Search

The main aim of the article is to prove that the symmetric function Φ n ( x , r ) = ∏ i 1 + i 2 + ⋯ + i n = r ( x 1 i 1 + x 2 i 2 + ⋯ + x n i n ) is Schur geometrically convex for x ∈ R + + n and fixed r ∈ N + = { 1 , 2 , ⋯ } , where i 1 , i 2 , ⋯ , i n are non-negative integers. Further, we obtain Φ n ( x , r ) is also Schur m-power convex for m ≤ 0 . As applications, a Klamkin–Newman type inequality is derived. Finally, we list a counter example to illustrate Φ n ( x , r ) is neither Schur convex nor Schur concave.

In this paper, we present various forms of the inverse of a special Schur complement C ⁢ D - 1 ⁢ B {CD^{-1}B} in the case where B, C, D are operators on Hilbert spaces. We show that when C ⁢ D - 1 ⁢ B {CD^{-1}B} is invertible, there always exist operators X and Y which belong to some special classes of generalized inverses of B and C, for which ( C ⁢ D - 1 ⁢ B ) - 1 = X ⁢ D ⁢ Y {(CD^{-1}B)^{-1}=XDY} . Some explicit expressions for such X and Y are given.

In 1926, I. Schur proved that the number of partitions of [Formula: see text] into parts [Formula: see text] equals the number of partitions of [Formula: see text] of the form [Formula: see text] such that [Formula: see text] with strictly inequality if [Formula: see text]. We prove that Schur’s partition function is related to a weight 3/2 Hecke eigenform modulo 2. As a consequence, we obtain some congruences for Schur’s partitions functions. Schur’s partitions theorem can be easily generalized to general moduli. We show that the generalized Schur’s partition functions satisfy beautiful transformation properties. We also find an unexpected relation between the generalized Schur’s partitions and singular overpartitions, which was introduced by G. E. Andrews recently.

The theory of Schur complement is very important in many fields such as control theory and computational mathematics. In this paper, by applying the properties of the Schur complement and some inequality techniques, some new estimates of the diagonally, -diagonally and product -diagonally dominant degree on the Schur complement of matrices are obtained, which improve some relative results. Further, as an application of these derived results, we present some distributions for the eigenvalues of the Schur complements. Finally, the numerical example is given to show the advantages of our derived results.

Spectral analysis of coupled PDEs and of their Schur complements via Generalized Locally Toeplitz sequences in 2D

This paper is concerned with root localization of a complex polynomial with respect to the unit circle in the more general case. The classical Schur-Cohn-Fujiwara theorem converts the inertia problem of a polynomial to that of an appropriate Hermitian matrix under the condition that the associated Bezout matrix is nonsingular. To complete it, we discuss an extended version of the Schur-Cohn-Fujiwara theorem to the singular case of that Bezout matrix. Our method is mainly based on a perturbation technique for a Bezout matrix. As an application of these results and methods, we further obtain an explicit formula for the number of roots of a polynomial located on the upper half part of the unit circle as well.

In this paper, we study the Schur decomposition for unitary symmetric matrices: The formula for the Schur decomposition is obtained, and the relationship of the unitary symmetric matrix with mother matrix is also obtained.

The theory of Schur complement plays an important role in many fields such as matrix theory, control theory and computational mathematics. In this paper, some new estimates of diagonally, α-diagonally and product α-diagonally dominant degree on the Schur complement of matrices are obtained, which improve some relative results. As an application, we present several new eigenvalue inclusion regions for the Schur complement of matrices. Finally, we give a numerical example to illustrate the advantages of our derived results.

[EN] In this paper we give a new characterization of when a Banach space E has the Schur property in terms of the disk algebra. We prove that E has the Schur property if and only if A(D, E) = A(D,E-w). (C) 2016 Elsevier Inc. All rights reserved., The first and third authors were supported by MINECO MTM2014-57838-C2-2-P and Prometeo II/2013/013. The second author was supported by MINECO MTM2013-43540-P.

We establish an unbounded version of Stinespring's Theorem and a lifting result for Stinespring representations of completely positive modular maps defined on the space of all compact operators. We apply these results to study positivity for Schur multipliers. We characterise positive local Schur multipliers, and provide a description of positive local Schur multipliers of Toeplitz type. We introduce local operator multipliers as a non-commutative analogue of local Schur multipliers, and obtain a characterisation that extends earlier results concerning operator multipliers and local Schur multipliers. We provide a description of the positive local operator multipliers in terms of approximation by elements of canonical positive cones., Comment: 31 pages

For , the symmetric functions are defined by where , and are non-negative integers. In this paper, the Schur convexity, geometric Schur convexity and harmonic Schur convexity of are investigated. As applications, Schur convexity for the other symmetric functions is obtained by a bijective transformation of independent variable for a Schur convex function, some analytic and geometric inequalities are established by using the theory of majorization, in particular, we derive from our results a generalization of Sharpiro's inequality, and give a new generalization of Safta's conjecture in the n-dimensional space and others.

