Quartic rings

In the first three parts of this series, we considered quadratic, cubic and quartic rings (i.e., rings free of ranks 2, 3, and 4 over Z) respectively, and found that various algebraic structures involving these rings could be completely parametrized by the integer orbits of an appropriate group representation on a vector space. These orbit results are summarized in Table 1.

43p dontetvui 17-01-2013 45 6   Download

Steiner symmetrization, one of the simplest and most powerful symmetrization processes ever introduced in analysis, is a classical and very well-known device, which has seen a number of remarkable applications to problems of geometric and functional nature. Its importance stems from the fact that, besides preserving Lebesgue measure, it acts monotonically on several geometric and analytic quantities associated with subsets of Rn. Among these, perimeter certainly holds a prominent position.

34p noel_noel 17-01-2013 41 7   Download

In the first two articles of this series, we investigated various higher analogues of Gauss composition, and showed how several algebraic objects involving orders in quadratic and cubic fields could be explicitly parametrized. In particular, a central role in the theory was played by the parametrizations of the quadratic and cubic rings themselves. These parametrizations are beautiful and easy to state.

33p tuanloccuoi 04-01-2013 53 7   Download