http://www.researchonline.mq.edu.au/vital/access/services/Feed ${session.getAttribute("locale")} 5 http://www.researchonline.mq.edu.au/vital/access/manager/Repository/mq:6462 Algebra has moved well beyond the topics discussed in standard undergraduate texts on ‘modern algebra’. Those books typically dealt with algebraic structures such as groups, rings and fields: still very important concepts! However Quantum Groups: A Path to Current Algebra is written for the reader at ease with at least one such structure and keen to learn the latest algebraic concepts and techniques. A key to understanding these new developments is categorical duality. A quantum group is a vector space with structure. Part of the structure is standard: a multiplication making it an ‘algebra’. Another part is not in those standard books at all: a comultiplication, which is dual to multiplication in the precise sense of category theory, making it a ‘coalgebra’. While coalgebras, bialgebras and Hopf algebras have been around for half a century, the term ‘quantum group’, along with revolutionary new examples, was launched by Drinfel'd in 1986. 2012-02-16T02:57:28.428Z ]]> http://www.researchonline.mq.edu.au/vital/access/manager/Repository/mq:6371 8 page(s) 2010-03-22T08:22:06.540Z ]]> http://www.researchonline.mq.edu.au/vital/access/manager/Repository/mq:6254 7 page(s) 2010-01-27T22:20:35.227Z ]]> http://www.researchonline.mq.edu.au/vital/access/manager/Repository/mq:6255 10 page(s) 2010-01-27T22:20:32.621Z ]]> http://www.researchonline.mq.edu.au/vital/access/manager/Repository/mq:6270 We show that, for every x exceeding some explicit bound depending only on k and N, there are at least C(k,N)x/log ¹⁷x positive and negative coefficients a(n) with n≤x in the Fourier expansion of any non-zero cuspidal Hecke eigenform of even integral weight k≥2 and squarefree level N that is a newform, where C(k,N) depends only on k and N. From this we deduce the existence of a sign change in a short interval. 2010-01-27T22:20:14.605Z ]]> http://www.researchonline.mq.edu.au/vital/access/manager/Repository/mq:6271 We use a result of Y. Furuta to show that for almost all positive integers m, the cyclotomic field Q(exp(2πi/m)) has an infinite Hilbert p-class field tower with high rank Galois groups at each step, simultaneously for all primes p of size up to about (log logm)[superscript]1+o(1). We also use a recent result of B. Schmidt to show that for infinitely many m there is an infinite Hilbert p-class field tower over Q(exp(2πi/m)) for some p≥m[superscript]0.3385+o(1). These results have immediate applications to the divisibility properties of the class number of Q(exp(2πi/m)). 2010-01-27T22:20:13.329Z ]]> http://www.researchonline.mq.edu.au/vital/access/manager/Repository/mq:6274 Following T. H. Chan, we consider the problem of approximation of a given rational fraction a/q by sums of several rational fractions a₁/q₁,…,an/qn with smaller denominators. We show that in the special cases of n=3 and n=4 and certain admissible ranges for the denominators q₁,…,qn, one can improve a result of T. H. Chan by using a different approach. 2010-01-27T22:20:11.705Z ]]> http://www.researchonline.mq.edu.au/vital/access/manager/Repository/mq:6276 Abstract unavailable due to use of unsupported mathematical characters in the formatting. 2010-01-27T22:20:09.992Z ]]>