554 research outputs found
A Graph Theoretical Approach to Network Encoding Complexity
Consider an acyclic directed network with sources and
distinct sinks . For , let denote the
min-cut between and . Then, by Menger's theorem, there exists a
group of edge-disjoint paths from to , which will be referred
to as a group of Menger's paths from to in this paper. Although
within the same group they are edge-disjoint, the Menger's paths from different
groups may have to merge with each other. It is known that by choosing Menger's
paths appropriately, the number of mergings among different groups of Menger's
paths is always bounded by a constant, which is independent of the size and the
topology of . The tightest such constant for the all the above-mentioned
networks is denoted by when all 's are
distinct, and by when all 's are in
fact identical. It turns out that and are closely
related to the network encoding complexity for a variety of networks, such as
multicast networks, two-way networks and networks with multiple sessions of
unicast. Using this connection, we compute in this paper some exact values and
bounds in network encoding complexity using a graph theoretical approach.Comment: 44 pages, 22 figure
Generalized Witten Genus and Vanishing Theorems
We construct a generalized Witten genus for spin manifolds, which takes
values in level 1 modular forms with integral Fourier expansion on a class of
spin manifolds called string manifolds. We also construct a mod 2
analogue of the Witten genus for dimensional spin manifolds. The
Landweber-Stong type vanishing theorems are proven for the generalized Witten
genus and the mod 2 Witten genus on string and string (generalized)
complete intersections in (product of) complex projective spaces respectively.Comment: 28 page
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