Let λ be a 2-regular partition of n into two parts and let D λ denote the corresponding irreducible F 2 Σ n-module. We say that D λ is of quadratic type if there is a non-degenerate Σ n-invariant quadratic form defined on this module. In this paper, we show that D λ is not of quadratic type precisely when the smaller part of λ is a power of 2, say 2 r , where r ≥ 0, and n ≡ k mod 2 r+2 , where k is one of the 2 r consecutive integers 2 r+1 + 2 r − 1,. .. , 2 r+2 − 2. Let Σ n denote the symmetric group of degree n and let F 2 denote the field of order 2. We say that a partition λ of n is 2-regular if λ has no repeated parts. Given a 2-regular partition of n, there is an absolutely irreducible F 2 Σ nmodule D λ that corresponds to λ. Moreover, every irreducible F 2 Σ n-module is isomorphic to some D µ for a suitable 2-regular partition µ of n. The module D (n) is the trivial module. Provided λ is different from (n), there is a nondegenerate Σ n-invariant alternating bilinear form f λ , say, defined on D λ × D λ. We say that D λ is of quadratic type if there is a non-degenerate Σ n-invariant quadratic form defined on D λ , whose polarization is f λ. A description in terms of the parts of λ of those D λ that are not of quadratic type does not appear to be known at present, and we may expect any complete solution of the quadratic type problem to involve some delicate combinatorial considerations. The purpose of this paper is to decide when D λ is not of quadratic type in the case that λ is a two-part partition, that is, a partition of n into exactly two non-zero parts. Our main result is that, when λ is a two-part partition, D λ is not of quadratic type precisely when the smaller part of λ is a power of 2, say 2 r , where r ≥ 0, and n ≡ k mod 2 r+2 , where k is one of the 2 r consecutive integers 2 r+1 + 2 r − 1,. .. , 2 r+2 − 2.

We analyse input-output tables to see what structural changes have occurred in the Irish economy over time. First we produce a consistent set of input-output tables by aligning classifications and deriving a sequence of supply tables. The resulting tables are then smoothed to make the underlying distributions symmetric. We then compare the smoothed tables by using biproportional adjustment. We identify and analyse structural change that has taken place in the Irish economy since 1975. Copyright (c) 2009 Royal Statistical Society.

Creators of mathematics: the Irish connection, Ken Houston (ed.). Pp. 160. £10.99. 2000. ISBN 1 900621 495 (University College Dublin Press). Ireland is rarely mentioned in histories of mathematics except to include Hamilton or perhaps Boole. It is not surprising, however, considering the proximity to England and the existence of Trinity College, Dublin since 1592 that Ireland has made some contribution to the world of mathematics. Creators of mathematics: the Irish connection is a collection of short biographies of the more famous mathematicians who were born or lived in Ireland for some of their working life. The earliest reference is to Thomas Harriot (1560-1621), a very accomplished Renaissance mathematician who worked for Walter Ralegh as navigator and surveyor in the New World and lived in Ireland from 1587 to 1598. The most recent biography is of Paddy Kennedy, from Cork, brilliant, hard-working and eccentric who died tragically in 1966. The greatest and best known of Irish mathematicians is, of course, William Rowan Hamilton (1805-1865). Hamilton grew up in Trim, Co. Meath and was widely read in oriental, classical and modern languages at a young age. He went on to have an illustrious undergraduate career and was awarded the position of Astronomer Royal of Ireland in his early twenties. He is remembered today in the areas of quantum physics, where he gives his name to the Hamiltonian function and in abstract algebra for quaternions. The discovery of the latter is the stuff of legend. The idea came to him in a moment of epiphany as Hamilton was walking from his observatory to a meeting at the Royal Irish Academy. He was so elated to have solved the problem he had been working on for fifteen years, that he there and then carved the famous identity i 2 =f = k 1 = ijk =-1 onto a stone of Brougham Bridge. George Boole (1815-64) was born and grew up in Lincoln. He was the son of a carpenter and was self-taught. Like Hamilton, Boole learnt Latin and Greek as a young boy. In the 1840s he gained renown as a mathematician after the publication of articles on calculus. He published his first book on Logic and was recommended for the position of Professor of Mathematics in the newly founded Queen's College Cork in 1847. Boole spent the rest of his life in Cork. The University is proud of this connection and has named the library after him. His biography is a very happy story-from poverty and disadvantage to success, respect and posthumous celebrity. It is told here by a frequent contributor to the Gazette and current Professor of Mathematics in the same University, Des MacHale.

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Classic algebra is a single-volume abridged version of Algebra by P. M. Cohn, published as a three volume set in 1973 and reprinted in 1981. It covers most of the algebra taught in mathematics courses at undergraduate level and some topics from graduate courses. There are eleven chapters in the book, each divided into sections. Many of these sections (particularly in the second half of the book) are self-contained, and in this way the book serves very well as a reference. Around a third of the book is devoted to linear algebra having comprehensive chapters on vector spaces, linear equations, determinants, quadratic forms and normal forms for matrices. Most of the material one would expect in a textbook at this level is present, with the notable exception of Galois Theory. On the other hand there is the inclusion of some topics that are not seen in the more course-driven textbooks. For example, Cohn discusses the theory of the Smith normal form in the section on principal ideal domains. This proves that given an integral matrix A there exist integral matrices P and Q with the property that PAQ is a diagonal matrix {d\, d 2 , ... , d r), such that dj divides dj + i for all j < r. There are also forty pages on polynomials which lead to Witt's proof of Wedderburn's theorem on finite division rings. The book has many merits, not least its style. Cohn is a very precise writer with a deep understanding of the language and structures of mathematics. The proofs are concise, written in clear and readable English and there are many examples and illustrations of the results. The solutions to the exercises are excellent and are complete. This is probably necessary as some of the exercises are quite challenging. The first two chapters introduce sets, mappings, the integers and rational numbers. By chapter 3 the algebra gets under way with groups, via monoids. This introduction to abstract algebra is quite complicated and, very soon, the reader meets homomorphisms, isomorphisms and endomorphisms. This may not be the best approach for a beginner, although the rest of the material in the book is presented in a more typical fashion. On the other hand, for those more experienced with abstract algebra, this approach is in fact clear and rewarding. Classic algebra is not the easiest book on undergraduate algebra, but it is a book that those interested in mathematics professionally, as well as the good student, will find valuable and reliable. Furthermore, as P. M. Cohn is such a respected writer in the field, it makes sense to have reproduced the previous work in a more accessible single volume form.