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Representations of the general linear group over symmetry classes of polynomials

Yousef Zamani, Mahin Ranjbari (2018)

Czechoslovak Mathematical Journal

Let V be the complex vector space of homogeneous linear polynomials in the variables x 1 , ... , x m . Suppose G is a subgroup of S m , and χ is an irreducible character of G . Let H d ( G , χ ) be the symmetry class of polynomials of degree d with respect to G and χ . For any linear operator T acting on V , there is a (unique) induced operator K χ ( T ) End ( H d ( G , χ ) ) acting on symmetrized decomposable polynomials by K χ ( T ) ( f 1 * f 2 * ... * f d ) = T f 1 * T f 2 * ... * T f d . In this paper, we show that the representation T K χ ( T ) of the general linear group G L ( V ) is equivalent to the direct sum of χ ( 1 ) copies of a representation...

Residual norm behavior for Hybrid LSQR regularization

Havelková, Eva, Hnětynková, Iveta (2023)

Programs and Algorithms of Numerical Mathematics

Hybrid LSQR represents a powerful method for regularization of large-scale discrete inverse problems, where ill-conditioning of the model matrix and ill-posedness of the problem make the solutions seriously sensitive to the unknown noise in the data. Hybrid LSQR combines the iterative Golub-Kahan bidiagonalization with the Tikhonov regularization of the projected problem. While the behavior of the residual norm for the pure LSQR is well understood and can be used to construct a stopping criterion,...

Robust stability of positive continuous-time linear systems with delays

Mikołaj Busłowicz (2010)

International Journal of Applied Mathematics and Computer Science

The paper is devoted to the problem of robust stability of positive continuous-time linear systems with delays with structured perturbations of state matrices. Simple necessary and sufficient conditions for robust stability in the general case and in the case of systems with a linear uncertainty structure in two sub-cases: (i) a unity rank uncertainty structure and (ii) nonnegative perturbation matrices are established. The problems are illustrated with numerical examples.

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