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This problem is meant to illustrate properties that can arise in non - random walks. One formulation of the following mathematical problem is in terms
This problem is meant to illustrate properties that can arise in nonrandom
walks. One formulation of the following mathematical problem is in terms of a
person visiting castles. Consider a network that is gridlike, such that nearest
neighbors are all connected and each node has three edges connected to it
Starting from some central location, imagine flow of material can happen
either to the "right" or "left". When the flow reaches the next node, it cannot
travel back to the node from which it came, and it must go the opposite
direction right versus left or vice versa than it did at the previous node. As
the flow of material continues to traverse the network, it must alternate
directions through edges forever after that. Prove that if this pattern continues
that the flow of material must eventually return to the starting node. This is
equivalent to proving that the flow cannot get stuck traveling forever through
a loop of edges and nodes of which the starting node is not a part. Hint:
Consider the "dual" network constructed by replacing each vessel with a node
at its midpoint and connecting these new nodes only if the original edges were
connected at an original node. Map the flow through the original network onto
what flow means through the new "dual" network and consider what this
means about loops. In the figure below, the first diagram is a path through the
original network, and the second diagram is a path through the dual network.
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