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.

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.

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.

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.

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.

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.

Two systems of queues are said to be isomorphic in the restricted sense if at least one of their output elements, for example waiting time, queue size, idle time, inter-departure time, and so on, has exactly or approximately the same probability distribution function, irrespective of differences in their input elements, for example, inter-arrival time, service time, and so on. This property has been used in deriving approximate solutions to systems which do not have exact solutions. The paper highlights the systems approach in queueing problems and its practical usefulness.

A monadicity theorem is established, for functors which satisfy the conditions of the “first isomorphism theorem” (following Linton's terminology). An application is made to the characterisation of certain types of algebraic categories generated by linear monads.

The first four paragraphs of [1, p. 258] are a mildly erroneous over simplification of the situation. A more accurate description follows.

The analysis of two-generator 3-groups of second maximal class goes along the following lines. We first define a class of group whose structure is particularly amenable to theoretical analysis.