77,305 research outputs found
Hyperbolic Polynomials and Generalized Clifford Algebras
We consider the problem of realizing hyperbolicity cones as spectrahedra,
i.e. as linear slices of cones of positive semidefinite matrices. The
generalized Lax conjecture states that this is always possible. We use
generalized Clifford algebras for a new approach to the problem. Our main
result is that if -1 is not a sum of hermitian squares in the Clifford algebra
of a hyperbolic polynomial, then its hyperbolicity cone is spectrahedral. Our
result also has computational applications, since this sufficient condition can
be checked with a single semidefinite program
A graph interpretation of the least squares ranking method
The paper aims at analyzing the least squares ranking method for generalized
tournaments with possible missing and multiple paired comparisons. The
bilateral relationships may reflect the outcomes of a sport competition,
product comparisons, or evaluation of political candidates and policies. It is
shown that the rating vector can be obtained as a limit point of an iterative
process based on the scores in almost all cases. The calculation is interpreted
on an undirected graph with loops attached to some nodes, revealing that the
procedure takes into account not only the given object's results but also the
strength of objects compared with it. We explore the connection between this
method and another procedure defined for ranking the nodes in a digraph, the
positional power measure. The decomposition of the least squares solution
offers a number of ways to modify the method
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