New proofs are given for the splitting lemma and two of Mather's theorems in the theory of C∞-functions. They use a Taylor series approach rather than the usual map-germ theory and are more elementary than previous proofs. A geometric characterisation of a class of degenerate functions is proved to hold in this account but not in the usual framework.

The purpose of this paper is to prove a result about the power and commutator structure of groups which generalises some results of Philip Hall. The results presented here are the key to determining the class of a nilpotent wreath product.

Let ƒ be a C1 diffeomorphism of a compact C∞ boundary–less manifold, and let ƒ# be the operator on the bounded or continuous sections of the tangent bundle (with supremum norm) defined by ƒ#η = Tƒ о η о ƒ−1. The main result of this paper is that ƒ is quasi-Anosov if and only if 1 – f# is injective and has closed range.

A monad T = (T, μ, η) on a category C is said to be linear with respect to a dense functor N: A → C if the operator T is the epimorphic image of a certain colimit of its values on A. The main aim of the article is to relate the concept of a linear monad to that of a monad with rank. A comparison is then made between linear monads and algebraic theories.

Let G be a group with a normal subgroup H whose index is a power of a prime p, and which is nilpotent with exponent a power of p. Gilbert Baumslag (Proc. Cambridge Philos. Soc. 55 (1959), 224–231) has shown that such a group is nilpotent; the main result of this paper is an upper bound on its nilpotency class in terms of parameters of H and G/H. It is shown that this bound is attained whenever G is a wreath product and H its base group.

A descending central series, here called the cpp-series, is involved in these calculations more closely than is the lower central series, and the class of the wreath product in terms of this series is also found.

Two tools used to obtain the main result, namely a useful basis for a finite p-group and a result about the augmentation ideal of the integer group ring of a finite p-group, may have some independent interest. The main result is applied to the construction of some two-generator groups of large nilpotency class with exponents 8, 9, and 25.

A class of replacement systems is studied which satisfies a “subcommutativity” condition. Examples of systems satisfying this condition are many of the systems of graph-like expressions which have recently been studied in connection with the efficient evaluation of recursive definitions. An optimality theory of subcommutative systems is developed and is used to give conditions which are sufficient to ensure that an evaluation (reduction) procedure is optimal. The optimality theory is also applied to develop conditions under which a given subcommutative system speeds up, in a natural sense, another replacement system.

Oscillation results are obtained for a first order functional differential equation, by transforming it to an equation for which oscillatory information exists in the literature already.

Consider a monopolistic firm selling each business period's production at the end of that period at a price determined by the buyers, and wishing to determine the production for which business period profit is maximal. In this paper we announce the results of our investigations into what happens when the firm employs a certain algorithm (based on linear approximations to its average contribution profit function) in an attempt to determine this optimal production. Our results are as follows: firstly, for many apparently feasible average contribution profit functions the algorithm generates production sequences globally convergent to the optimal production, the convergence being linear with convergence ratio dependent on the average contribution profit function; secondly, in certain cases a lower bound for the initial rate of convergence of the algorithm can be obtained. Proofs are for the most part given only in outline, and will be published in full later.

Several theorems concerning the norming constants {aṇ} and {bn} making a normed Markov chain {an (Xn+bn): n ≥ 0} convergent in distribution (or in probability) are given. It is shown that if Rényi's mixing conditions holds, and , whereas in the general case with α ≠ 0 and exists and are finite. Examples regarding maxima of independent and identically distributed random variables, random walk, and branching processes are considered.

Roughly speaking, a system of linear differential equations has an exponential dichotomy if it has a subspace of solutions shrinking exponentially and a complementary subspace of solutions growing exponentially. In the case of constant coefficients, this happens if and only if the eigenvalues of the coefficient matrix have nonzero real parts. In the general case, Lazer has shown that if the coefficient matrix function is bounded and satisfies a diagonal dominance condition (which, in the constant case, is a sufficient but not necessary condition that the eigenvalues have nonzero real parts) then the system has an exponential dichotomy. In this paper we prove the same result with a weaker diagonal dominance condition, thus generalizing a theorem of Nakajima.

We present in this note a useful extension of the criteria given in a recent paper [Advances in Appl. Probability 8 (1976), 737–771] for the finiteness of hitting times and mean hitting times of a Markov chain on sets in its (general) state space. We illustrate our results by giving conditions for the finiteness of the mean number of customers in the busy period of a queue in which both the service-times and the arrival process may depend on the waiting time in the queue. Such conditions also suffice for the embedded waiting time chain to have a unique stationary distribution.

We introduce primitive recursion as a generation process for arrows of algebraic theories in the sense of Lawvere and carry over important results on algebraic theories and functorial semantics to the enriched setting of “primitive recursive algebra”: existence of free primitive recursive theories and of theories presented by operations and equations on primitive recursive functions; existence of free models presented by generators and equations. Finally semantical correctness of translations is reduced to correctness for the basic operations. There is a connection to the theory of program schemes: program schemes involving primitive recursion correspond to arrows of a primitive recursive theory freely generated over a graph of basic operations. This theory T can be viewed as a programming language with “arithmetics” given by the basic operations and with DO-loops. A machine loaded with a compiler for T can be interpreted as a T-model in Lawvere's sense, preserving primitive recursion.

The aim of this paper is to completely classify the finite simple groups with no elements of order 6. The proof is by induction and involves an analysis of the structure of the 2-local subgroups of a counterexample of minimum order. A recent result of Glauberman plays an essential role.

Let Λ be a complete R-order in the semi-simple K-algebra A. Then it has been shown that for each indecomposable Λ-lattice M which is not projective, there exists a unique almost split sequence 0 → N → E → M → 0. Here we study the middle term E and characterize those almost split sequences where E has a projective direct summand. In the case where Λ is the group-ring RG for a finite group G, we get information about the almost split sequences for the syzygies and apply our results in an example.

It is shown how George D. Birkhoff's proof of the Poincaré Birkhoff theorem can be modified using ideas of H. Poincaré to give a rather precise lower bound on the number of components of the set of periodic points of the annulus. Some open problems related to this theorem are discussed.

Every multiplicative linear functional on a pseudocomplete locally convex algebra satisfying the “sequential” property of Husain and Ng is bounded (a topological algebra is called “sequential” if every null sequence contains an element whose powers converge to zero). Characterizations of such algebras are given, with some examples.

In this article we examine the problem of when a left Kan extension of a finite-product-preserving functor is finite-product preserving. This extension property is of significance in the development of finitary universal algebra in a closed category, details of which will appear elsewher. We give a list of closed categories with the required extension property.

Let X be a finite set of cardinality n, and let F be a family of k-subsets of X. In this paper we prove the following conjecture of P. Erdös and V.T. Sós.

If n > n0(k), k ≥ 4, then we can find two members F and G in F such that |F ∩ G| = 1.

A classification of two-generator 3-groups of second maximal class and low order is presented. All such groups with orders up to 38 are described, and in some cases with orders up to 310. The classification is based on computer aided computations. A description of the computations and their results are presented, together with an indication of their significance.

In the category of locally compact groups not all families of groups have a product. Precisely which families do have a product and a description of the product is a corollary of the main theorem proved here. In the category of locally compact abelian groups a family {Gj; j ∈ J} has a product if and only if all but a finite number of the Gj are of the form Kj × Dj, where Kj is a compact group and Dj is a discrete torsion free group. Dualizing identifies the families having coproducts in the category of locally compact abelian groups and so answers a question of Z. Semadeni.