in the sense of Bryant et al. [5] (Example 4·4). This gives an alternative solution of a problem in [5] which has already been resolved by Papistas [15] in the negative. In our final section, we consider groups of the type F/[R′,F] and, in contrast to groups of the type F/R′, we show that the Nielsen's commutator test does hold in most of these groups (Theorem 5·1). We conclude with a sufficiency condition under which Nielsen's commutator test is valid for a given pair of generating elements ofF modulo [R′,F] (Proposition 5·2).

In 1966, J. M. Howie characterized the self-maps of a set which can be written as a product (under composition) of idempotent self-maps of the same set. In 1967, J. A. Erdos considered the analogous question for linear maps of a finite dimensional vector space and in 1985, Reynolds and Sullivan solved the problem for linear maps of an infinite dimensional vector space. Using the concept of independence algebra, the authors gave a common generalization of the results of Howie and Erdos for the cases of finite sets and finite dimensional vector spaces. In the present paper we introduce strong independence algebras and provide a common generalization of the results of Howie and Reynolds and Sullivan for the cases of infinite sets and infinite dimensional vector spaces.

The construction of geometric surfaces via labelled complexes was introduced by Thurston[16, chapter 13], and subsequent applications and developments have appeared in [1, 3, 4, 5, 14, 15]. The basic idea of using labelled complexes to produce geometric structures is that the vertices of a simplicial triangulation of a surface can be labelled with positive real numbers that collectively determine a metric of constant curvature ±1 or 0, with possible singularities at vertices, by using the label values to identify 2-simplices of the triangulation with geometric triangles. Beardon and Stephenson[1] developed a particularly simple method for producing non-singular surfaces via labelled complexes that is modelled after the classical Perron method for producing harmonic functions, and they applied their method in [2] to construct a fairly comprehensive theory of circle packings in general Riemann surfaces. This Perron method was developed more fully by Stephenson and the author in [3, 4] and applied to the study of circle packing points in moduli space. At about the same time and independently of Beardon, Stephenson, and Bowers, Carter and Rodin [5] and Doyle [8] developed the method for flat surfaces and Minda and Rodin [14] developed the method for finite type surfaces. Minda and Rodin [14] applied their development to give partial solutions to the labelled complex version of the classical Schwarz-Picard problem that concerns the construction of singular hyperbolic metrics on surfaces with prescribed singularities. In this paper, we modify the aforementioned approaches and examine the upper Perron method for producing non-singular geometric surfaces. This upper method has several advantages over the Perron method as developed previously and provides a complete solution to the labelled complex version of the Schwarz-Picard problem.

We consider random monic polynomials of degree n over a finite field of q elements, chosen with all qn possibilities equally likely, factored into monic irreducible factors. More generally, relaxing the restriction that q be a prime power, we consider that multiset construction in which the total number of possibilities of weight n is qn. We establish various approximations for the joint distribution of factors, by giving upper bounds on the total variation distance to simpler discrete distributions. For example, the counts for particular factors are approximately independent and geometrically distributed, and the counts for all factors of sizes 1, 2, …, b, where b = O(n/log n), are approximated by independent negative binomial random variables. As another example, the joint distribution of the large factors is close to the joint distribution of the large cycles in a random permutation. We show how these discrete approximations imply a Brownian motion functional central limit theorem and a Poisson-Dirichiet limit theorem, together with appropriate error estimates. We also give Poisson approximations, with error bounds, for the distribution of the total number of factors.

The non-negative harmonic functions of a transient Markov process yield a great deal of information about the ‘behaviour at infinity’ of the process, and can be used to h-transform the process to behave in a certain way at infinity. The traditional analytic way of studying the non-negative harmonic functions is to construct the Martin boundary of the process (see, for example, Meyer [4], Kunita and T. Watanabe[3], and Kemeny, Snell & Knapp[2], Williams [7] for the chain case). However, certain conditions on the process need to be satisfied, one of the most basic of which is that there exists a reference measure η such that Uλ (x, ·) ≪ η for all λ > 0, all x ∈ E, the state space of the Markov process. (Here, (Uλ)λ>0 is the resolvent of the process.)

Cosmic censorship hypothesis [1] is a major unsolved problem in classical general relativity. According to this hypothesis singularities occurring in generic space-times should not be naked. This means that there should not be singularities to the future of a regular initial surface that are visible to observers at infinity A mathematically precise statement of the hypothesis is that a space-time should be future asymptotically predictable from a partial Cauchy surface ([2] p. 310).