20,176 research outputs found
Numerical Schubert calculus
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
Integrable Euler top and nonholonomic Chaplygin ball
We discuss the Poisson structures, Lax matrices, -matrices, bi-hamiltonian
structures, the variables of separation and other attributes of the modern
theory of dynamical systems in application to the integrable Euler top and to
the nonholonomic Chaplygin ball.Comment: 25 pages, LaTeX with AMS fonts, final versio
Spectral curve, Darboux coordinates and Hamiltonian structure of periodic dressing chains
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 has a control parameter .
If , the -periodic dressing chain may be thought of as a
generalization of the fourth or fifth (depending on the parity of )
Painlev\'e equations . The -periodic dressing chain has two different Lax
representations due to Adler and to Noumi and Yamada. Adler's 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
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
The Schlesinger equations describe monodromy preserving
deformations of order Fuchsian systems with poles. They can be
considered as a family of commuting time-dependent Hamiltonian systems on the
direct product of copies of matrix algebras equipped with the
standard linear Poisson bracket. In this paper we present a new canonical
Hamiltonian formulation of the general Schlesinger equations for
all , 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
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
For each pair of complex symmetric matrices we provide a normal form
with a minimal number of independent parameters, to which all pairs of complex
symmetric matrices , close to can be
reduced by congruence transformation that smoothly depends on the entries of
and . Such a normal form is called a miniversal
deformation of 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 to
its miniversal deformation.Comment: arXiv admin note: text overlap with arXiv:1104.249
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