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Motivic Hopf elements and relations
We use Cayley-Dickson algebras to produce Hopf elements eta, nu and sigma in
the motivic stable homotopy groups of spheres, and we prove via geometric
arguments that the the products eta*nu and nu*sigma both vanish. Along the way
we develop several basic facts about the motivic stable homotopy ring
List rankings and on-line list rankings of graphs
A -ranking of a graph is a labeling of its vertices from
such that any nontrivial path whose endpoints have the same
label contains a larger label. The least for which has a -ranking is
the ranking number of , also known as tree-depth. The list ranking number of
is the least such that if each vertex of is assigned a set of
potential labels, then can be ranked by labeling each vertex with a label
from its assigned list. Rankings model a certain parallel processing problem in
manufacturing, while the list ranking version adds scheduling constraints. We
compute the list ranking number of paths, cycles, and trees with many more
leaves than internal vertices. Some of these results follow from stronger
theorems we prove about on-line versions of list ranking, where each vertex
starts with an empty list having some fixed capacity, and potential labels are
presented one by one, at which time they are added to the lists of certain
vertices; the decision of which of these vertices are actually to be ranked
with that label must be made immediately.Comment: 16 pages, 3 figure
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