424 research outputs found
Local-global principles for Galois cohomology
This paper proves local-global principles for Galois cohomology groups over
function fields of curves that are defined over a complete discretely
valued field. We show in particular that such principles hold for , for all . This is motivated by work of Kato and others, where
such principles were shown in related cases for . Using our results in
combination with cohomological invariants, we obtain local-global principles
for torsors and related algebraic structures over . Our arguments rely on
ideas from patching as well as the Bloch-Kato conjecture.Comment: 32 pages. Some changes of notation. Statement of Lemma 2.4.4
corrected. Lemma 3.3.2 strengthened and made a proposition. Some proofs
modified to fix or clarify specific points or to streamline the presentatio
Revolutionaries and spies: Spy-good and spy-bad graphs
We study a game on a graph played by {\it revolutionaries} and
{\it spies}. Initially, revolutionaries and then spies occupy vertices. In each
subsequent round, each revolutionary may move to a neighboring vertex or not
move, and then each spy has the same option. The revolutionaries win if of
them meet at some vertex having no spy (at the end of a round); the spies win
if they can avoid this forever.
Let denote the minimum number of spies needed to win. To
avoid degenerate cases, assume |V(G)|\ge r-m+1\ge\floor{r/m}\ge 1. The easy
bounds are then \floor{r/m}\le \sigma(G,m,r)\le r-m+1. We prove that the
lower bound is sharp when has a rooted spanning tree such that every
edge of not in joins two vertices having the same parent in . As a
consequence, \sigma(G,m,r)\le\gamma(G)\floor{r/m}, where is the
domination number; this bound is nearly sharp when .
For the random graph with constant edge-probability , we obtain constants
and (depending on and ) such that is near the
trivial upper bound when and at most times the trivial lower
bound when . For the hypercube with , we have
when , and for at least spies are
needed.
For complete -partite graphs with partite sets of size at least , the
leading term in is approximately
when . For , we have
\sigma(G,2,r)=\bigl\lceil{\frac{\floor{7r/2}-3}5}\bigr\rceil and
\sigma(G,3,r)=\floor{r/2}, and in general .Comment: 34 pages, 2 figures. The most important changes in this revision are
improvements of the results on hypercubes and random graphs. The proof of the
previous hypercube result has been deleted, but the statement remains because
it is stronger for m<52. In the random graph section we added a spy-strategy
resul
Automated Conjecturing Approach for Benzenoids
Benzenoids are graphs representing the carbon structure of molecules, defined by a closed path in the hexagonal lattice. These compounds are of interest to chemists studying existing and potential carbon structures. The goal of this study is to conjecture and prove relations between graph theoretic properties among benzenoids. First, we generate conjectures on upper bounds for the domination number in benzenoids using invariant-defined functions. This work is an extension of the ideas to be presented in a forthcoming paper. Next, we generate conjectures using property-defined functions. As the title indicates, the conjectures we prove are not thought of on our own, rather generated by a process of automated conjecture-making. This program, named Cᴏɴᴊᴇᴄᴛᴜʀɪɴɢ, is developed by Craig Larson and Nico Van Cleemput
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