If the characteristic polynomial of a discrete-time system has all its roots in the open unit disc of the complex plane, the system is called Schur stable. In this paper, the Schur stabilization problem of closed loop discrete-time system by affine compensator is considered. For this purpose, the distance function between the Schur stability region and the affine controller subset is investigated.

The monotonicity and the Schur-convexity with parameters (s, t) in R2 for fixed (x, y) and the Schur-convexity and the Schur-geometrically convexity with variables (x, y) in R++ for fixed (s, t) of Gini mean G(r, s;x, y) are discussed. Some new inequalities are obtained.

Confirming a conjecture of Mark Shimozono, we identify polynomial representatives for the Schubert classes of the affine Grassmannian as the k-Schur functions in homology and affine Schur functions in cohomology. Our results rely on Kostant and Kumar’s nilHecke ring, work of Peterson on the homology of based loops on a compact group, and earlier work of ours on non-commutative k-Schur functions.

In this paper, we obtain some formulas for compound matrices of generalized Schur complements of matrices. Further, we give some Löwner partial orders for compound matrices of Schur complements of positive semidefinite Hermitian matrices, and obtain some estimates for eigenvalues of Schur complements of sums of positive semidefinite Hermitian matrices.

X iv :0 90 5. 25 57 v2 [ m at h. R T ] 1 0 Ju n 20 09 JACOBI–TRUDY FORMULA FOR GENERALISED SCHUR POLYNOMIALS A.N. SERGEEV AND A.P. VESELOV Abstract. Jacobi–Trudy formula for a generalisation of Schur polynomials related to any sequence of orthogonal polynomials in one variable is given. As a corollary we have Giambelli formula for generalised Schur polynomials.

Stewart’s recently introduced Krylov-Schur algorithm is a modification of the implicitly restarted Arnoldi algorithm which employs reordered Schur decompositions to perform restarts and deflations in a numerically reliable manner. This paper describes a variant of the Krylov-Schur algorithm suitable for addressing eigenvalue problems associated with products of large and sparse matrices. It per...

We prove that the duality operator preserves the Frobenius–Schur indicators of characters of connected reductive groups of Lie type with connected center. This allows us to extend a result of D. Prasad which relates the Frobenius–Schur indicator of a regular real-valued character to its central character. We apply these results to compute the Frobenius–Schur indicators of certain real-valued, i...

This paper introduces a generalized Schur algorithm in the Krein space with an indefinite inner product. Concepts such as Caratheodory classes and Schur classes used in the classical Schur algorithm cannot be applied in the Krein space since the positivedefiniteness corresponds merely to the nonsingularity in the Krein space. We note also that these problems appear when fast algorithms for subo...

Introduction 1 1. Representation Theory of Finite Groups 2 1.1. Preliminaries 2 1.2. Group Algebra 4 1.3. Character Theory 5 2. Irreducible Representations of the Symmetric Group 8 2.1. Specht Modules 8 2.2. Dimension Formulas 11 2.3. The RSK-Correspondence 12 3. Schur-Weyl Duality 13 3.1. Representations of Lie Groups and Lie Algebras 13 3.2. Schur-Weyl Duality for GL(V ) 15 3.3. Schur Functor...