1,441 research outputs found
Countable locally 2-arc-transitive bipartite graphs
We present an order-theoretic approach to the study of countably infinite
locally 2-arc-transitive bipartite graphs. Our approach is motivated by
techniques developed by Warren and others during the study of cycle-free
partial orders. We give several new families of previously unknown countably
infinite locally-2-arc-transitive graphs, each family containing continuum many
members. These examples are obtained by gluing together copies of incidence
graphs of semilinear spaces, satisfying a certain symmetry property, in a
tree-like way. In one case we show how the classification problem for that
family relates to the problem of determining a certain family of highly
arc-transitive digraphs. Numerous illustrative examples are given.Comment: 29 page
Rings of small rank over a Dedekind domain and their ideals
In 2001, M. Bhargava stunned the mathematical world by extending Gauss's
200-year-old group law on integral binary quadratic forms, now familiar as the
ideal class group of a quadratic ring, to yield group laws on a vast assortment
of analogous objects. His method yields parametrizations of rings of degree up
to 5 over the integers, as well as aspects of their ideal structure, and can be
employed to yield statistical information about such rings and the associated
number fields.
In this paper, we extend a selection of Bhargava's most striking
parametrizations to cases where the base ring is not Z but an arbitrary
Dedekind domain R. We find that, once the ideal classes of R are properly
included, we readily get bijections parametrizing quadratic, cubic, and quartic
rings, as well as an analogue of the 2x2x2 cube law reinterpreting Gauss
composition for which Bhargava is famous. We expect that our results will shed
light on the analytic distribution of extensions of degree up to 4 of a fixed
number field and their ideal structure.Comment: 39 pages, 1 figure. Harvard College senior thesis, edite
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