A composition sequence for a torsion-free abelian group A is an increasing sequenceof pure subgroups with rank 1 quotients and union A. Properties of A can be described by the sequence of types of these quotients. For example, if A is uniform, that is all the types in some sequence are equal, then A is complete decomposable if it is homogeneous. If A has finite rank and all permutations ofone of its type sequences can be realized, then A is quasi-isomorphic to a direct sum of uniform groups.

Vertex-primitive self-complementary graphs were proved to be affine or in product action by Guralnick et al. [‘On orbital partitions and exceptionality of primitive permutation groups’, Trans. Amer. Math. Soc.356 (2004), 4857–4872]. The product action type is known in some sense. In this paper, we provide a generic construction for the affine case and several families of new self-complementary Cayley graphs are constructed.

We prove, by constructing a function with given parameters, that the estimate by G. V. Chudnovsky of the number of points at which a meromorphic function has algebraic Taylor coefficients is optimal. The construction is carried out by the use of interpolation series.

Reynolds (1972), using character-theory, showed that the p-section sums span an ideal of the centre Z(kG) of the group algebra of a finite group G over a field k of characteristic dividing the order of G. In O'Reilly (1973) a character-free proof was given. Here we extend these techniques to show the existence of a wider class of ideals of Z(kG).

The purpose of this note is to determine the automorphism group of the doubly regular tournament of Szekeres type, and to use it to show that the corresponding skew Hadamard matrix H of order 2(q + 1), where q ≡5(mod 8) and q > 5, is not equivalent to the skew Hadamard matrix H(2q + 1) of quadratic residue type when 2q + 1 is a prime power.

The surjective additive maps on the Lie ring of skew-Hermitian linear transformations on a finite-dimensional vector space over a division ring which preserve the set of rank 1 elements are determined. As an application, maps preserving commuting pairs of transformations are determined.

It is our purpose to show that the constructible orthomodular lattices defined by Janowitz in [2], are embeddable into Boolean lattices. In fact they are subdirect products of Boolean lattices, where the subdirect products are taken in the class of orthomodular posets. We shall make these notions precise. Other concepts, such as disjoint sum and constructible lattice, are defined in [1] and [2].

Let n be a positive integer and let T be any nonempty set of positive integers. By (n, T) = 1 we men n is relatively prime to each element of T. Hence n can be written as n = n1n2 where n1 is the largest divisor of n such that (n1, T)= 1. Let P be a property associated with positive integers. We shall say that a positive integer n is a P-number if it satisfies the property P. If in the above representation of n, the integer n1 is a P-number. then we shall say that n is a quasi P-number relative to T, or, simply, a quasi P-number. In particular, for k a positive integer >1, n is quasi k-free for given set T) if n1 is k-free.

Let S be a bisimple semigroup, let Es denote the set of idempotents of S, and let ≦ denote the natural partial order relation on Es. Let ≤ * denote the inverse of ≦. The idempotents of S are said to be well-ordered if (Es, ≦ *) is a well-ordered set.

Let E, Ê, and E′ denote a locally convex linear Hausdorff space, completion of E and the dual of E, respectively. It has been observed that Ê is a subspace of E″ under certain conditions on E. It is the primary goal of this paper to give necessary and sufficient conditions for the Ê ⊂ E″ to be valid. Such conditions are found and are given Theorem 4. With a variation of the technique used, several equivalent characterizations of semi-reflexive spaces are given in Theorem 5. The nationa throughtout will follow that in [2].

Let A be a finite dimensional algebra over a field F. Let R and S be biregular algebras over F such that 1R ∈ R and 1S ∈ S. We show that if R/P≃A≃ S/M for each primitive ideal P in A and each primitive ideal M in S then End FR≃ End S implies R≃S.

Let f and g be two permutable transcendental entire functions. Assume that f is a solution of a linear differential equation with polynomial coefficients. We prove that, under some restrictions on the coefficients and the growth of f and g, there exist two non-constant rational functions R1 and R2 such that R1 (f) = R(g). As a corollary, we show that f and g have the same Julia set: J(f) = J(g). As an application, we study a function f which is a combination of exponential functions with polynomial coefficients. This research addresses an open question due to Baker.

Even if in a decomposition of a group the Ai are completely indecomposable, there may be another decomposition with each Cj properly contained in some Ai a proper subgroup of B. The example of Bryce ([1], p. 636) may be modified, at the cost of having one Ai = B, so that I = J and Ci > Ai for all i. It is our object to study this relationship between decompositions of a group.

The Kasparov groups are extended to the setting of inverse limits of G-C*-algebras, where G is assumed to be a locally compact group. The K K-product and other important features of the theory are generalized to this setting.

We introduce the class of p-adic F-functions which contains both the p-adic E-function and p-adic G-functions, as well as other functions. In this paper we obtain lower bounds for polynomials in the values at algebraic points of a class of p-adic F-functions defined over the completion of the algebraic closure of a p-adic field.

We consider ineducible Goppa codes of length qm over Fq defined by polynomials of degree r, where q = pt and p, m, r are distinct primes. The number of such codes, inequivalent under coordinate permutations and field automorphisms, is determined.

In part I, I considered the problem of discovering when, given an irrational α which has a simple continued fraction representation with convergents pn/qn, there exists α' for which the denominator sequence for convergents is a subsequence of (qn). It was shown that such an α' exists if the continued fraction representation was “nearly periodic” with odd period. The following is a generalization of the results of part I to semi-regular continued fractions, where the problem seems to fit more naturally.

In this paper we solve the equation f(g(x))=f(x)hm(x) where f(x), g(x) and h(x) are unknown polynomials with coefficients in an arbitrary field K, f(x) is nonconstant and separable, deg g≥2, the polynomial g(x) has nonzero derivative g′(x)≠0in K[x]and the integer m≥2is not divisible by the characteristic of the field K. We prove that this equation has no solutions if deg f≥3 . If deg f=2 , we prove that m=2and give all solutions explicitly in terms of Chebyshev polynomials. The Diophantine applications for such polynomials f(x) , g(x) , h(x)with coefficients in ℚ or ℤ are considered in the context of the conjecture of Cassaigne et al. on the values of Liouville’s λ function at points f(r) , r∈ℚ.

In light of the recent work by Maynard and Tao on the Dickson $k$-tuples conjecture, we show that with a small improvement in the known bounds for this conjecture, we would be able to prove that for some fixed $R$, there are infinitely many Carmichael numbers with exactly $R$ factors for some fixed $R$. In fact, we show that there are infinitely many such $R$.

Let Bd(e) denote the Burnside group with d ≥ 2 generators a1,.a2, …, a, and exponent e > 0, i.e., the free group of rank d of the Burnside variety of exponent e. It is known that Bd(e) is infinite for all sufficiently large odd values of e; cf. Novikov and Adyan [3] or Britton [1]. In particular Bd(pk), where p is an odd prime, is infinite for all sufficiently large k. It is not known whether or not Bd(2k) is infinite for all sufficiently large k infiniteness would imply that Bd(n) is infinite for all sufficiently large n, as has been conjectured by Novikov [2].