667 research outputs found
Orthogonal Laurent polynomials in unit circle, extended CMV ordering and 2D Toda type integrable hierarchies
Orthogonal Laurent polynomials in the unit circle and the theory of Toda-like
integrable systems are connected using the Gauss--Borel factorization of a
Cantero-Moral-Velazquez moment matrix, which is constructed in terms of a
complex quasi-definite measure supported in the unit circle. The factorization
of the moment matrix leads to orthogonal Laurent polynomials in the unit circle
and the corresponding second kind functions. Jacobi operators, 5-term recursion
relations and Christoffel-Darboux kernels, projecting to particular spaces of
truncated Laurent polynomials, and corresponding Christoffel-Darboux formulae
are obtained within this point of view in a completely algebraic way.
Cantero-Moral-Velazquez sequence of Laurent monomials is generalized and
recursion relations, Christoffel-Darboux kernels, projecting to general spaces
of truncated Laurent polynomials and corresponding Christoffel-Darboux formulae
are found in this extended context. Continuous deformations of the moment
matrix are introduced and is shown how they induce a time dependant
orthogonality problem related to a Toda-type integrable system, which is
connected with the well known Toeplitz lattice. Using the classical
integrability theory tools the Lax and Zakharov-Shabat equations are obtained.
The dynamical system associated with the coefficients of the orthogonal Laurent
polynomials is explicitly derived and compared with the classical Toeplitz
lattice dynamical system for the Verblunsky coefficients of Szeg\H{o}
polynomials for a positive measure. Discrete flows are introduced and related
to Darboux transformations. Finally, the representation of the orthogonal
Laurent polynomials (and its second kind functions), using the formalism of
Miwa shifts, in terms of -functions is presented and bilinear equations
are derived
Orbitopes
An orbitope is the convex hull of an orbit of a compact group acting linearly
on a vector space. These highly symmetric convex bodies lie at the crossroads
of several fields, in particular convex geometry, optimization, and algebraic
geometry. We present a self-contained theory of orbitopes, with particular
emphasis on instances arising from the groups SO(n) and O(n). These include
Schur-Horn orbitopes, tautological orbitopes, Caratheodory orbitopes, Veronese
orbitopes and Grassmann orbitopes. We study their face lattices, their
algebraic boundary hypersurfaces, and representations as spectrahedra or
projected spectrahedra.Comment: 37 pages. minor revisions of origina
New Structured Matrix Methods for Real and Complex Polynomial Root-finding
We combine the known methods for univariate polynomial root-finding and for
computations in the Frobenius matrix algebra with our novel techniques to
advance numerical solution of a univariate polynomial equation, and in
particular numerical approximation of the real roots of a polynomial. Our
analysis and experiments show efficiency of the resulting algorithms.Comment: 18 page
Matrix models for circular ensembles
We describe an ensemble of (sparse) random matrices whose eigenvalues follow
the Gibbs distribution for n particles of the Coulomb gas on the unit circle at
inverse temperature beta. Our approach combines elements from the theory of
orthogonal polynomials on the unit circle with ideas from recent work of
Dumitriu and Edelman. In particular, we resolve a question left open by them:
find a tri-diagonal model for the Jacobi ensemble.Comment: 28 page
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