It is the purpose of this paper to exhibit an infinite collection of prime, noninvertible links; each link in the collection is the union of two invertible knots. In fact, we present two such families of links. Our examples heuristically indicate that collections of links of this type exist in many varieties.

This paper establishes the radii of convexity of the following two classes of regular functions,

where

.

Suppose T is the natural transformation

where the variables are abelian groups. We find conditions on A, B, and C that guarantee that T is injective.

In this paper, a sharp version of the Schwarz–Pick Lemma for hyperbolic derivatives is provided for holomorphic selfmappings on the unit disk with fixed multiplicity for the zero at the origin. This extends a recent result due to Beardon. A property of preserving hyperbolic distances also studied by Beardon is here completely characterised.

We consider some subgroups of Brauer groups arising from twists of matrix algebras by some continuous characters. Explicit descriptions of these subgroups are given in terms of Gauss sums of Dirichlet characters.

There is a numerical error in the statement of Theorem 1 of [1]. The expression

which occurs in (6) must be replaced by

and a similar change must be made in line 5 of page 46. As a consequence, equation (7) must be changed to

and the definition of x to

in the statement of Theorem 2. These changes are purely arithmetical, and do not entail any further changes in the paper.

We show that there are biases in the number of appearances of the parts in two residue classes in the set of ordinary partitions. More precisely, let $p_{j,k,m} (n)$ be the number of partitions of n such that there are more parts congruent to j modulo m than parts congruent to k modulo m for $m \geq 2$ . We prove that $p_{1,0,m} (n)$ is in general larger than $p_{0,1,m} (n)$ . We also obtain asymptotic formulas for $p_{1,0,m}(n)$ and $p_{0,1,m}(n)$ for $m \geq 2$ .

Some problems in the behavioural and physical sciences arise in the context of an incomplete knowledge of the fine detail of underlying practical situations. This paper presents a general mathematical framework for the discussion of such problems. This framework provides an algebraic language for the discussion of ecological analysis in the social sciences, aggregation in economics and macroscopic descriptions in statistical physics. Here, however, only the mathematical framework is presented; detailed applications will be presented elsewhere.

Higher-order necessary and sufficient optimality conditions for a nonsmooth minimax problem with infinitely many constraints of inequality type are established under suitable basic assumptions and regularity conditions.

We determine the minimal relational sets related to finite groups. With the help of this result we prove that two categorically equivalent finite groups are weakly isomorphic.

We give a necessary and sufficient condition for the existence of an Edwards-Walsh resolution of a complex. Our theorem is an extension of Dydak-Walsh's theorem to all simplicial complexes of dimension ≥ n + 2. We also determine the structure of an Abelian group with the Edwards-Walsh condition, (which was introduced by Koyama and the author).

Suppose that A is a closed linear operator in a Fréchet space X. We show that there always is a maximal subspace Z containing all x ∈ X for which the abstract Cauchy problem has a mild solution, which is a Fréchet space for a stronger topology. The space Z is isomorphic to a quotient of a Fréchet space F, and the part Az of A in Z is similar to the quotient of a closed linear operator B on F for which the abstract Cauchy problem is well-posed. If mild solutions of the Cauchy problem for A in X are unique it is not necessary to pass to a quotient, and we reobtain a result due to R. deLaubenfels.

Moreover, we obtain a continuous selection operator for mild solutions of the inhomogeneous equation.

We describe semigroup rings which belong to self-dual varieties generated by a finite number of finite fields.

Gilbert Baums lag has given sufficient conditions for a group G to be nilpotent in terms of properties of a normal subgroup H and the quotient G/H, and has shown these conditions to be necessary when G is a wreath product and H its base group. His results are extended here to give corresponding conditions for weaker properties ranging from “bounded local nilpotency” to “being an Engel group”. Again we ask whether the sufficient conditions found are necessary when G is non-trivially a wreath product, or is of the form the answer is “yes” for the stronger properties considered, but for the weaker ones the question remains open.

Let n be a positive integer and let $\mathbb{F} _{q^n}$ be the finite field with $q^n$ elements, where q is a prime power. We introduce a natural action of the projective semilinear group ${\mathrm{P}\Gamma\mathrm{L}} (2, q^n)={\mathrm{PGL}} (2, q^n)\rtimes {\mathrm{Gal}} ({\mathbb F_{q^n}} /\mathbb{F} _q)$ on the set of monic irreducible polynomials over the finite field $\mathbb{F} _{q^n}$ . Our main results provide information on the characterisation and number of fixed points.

An asymptotic relation for the expected number of excursions above a boundary g(n) by a random walk Sn, n = 1,2, ‥, N is given in terms of an integral involving g. An integral test is given to determine whether the total excursion time has finite expectation. If some moment assumptions hold then the expectation of the total excursions is finite if and only if .

Let {Qn} denote the orthogonal polynomials associated with the weight function p on [−1, 1] and let denote the zeros of (1−x2) Qn (x). Consider the Lagrange polynomials which interpolate a given continuous function at these points. It is shown that, as n → ∞, the Lagrange polynomial converges to the function in the W weighted mean square sense, where w (x) = ρ(x)\(1−x2), provided that W is integrable. An application to numerical product integration is noted.

An algebraic formulation of the density theorem for loop near-rings is presented, which generalizes the same result for near-rings.

We improve S. Yamashita's hyperbolic version of the well-known Hardy-Littlewood theorem. Let f be holomorphic and bounded by one in the unit disc D. If (f#)p has a harmonic mojorant in D for some p, p > 0, then so does σ(f)q for all q, 0 < q < ∞. Here