9,454 research outputs found
The remaining cases of the Kramer-Tunnell conjecture
For an elliptic curve over a local field and a separable quadratic
extension of , motivated by connections to the Birch and Swinnerton-Dyer
conjecture, Kramer and Tunnell have conjectured a formula for computing the
local root number of the base change of to the quadratic extension in terms
of a certain norm index. The formula is known in all cases except some when
is of characteristic , and we complete its proof by reducing the positive
characteristic case to characteristic . For this reduction, we exploit the
principle that local fields of characteristic can be approximated by finite
extensions of --we find an elliptic curve defined over a
-adic field such that all the terms in the Kramer-Tunnell formula for
are equal to those for .Comment: 13 pages; final version, to appear in Compositio Mathematic
Rendezvous of Distance-aware Mobile Agents in Unknown Graphs
We study the problem of rendezvous of two mobile agents starting at distinct
locations in an unknown graph. The agents have distinct labels and walk in
synchronous steps. However the graph is unlabelled and the agents have no means
of marking the nodes of the graph and cannot communicate with or see each other
until they meet at a node. When the graph is very large we want the time to
rendezvous to be independent of the graph size and to depend only on the
initial distance between the agents and some local parameters such as the
degree of the vertices, and the size of the agent's label. It is well known
that even for simple graphs of degree , the rendezvous time can be
exponential in in the worst case. In this paper, we introduce a new
version of the rendezvous problem where the agents are equipped with a device
that measures its distance to the other agent after every step. We show that
these \emph{distance-aware} agents are able to rendezvous in any unknown graph,
in time polynomial in all the local parameters such the degree of the nodes,
the initial distance and the size of the smaller of the two agent labels . Our algorithm has a time complexity of
and we show an almost matching lower bound of
on the time complexity of any
rendezvous algorithm in our scenario. Further, this lower bound extends
existing lower bounds for the general rendezvous problem without distance
awareness
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