23,006 research outputs found
Heegaard Floer correction terms and rational genus bounds
Given an element in the first homology of a rational homology 3-sphere ,
one can consider the minimal rational genus of all knots in this homology
class. This defines a function on , which was
introduced by Turaev as an analogue of Thurston norm. We will give a lower
bound for this function using the correction terms in Heegaard Floer homology.
As a corollary, we show that Floer simple knots in L-spaces are genus
minimizers in their homology classes, hence answer questions of Turaev and
Rasmussen about genus minimizers in lens spaces.Comment: 21 pages. V2: corrects a mistake in the proof of Proposition 1.5,
incorporates the referee's comment
Learning Loosely Connected Markov Random Fields
We consider the structure learning problem for graphical models that we call
loosely connected Markov random fields, in which the number of short paths
between any pair of nodes is small, and present a new conditional independence
test based algorithm for learning the underlying graph structure. The novel
maximization step in our algorithm ensures that the true edges are detected
correctly even when there are short cycles in the graph. The number of samples
required by our algorithm is C*log p, where p is the size of the graph and the
constant C depends on the parameters of the model. We show that several
previously studied models are examples of loosely connected Markov random
fields, and our algorithm achieves the same or lower computational complexity
than the previously designed algorithms for individual cases. We also get new
results for more general graphical models, in particular, our algorithm learns
general Ising models on the Erdos-Renyi random graph G(p, c/p) correctly with
running time O(np^5).Comment: 45 pages, minor revisio
Chaos in two black holes with next-to-leading order spin-spin interactions
We take into account the dynamics of a complete third post-Newtonian
conservative Hamiltonian of two spinning black holes, where the orbital part
arrives at the third post-Newtonian precision level and the spin-spin part with
the spin-orbit part includes the leading-order and next-to-leading-order
contributions. It is shown through numerical simulations that the
next-to-leading order spin-spin couplings play an important role in chaos. A
dynamical sensitivity to the variation of single parameter is also
investigated. In particular, there are a number of \textit{observable} orbits
whose initial radii are large enough and which become chaotic before
coalescence
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