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Cost-allocation principles for pipeline capacity and usage
This paper applies principles f rom game theory to the problem o f allocating the cost o f a shared facility, such as a pipeline. The theory o f cooperative games s t r ongl y suggest s t hat no method e x i s t s for allocating costs that wi l l achieve all major policy goals. We apply results from the theory o f cooperative games a n d principles o f cost allocation to assess some c o mmo n l y adopted rules for allocating costs and def i ni ng u n i t charges. Mos t notably, the postage-stamp toll is f o u n d to fail a mi ni mal set o f commonly applied principles.cost allocation; pipeline
Functional inversion for potentials in quantum mechanics
Let E = F(v) be the ground-state eigenvalue of the Schroedinger Hamiltonian H
= -Delta + vf(x), where the potential shape f(x) is symmetric and monotone
increasing for x > 0, and the coupling parameter v is positive.
If the 'kinetic potential' bar{f}(s) associated with f(x) is defined by the
transformation: bar{f}(s) = F'(v), s = F(v)-vF'(v),then f can be reconstructed
from F by the sequence: f^{[n+1]} = bar{f} o bar{f}^{[n]^{-1}} o f^{[n]}.
Convergence is proved for special classes of potential shape; for other test
cases it is demonstrated numerically. The seed potential shape f^{[0]} need not
be 'close' to the limit f.Comment: 14 pages, 2 figure
Constrained Ramsey Numbers
For two graphs S and T, the constrained Ramsey number f(S, T) is the minimum
n such that every edge coloring of the complete graph on n vertices, with any
number of colors, has a monochromatic subgraph isomorphic to S or a rainbow
(all edges differently colored) subgraph isomorphic to T. The Erdos-Rado
Canonical Ramsey Theorem implies that f(S, T) exists if and only if S is a star
or T is acyclic, and much work has been done to determine the rate of growth of
f(S, T) for various types of parameters. When S and T are both trees having s
and t edges respectively, Jamison, Jiang, and Ling showed that f(S, T) <=
O(st^2) and conjectured that it is always at most O(st). They also mentioned
that one of the most interesting open special cases is when T is a path. In
this work, we study this case and show that f(S, P_t) = O(st log t), which
differs only by a logarithmic factor from the conjecture. This substantially
improves the previous bounds for most values of s and t.Comment: 12 pages; minor revision
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