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ErdősPósa Property and Its Algorithmic Applications  Parity Constraints, Subset Feedback Set, and Subset Packing
"... The wellknown ErdősPósa theorem says that for any integer k and any graph G, either G contains k vertexdisjoint cycles or a vertex set X of order at most c·k log k (for some constant c) such that G−X is a forest. Thomassen [39] extended this result to the even cycles, but on the other hand, it is ..."
Abstract

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. For fixed k, we can either find a vertex set X of size k such that G − X has no Scycle, or conclude that such a vertex set does not exist in O(n 2 m) time (independently obtained in [7]). 2. For fixed k, we can either find k vertexdisjoint Scycles, or conclude that such k disjoint cycles do not exist
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"... , and we can say the Campbell bandwidth is the minimum average bandwidth for encoding the process across all possible distortion levels. IX. CONCLUSION We have presented two new derivations of the coefficient rate introduced by Campbell. One derivation solidifies its interpretation as a coefficien ..."
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, and we can say the Campbell bandwidth is the minimum average bandwidth for encoding the process across all possible distortion levels. IX. CONCLUSION We have presented two new derivations of the coefficient rate introduced by Campbell. One derivation solidifies its interpretation as a coefficient rate, and shows that the spectral entropy of a random process is proportional to the logarithm of the equivalent bandwidth of the smallest frequency band that contains most of the energy. The second derivation implies that the number of samples of a particular component should be proportional to the variance of that component. We discussed the implications of the latter result for realizationadaptive source coding and provided a connection with the familiar reverse waterfilling result from rate distortion theory. From the coefficient rate, we defined a quantity called the Campbell bandwidth of a random process, and we contrasted Fourier bandwidth, Shannon bandwidth, and Campbell bandwidth. ACKNOWLEDGMENT The authors are indebted to the referees for their constructive comments and insights.
Physics
, 2011
"... Ultracold atoms in optical lattices provide a highly controllable environment for the clean experimental realization of various model Hamiltonians from condensed matter and statistical physics. For example, the twocomponent BoseHubbard model, which reduces to an anisotropic spin1/2 Heisenberg m ..."
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Ultracold atoms in optical lattices provide a highly controllable environment for the clean experimental realization of various model Hamiltonians from condensed matter and statistical physics. For example, the twocomponent BoseHubbard model, which reduces to an anisotropic spin1/2 Heisenberg