20,176 research outputs found

    Numerical Schubert calculus

    Full text link
    We develop numerical homotopy algorithms for solving systems of polynomial equations arising from the classical Schubert calculus. These homotopies are optimal in that generically no paths diverge. For problems defined by hypersurface Schubert conditions we give two algorithms based on extrinsic deformations of the Grassmannian: one is derived from a Gr\"obner basis for the Pl\"ucker ideal of the Grassmannian and the other from a SAGBI basis for its projective coordinate ring. The more general case of special Schubert conditions is solved by delicate intrinsic deformations, called Pieri homotopies, which first arose in the study of enumerative geometry over the real numbers. Computational results are presented and applications to control theory are discussed.Comment: 24 pages, LaTeX 2e with 2 figures, used epsf.st

    Spectral curve, Darboux coordinates and Hamiltonian structure of periodic dressing chains

    Full text link
    A chain of one-dimensional Schr\"odinger operators connected by successive Darboux transformations is called the ``Darboux chain'' or ``dressing chain''. The periodic dressing chain with period NN has a control parameter α\alpha. If α≠0\alpha \not= 0, the NN-periodic dressing chain may be thought of as a generalization of the fourth or fifth (depending on the parity of NN) Painlev\'e equations . The NN-periodic dressing chain has two different Lax representations due to Adler and to Noumi and Yamada. Adler's 2×22 \times 2 Lax pair can be used to construct a transition matrix around the periodic lattice. One can thereby define an associated ``spectral curve'' and a set of Darboux coordinates called ``spectral Darboux coordinates''. The equations of motion of the dressing chain can be converted to a Hamiltonian system in these Darboux coordinates. The symplectic structure of this Hamiltonian formalism turns out to be consistent with a Poisson structure previously studied by Veselov, Shabat, Noumi and Yamada.Comment: latex2e, 41 pages, no figure; (v2) some minor errors are corrected; (v3) fully revised and shortend, and some results are improve

    Apparent singularities of Fuchsian equations, and the Painlev\'e VI equation and Garnier systems

    Full text link
    We study movable singularities of Garnier systems using the connection of the latter with isomonodromic deformations of Fuchsian systems. Questions on the existence of solutions for some inverse monodromy problems are also considered.Comment: 24 page

    Canonical structure and symmetries of the Schlesinger equations

    Full text link
    The Schlesinger equations S(n,m)S_{(n,m)} describe monodromy preserving deformations of order mm Fuchsian systems with n+1n+1 poles. They can be considered as a family of commuting time-dependent Hamiltonian systems on the direct product of nn copies of m×mm\times m matrix algebras equipped with the standard linear Poisson bracket. In this paper we present a new canonical Hamiltonian formulation of the general Schlesinger equations S(n,m)S_{(n,m)} for all nn, mm and we compute the action of the symmetries of the Schlesinger equations in these coordinates.Comment: 92 pages, no figures. Theorem 1.2 corrected, other misprints removed. To appear on Comm. Math. Phy

    Multi-objective/loading optimization for rotating composite flexbeams

    Get PDF
    With the evolution of advanced composites, the feasibility of designing bearingless rotor systems for high speed, demanding maneuver envelopes, and high aircraft gross weights has become a reality. These systems eliminate the need for hinges and heavily loaded bearings by incorporating a composite flexbeam structure which accommodates flapping, lead-lag, and feathering motions by bending and twisting while reacting full blade centrifugal force. The flight characteristics of a bearingless rotor system are largely dependent on hub design, and the principal element in this type of system is the composite flexbeam. As in any hub design, trade off studies must be performed in order to optimize performance, dynamics (stability), handling qualities, and stresses. However, since the flexbeam structure is the primary component which will determine the balance of these characteristics, its design and fabrication are not straightforward. It was concluded that: pitchcase and snubber damper representations are required in the flexbeam model for proper sizing resulting from dynamic requirements; optimization is necessary for flexbeam design, since it reduces the design iteration time and results in an improved design; and inclusion of multiple flight conditions and their corresponding fatigue allowables is necessary for the optimization procedure

    Miniversal deformations of pairs of symmetric matrices under congruence

    Full text link
    For each pair of complex symmetric matrices (A,B)(A,B) we provide a normal form with a minimal number of independent parameters, to which all pairs of complex symmetric matrices (A~,B~)(\widetilde{A},\widetilde{B}), close to (A,B)(A,B) can be reduced by congruence transformation that smoothly depends on the entries of A~\widetilde{A} and B~\widetilde{B}. Such a normal form is called a miniversal deformation of (A,B)(A,B) under congruence. A number of independent parameters in the miniversal deformation of a symmetric matrix pencil is equal to the codimension of the congruence orbit of this symmetric matrix pencil and is computed too. We also provide an upper bound on the distance from (A,B)(A,B) to its miniversal deformation.Comment: arXiv admin note: text overlap with arXiv:1104.249
    • …
    corecore