The authors present their long awaited account of Artin algebras, almostsplit sequences and Auslander-Reiten quivers, which is refreshingly down-toearth. Hereditary algebras, short cycles and stable equivalence are also treated, but many ramifications such as tilting, bocses, derived functors etc. had to be omitted. Exercises add to the readability of what is sure to become a classic.M. J. BAINES. Moving Finite Elements. Clarendon Press. 1994, ISBN 0 19 853467 1, 226 pp., £45.Mathematical in approach and assuming knowledge of standard finite element techniques, this research monograph describes the moving finite element method for solving partial differential equations. (This is an adaptive grid method with a fixed number of grid points.) Thorough presentation of the underlying theory and current understanding of the method are given, with descriptions presented from several different viewpoints. Simple examples are used throughout.

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Abstract. In a series of papers additive subbifunctors F of the bifunctor Ext Λ ( , ) are studied in order to establish a relative homology theory for an artin algebra Λ. On the other hand, one may consider the elements of F (X, Y ) as short exact sequences. We observe that these exact sequences make mod Λ into an exact category if and only if F is closed in the sense of Butler and Horrocks.Concerning the axioms for an exact category we refer to Gabriel and Roiter's book. In fact, for our general results we work with subbifunctors of the extension functor for arbitrary exact categories.In order to study projective and injective objects for exact categories it turns out to be convenient to consider categories with almost split exact pairs, because many earlier results can easily be adapted to this situation.Exact categories arise in representation theory for example if one studies categories of representations of bimodules. Representations of bimodules gained their importance in studying questions about representation types. They appear as domains of certain reduction functors defined on categories of modules. These reduction functors are often closely related to the functor Ext Λ ( , ) and in general do not preserve at all the usual exact structure of mod Λ.By showing the closedness of suitable subbifunctors of Ext Λ ( , ) we can equip mod Λ with an exact structure such that some reduction functors actually become 'exact'. This allows us to derive information about the projective and injective objects in the respective categories of representations of bimodules appearing as domains, and even show that almost split sequences for them.Examples of such domains appearing in practice are the subspace categories of a vector space category with bonds. We provide an example showing that existence of almost split sequences for them is not a general fact but may even fail if the vector space category is finite. Exact categoriesThis section is devoted to transferring some definitions and basic results on relative theory developed for abelian categories in [BH] and for finitely generated modules over artin algebras in [ASo] to the context of exact categories. Furthermore, we show how one can construct a new exact structure on a category from closed subbifunctors of the extension bifunctor induced by a given exact structure.