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The hyperbolic boundary () of the free factor graph can be identified with the set of equivalence classes of "arational" -trees in the boundary of the Outer space . [ 8 ] The free factor complex is a key tool in studying the behavior of random walks on Out ( F n ) {\displaystyle \operatorname {Out} (F_{n})} and in identifying the Poisson ...
A k-factor of a graph is a spanning k-regular subgraph, and a k-factorization partitions the edges of the graph into disjoint k-factors. A graph G is said to be k-factorable if it admits a k-factorization. In particular, a 1-factor is a perfect matching, and a 1-factorization of a k-regular graph is a proper edge coloring with k colors.
with a corresponding factor graph shown on the right. Observe that the factor graph has a cycle. If we merge (,) (,) into a single factor, the resulting factor graph will be a tree. This is an important distinction, as message passing algorithms are usually exact for trees, but only approximate for graphs with cycles.
Dependency, in contrast, is a one-to-one relation; every word in the sentence corresponds to exactly one node in the tree diagram. Both parse trees employ the convention where the category acronyms (e.g. N, NP, V, VP) are used as the labels on the nodes in the tree.
Centuries later, Leonardo da Vinci (1452–1519) noted the spiral arrangement of leaf patterns, that tree trunks gain successive rings as they age, and proposed a rule purportedly satisfied by the cross-sectional areas of tree-branches. [4] [3] In 1202, Leonardo Fibonacci introduced the Fibonacci sequence to the western world with his book ...
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