In this paper we study properties of the partial Schur multiplier. As an application a characterization of components of the partial Schur multiplier for some groups is given. We also describe the classical Schur multiplier of elementary abelian 2-groups by using partial factor sets.

By the properties of Schur-convex function, Schur geometrically convex function and Schur harmonically convex function, Schur-convexity, Schur geometric and Schur harmonic con- vexities of the dual form for a class of symmetric functions are simply proved. As an application, several inequalities are obtained, some of which extend the known ones.

We present sucient conditions in order (the space of) a Riesz op- erator T in a Hilbert space H have a Jordan-Schur basis with respect to a scalar product equivalent to the original one. This is related to Schur's lemma for a compact operator, which is an extension of Schur's classical theorem on unitary triangularization in a finite dimensional space. The finite dimensional case is also studied.

In this paper, we study the Schur factoization for unitary extended matrices: The formula for the Schur-type factorization is obtained, and the relationship of the unitary extended matrix with mother matrix is also obtained.

In a letter written to Landau in 1935, Schur stated that for any integer t > 2, there are primes p 1 < p 2 < … < p t such that p 1 + p 2 > pt. In this note, we use the Prime Number Theorem and extend Schur’s result to show that for any integers t ⩾ k ⩾ 1 and real ɛ > 0, there exist primes p 1 < p 2 < … < p t such that p 1 + p 2 + … + p k > (k − ɛ)p t .

In this paper, the Schur D-stability problem for a class of linear grey discrete dynamic systems is studied in terms of the matrix eigenvalues theory and spectral radius approach. Several necessary and sufficient conditions and some sufficient conditions are obtained which can guarantee the Schur D-stability of linear grey discrete dynamic systems. The equivalence relation between the Schur D-stability and Schur stability of linear grey discrete dynamic systems is established.

We examine when a matrix whose elements are differentiable functions in one variable commutes with its derivative. This problem was discussed in a letter from Issai Schur to Helmut Wielandt written in 1934, which we found in Wielandt's Nachlass. We present this letter and its translation into English. The topic was rediscovered later and partial results were proved. However, there are many subtle observations in Schur's letter which were not obtained in later years. Using an algebraic setting, we put these into perspective and extend them in several directions. We present in detail the relationship between several conditions mentioned in Schur's letter and we focus in particular on the characterization of matrices called Type 1 by Schur. We also present several examples that demonstrate Schur's observations., 21 pages, 1 facsimile, some inaccuracies corrected, historical comments and acknowledgments added; final version to appear in Linear Algebra and its Applications

In this paper we classify all Schur functions and skew Schur functions that are multiplicity free when expanded in the basis of fundamental quasisymmetric functions, termed F-multiplicity free. Combinatorially, this is equivalent to classifying all skew shapes whose standard Young tableaux have distinct descent sets. We then generalize our setting, and classify all F-multiplicity free quasisymmetric Schur functions with one or two terms in the expansion, or one or two parts in the indexing composition. This identifies composition shapes such that all standard composition tableaux of that shape have distinct descent sets. We conclude by providing such a classification for quasisymmetric Schur function families, giving a classification of Schur functions that are in some sense almost F-multiplicity free., Comment: 20 pages; final version to appear in Ann. Comb

We introduce the notion of a Schur-finite element in a λ-ring.

By applying the properties of Schur complement and some inequality techniques, some new estimates of diagonally and doubly diagonally dominant degree of the Schur complement of Ostrowski matrix are obtained, which improve the main results of Liu and Zhang (2005) and Liu et al. (2012). As an application, we present new inclusion regions for eigenvalues of the Schur complement of Ostrowski matrix. In addition, a new upper bound for the infinity norm on the inverse of the Schur complement of Ostrowski matrix is given. Finally, we give numerical examples to illustrate the theory results.

Characterization of generalized Schur functions in terms of their Taylor coefficients was established by Krein and Langer [‘Über einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raume ${\Pi }_{\kappa } $ zusammenhängen. I. Einige Funktionenklassen und ihre Darstellungen’, Math. Nachr. 77 (1977), 187–236]. We establish a boundary analogue of this characterization.

In this paper we present a framework in which the Schur transformation and the basic interpolation problem for generalized Schur functions, generalized Nevanlinna functions and the like can be studied in a unified way. The basic object is a general class of functions for which a certain kernel has a finite number of negative squares. The results are based on and generalize those in previous papers of the first three authors on the Schur transformation in an indefinite setting. (c) 2012 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.

