228 research outputs found
Poisson structures on affine spaces and flag varieties. I. Matrix affine Poisson space
The standard Poisson structure on the rectangular matrix variety Mm,n(C) is
investigated, via the orbits of symplectic leaves under the action of the maximal torus T ⊂
GLm+n(C). These orbits, finite in number, are shown to be smooth irreducible locally closed
subvarieties of Mm,n(C), isomorphic to intersections of dual Schubert cells in the full flag
variety of GLm+n(C). Three different presentations of the T-orbits of symplectic leaves in
Mm,n(C) are obtained ā (a) as pullbacks of Bruhat cells in GLm+n(C) under a particular
map; (b) in terms of rank conditions on rectangular submatrices; and (c) as matrix products
of sets similar to double Bruhat cells in GLm(C) and GLn(C). In presentation (a), the orbits
of leaves are parametrized by a subset of the Weyl group Sm+n, such that inclusions of
Zariski closures correspond to the Bruhat order. Presentation (b) allows explicit calculations
of orbits. From presentation (c) it follows that, up to Zariski closure, each orbit of leaves is
a matrix product of one orbit with a fixed column-echelon form and one with a fixed rowechelon
form. Finally, decompositions of generalized double Bruhat cells in Mm,n(C) (with
respect to pairs of partial permutation matrices) into unions of T-orbits of symplectic leaves
are obtained
Model Reduction of Multi-Dimensional and Uncertain Systems
We present model reduction methods with guaranteed error bounds for systems represented by a Linear Fractional Transformation (LFT) on a repeated scalar uncertainty structure. These reduction methods can be interpreted either as doing state order reduction for multi-dimensionalsystems, or as uncertainty simplification in the case of uncertain systems, and are based on finding solutions to a pair of Linear Matrix Inequalities (LMIs). A related necessary and sufficient condition for the exact reducibility of stable uncertain systems is also presented
Algebraic properties of Manin matrices 1
We study a class of matrices with noncommutative entries, which were first
considered by Yu. I. Manin in 1988 in relation with quantum group theory. They
are defined as "noncommutative endomorphisms" of a polynomial algebra. More
explicitly their defining conditions read: 1) elements in the same column
commute; 2) commutators of the cross terms are equal: (e.g. ). The basic claim
is that despite noncommutativity many theorems of linear algebra hold true for
Manin matrices in a form identical to that of the commutative case. Moreover in
some examples the converse is also true. The present paper gives a complete
list and detailed proofs of algebraic properties of Manin matrices known up to
the moment; many of them are new. In particular we present the formulation in
terms of matrix (Leningrad) notations; provide complete proofs that an inverse
to a M.m. is again a M.m. and for the Schur formula for the determinant of a
block matrix; we generalize the noncommutative Cauchy-Binet formulas discovered
recently [arXiv:0809.3516], which includes the classical Capelli and related
identities. We also discuss many other properties, such as the Cramer formula
for the inverse matrix, the Cayley-Hamilton theorem, Newton and
MacMahon-Wronski identities, Plucker relations, Sylvester's theorem, the
Lagrange-Desnanot-Lewis Caroll formula, the Weinstein-Aronszajn formula, some
multiplicativity properties for the determinant, relations with
quasideterminants, calculation of the determinant via Gauss decomposition,
conjugation to the second normal (Frobenius) form, and so on and so forth. We
refer to [arXiv:0711.2236] for some applications.Comment: 80 page
Robust Multiple Signal Classification via Probability Measure Transformation
In this paper, we introduce a new framework for robust multiple signal
classification (MUSIC). The proposed framework, called robust
measure-transformed (MT) MUSIC, is based on applying a transform to the
probability distribution of the received signals, i.e., transformation of the
probability measure defined on the observation space. In robust MT-MUSIC, the
sample covariance is replaced by the empirical MT-covariance. By judicious
choice of the transform we show that: 1) the resulting empirical MT-covariance
is B-robust, with bounded influence function that takes negligible values for
large norm outliers, and 2) under the assumption of spherically contoured noise
distribution, the noise subspace can be determined from the eigendecomposition
of the MT-covariance. Furthermore, we derive a new robust measure-transformed
minimum description length (MDL) criterion for estimating the number of
signals, and extend the MT-MUSIC framework to the case of coherent signals. The
proposed approach is illustrated in simulation examples that show its
advantages as compared to other robust MUSIC and MDL generalizations
Matrix algebras and displacement decompositions
A class xi of algebras of symmetric nxn matrices, related to Toeplitz-plus-Hankel structures and including the well-known algebra H diagonalized by the Hartley transform, is investigated. The algebras of xi are then exploited in a general displacement decomposition of an arbitrary nxn matrix A. Any algebra of xi is a 1-space, i.e., it is spanned by n matrices having as first rows the vectors of the canonical basis. The notion of 1-space (which generalizes the previous notions of L1 space [Bevilacqua and Zellini, Linear and Multilinear Algebra, 25 (1989), pp.1-25] and Hessenberg algebra [Di Fiore and Zellini, Linear Algebra Appl., 229 (1995), pp.49-99]) finally leads to the identification in xi of three new (non-Hessenberg) matrix algebras close to H, which are shown to be associated with fast Hartley-type transforms. These algebras are also involved in new efficient centrosymmetric Toeplitz-plus-Hankel inversion formulas
A theory of linear estimation
Theory of linear estimation and applicability to problems of smoothing, filtering, extrapolation, and nonlinear estimatio
Frequency Dispersion of Sound Propagation in Rouse Polymer Melts via Generalized Dynamic Random Phase Approximation
An extended generalization of the dynamic random phase approximation (DRPA)
for L-component polymer systems is presented. Unlike the original version of
the DRPA, which relates the (LxL) matrices of the collective density-density
time correlation fumctions and the corresponding susceptibilities of polymer
concentrated systems to those of the tracer macromolecules and so-called broken
links system (BLS), our generalized DRPA solves this problem for (5xL)x(5xL)
matrices of the coupled susceptibilities and time correlation functions of the
component number, kinetic energy and flux densities. The presented technique is
used to study propagation of sound and dynamic form-factor in disentangled
(Rouse) monodisperse homopolymer melt. The calculated sound velocity and
absorption coefficient reveal substantial frequency dispersion. The relaxation
time is found to be N times less than the Rouse time (N is the degree of
polymerization), which evidences strong dynamic screening because of interchain
interaction. We discuss also some peculiarities of the Brillouin scattering in
polymer melts. Besides, a new convenient expression for the dynamic structural
function of the Rouse chain in (q,p)-representation is found.Comment: 37 pages, 2 appendices, 48 references, 1 figur
Rational interpolation and state-variable realizations
AbstractThe problem is considered of passing from interpolation data for a real rational transfer-function matrix to a minimal state-variable realization of the transfer-function matrix. The tool is a Loewner matrix, which is a generalization of the Standard Hankel matrix of linear system realization theory, and which possesses a decomposition into a product of generalized observability and controllability matrices
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