15-vertex triangulations of an 8-manifold by Brehm U., Kuhnel W.

By Brehm U., Kuhnel W.

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In keeping with papers on the Intl Workshop on Differential Equations and optimum keep an eye on held lately at Ohio college, Athens.

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Let u ∈ Vk−1 (x). Then u has access to N1,k−1 inputs including x. It follows that N1,k−1 − 1 inputs can generate an intersecting connection entering at or before stage k − 1. Similarly, Mk+1,s − 1 outputs can generate an intersecting connection exiting at r after stage k + 1. Thus N1,k−1 − 1 + Mk+1,s − 1 = N1,k−1 + Mk+1,s − 2 is an upper bound. Define ω = min {ω(k) is def ined : k = 2, . . , s − 1, N − 1, M − 1} . The next step is to assign these ω intersecting connections to the entry stages and the exit stages to maximize the number B of paths blocked in G(x, y).

Ia } and {o1 , . . , ob }. Then in a state containing M but i and o are idle, the request (i, o) cannot be connected, contradicting the assumption that G is SNB. Therefore either d(vj ) ≤ j for some 1 ≤ j ≤ a or d (uk ) ≤ k for some 1 ≤ k ≤ b. Assume the former. Then j k=1 1 ≥ d(vk ) j k=1 1 = 1. 2. Let G(V, E) be a SNB s-stage network such that d(v)≤∆ and d (v)≤∆ for all v ∈ V . Then N < 2∆s−1 . In particular, N 2 < |E| f or s = 2. Proof. Let Ao denote the set of stage-2 nodes which has a path to o, and let Bi denote the set of stage-(s − 1) nodes which i has a path to.

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