The inverse of a Schur complement is a very useful tool in many algorithms for the computation of the matrix inversion. In this paper we study the inverse of a special Schur complement CD −1 B . We proved that always exist some X and Y such that XDY is a inverse of CD −1 B . Furthermore, using minimal rank properties, we give some explicit expressions for X , Y and the inverse of CD −1 B . Numerical examples also given.

The theory of Schur complement plays an important role in many fields, as well as the theory of H -matrices. In this paper, we obtain some bounds for the eigenvalues of Schur complement by the entries of the original matrix instead of those of the Schur complement, for some special H -matrices. Also, we show how this result for a wider class can be applied for estimating bounds for the eigenvalues of Schur complement of an SDD matrix.

We firstly consider the block dominant degree for I-(II-)block strictly diagonally dominant matrix and their Schur complements, showing that the block dominant degree for the Schur complement of an I-(II-)block strictly diagonally dominant matrix is greater than that of the original grand block matrix. Then, as application, we present some disc theorems and some bounds for the eigenvalues of the Schur complement by the elements of the original matrix. Further, by means of matrix partition and the Schur complement of block matrix, based on the derived disc theorems, we give a kind of iteration called the Schur-based iteration, which can solve large scale linear systems though reducing the order by the Schur complement and the numerical example illustrates that the iteration can compute out the results faster.

In a recent article Gowda and Sznajder (Linear Algebra Appl 432:1553---1559, 2010) studied the concept of Schur complement in Euclidean Jordan algebras and described Schur determinantal and Haynsworth inertia formulas. In this article, we establish some more results on the Schur complement. Specifically, we prove, in the setting of Euclidean Jordan algebras, an analogue of the Crabtree-Haynsworth quotient formula and show that any Schur complement of a strictly diagonally dominant element is strictly diagonally dominant. We also introduce the concept of Schur product of a real symmetric matrix and an element of a Euclidean Jordan algebra when its Peirce decomposition with respect to a Jordan frame is given. An Oppenheim type inequality is proved in this setting.

For Schur stable linear difference equation system with periodic coefficients, we prove continuity theorems on monodromy matrix which show how much change is permissible without disturbing the Schur stability, and some examples illustrating the efficiency of the theorems are given.

The Jacobi–Trudi identity expresses a skew Schur function as a determinant of complete symmetric functions. Bressoud and Wei extend this idea, introducing an integer parameter t⩾−1 and showing that signed sums of skew Schur functions of a certain shape are expressible once again as a determinant of complete symmetric functions. Koike provides a Jacobi–Trudi-style definition of universal rational characters of the general linear group and gives their expansion as a signed sum of products of Schur functions in two distinct sets of variables. Here we extend Bressoud and Wei's formula by including an additional parameter and extending the result to the case of all integer t. Then we introduce this parameter idea to the Koike formula, extending it in the same way. We prove our results algebraically using Laplace determinantal expansions.

Let S1 = A ? BD?C and S2 = D?CA?B be the associated Schur complements of M = [A B/C D]. In this paper, we derive necessary and sufficient conditions for S1 = 0 imply S2 = 0 by using generalized inverses of matrices and singular value decompositions.

The Pieri rule expresses the product of a Schur function and a single row Schur function in terms of Schur functions. We extend the classical Pieri rule by expressing the product of a skew Schur function and a single row Schur function in terms of skew Schur functions. Like the classical rule, our rule involves simple additions of boxes to the original skew shape. Our proof is purely combinatorial and extends the combinatorial proof of the classical case., La règle de Pieri exprime le produit d'une fonction de Schur et de la fonction de Schur d'une seule ligne en termes de fonctions de Schur. Nous étendons la règle classique de Pieri en exprimant le produit d'un fonction gauche de Schur et de la fonction de Schur d'une ligne en termes de fonctions gauches de Schur. Comme la règle classique, notre règle implique l'ajout de cases à la forme gauche initiale. Notre preuve est purement combinatoire et étend celle du cas classique.

In this paper, using Schur complements, we prove various inequalities in Euclidean Jordan algebras. Specifically, we study analogues of the inequalities of Fischer, Hadamard, Bergstrom, Oppenheim, and other inequalities related to determinants, eigenvalues, and Schur complements.

We study Schur algebras of classical groups over an algebraically closed field of characteristic different from 2. We prove that Schur algebras are generalized Schur algebras (in Donkin's sense) in types A, C, and D, while this does not hold in type B. Consequently Schur algebras of types A, C, and D are integral quasi-hereditary by Donkin [7, 9]. By using the coalgebra approach we put Schur algebras of a fixed classical group into a certain inverse system. We find that the corresponding hyperalgebra is contained in the inverse limit as a subalgebra. Moreover in types A, C, and D, the surjections in the inverse systems are compatible with the integral quasi-hereditary structure of Schur algebras.

In this article, we study the concept of Schur complement in the setting of Euclidean Jordan algebras and describe Schur determinantal and Haynsworth inertia formulas.

The theory of Schur complement plays an important role in many fields such as control theory and computational mathematics. In this paper, we obtain some estimates for the diagonally, @c-diagonally and product @c-diagonally dominant degree on the Schur complement of matrices, which improve some relative results. As application we present some bounds for the eigenvalues of Schur complement by the entries of the original matrix instead of those of the Schur complement. Particularly, we obtain that the eigenvalues of the Schur complements are located in the Gerschgorin Circles of the original matrices under certain conditions.

In this paper, we obtain some estimates for the γ-diagonally and product γ-diagonally dominant degree of the Schur complement of matrices. As application we present some bounds for the eigenvalues of Schur complement by the entries of the original matrix. Based on these results, we give a kind of iteration called the Schur-based iteration, which can solve large scale linear systems though reducing the order by the Schur complement and can compute out the results faster.

An algorithm for computing primary roots of a nonsingular matrix A is presented. In particular, it computes the principal root of a real matrix having no nonpositive real eigenvalues, using real arithmetic. The algorithm is based on the Schur decomposition of A and has an order of complexity lower than the customary Schur based algorithm, namely the Smith algorithm.

We compute the norm of the restriction of a Schur multiplier, arising from a multiplication operator, to a coordinate subspace. This result is used to generalize Wielandt's minimax inequality. Furthermore, we compute various s-numbers of an elementary Schur multiplier and determine criteria for membership of such multipliers in certain operator ideals.

Schur multipliers were introduced by Schur in the early 20th century and have since then found a considerable number of applications in Analysis and enjoyed an intensive development. Apart from the beauty of the subject in itself, sources of interest in them were connections with Perturbation Theory, Harmonic Analysis, the Theory of Operator Integrals and others. Advances in the quantisation of Schur multipliers were recently made in [29]. The aim of the present article is to summarise a part of the ideas and results in the theory of Schur and operator multipliers. We start with the classical Schur multipliers defined by Schur and their characterisation by Grothendieck, and make our way through measurable multipliers studied by Peller and Spronk, operator multipliers defined by Kissin and Shulman and, finally, multidimensional Schur and operator multipliers developed by Juschenko and the authors. We point out connections of the area with Harmonic Analysis and the Theory of Operator Integrals.

Lower bounds for the representation dimension of Schur algebras for GL n in characteristic p ≥ 2n − 1 are established. In particular it is shown that for fixed n the representation dimensions of the Schur algebras get arbitrarily large.

Classical Schur's matrix is a different evaluation, provided by Schur, of the quadratic Gaussian sum from Gauss. The advanced information was studied by L. Carlitz who determined its eigenvalues, and by P. Morton who determined its eigenvectors. In this paper, we generalize the classical Schur's matrix to the case in polynomial rings over finite fields, and what is more, we give explicit formulas for the determinant, inverse matrix, eigenvalues, multiplicity and eigenvectors with respect to each eigenvalue of the polynomial Schur's matrix.

PurposeThe purpose of this paper is to discuss the Schur D‐stability and the vertex stability of interval matrices (including point matrix obviously). Some new sufficient conditions (criteria) are proposed which guarantee the interval matrix is Schur D‐stable. This results are shown to be less conservative than those in recent literatures. In addition, two equivalence relations between the Schur D‐stability and the vertex stability of interval matrices will be proposed and a new Schur D‐stability range of an interval matrix presented.Design/methodology/approachMatrix eigenvalues theory and matrix measure approach.FindingsSeveral simple sufficient conditions (criteria) for guaranteeing the Schur D‐stability of interval matrices are derived, two equivalence relations between the Schur D‐stability and the vertex stability of interval matrices are proposed, and a new Schur D‐stability range of an interval matrix is presented.Research limitations/implicationsControl theory or stability theory. These stability criterion possess simple forms and provide useful tools to check Schur D‐stability of interval matrices (including point matrix) at first stage.Practical implicationsThe paper provides useful tools to check Schur D‐stability of interval matrices (including point matrix) at first stage.Originality/valueTwo equivalence relations between the Schur D‐stability and the vertex stability for general interval matrices (including point matrix) are proposed, such that the conditional limitations for tridiagonal matrix in recent papers are broken. A new Schur D‐stability range of an interval matrix is presented, and several simple sufficient conditions are obtained which guarantee the Schur D‐stability of interval matrices (including point matrix).

The purpose of this paper is twofold: We first present a sufficient condition for testing strictly generalized diagonally dominant matrices (i.e., H-matrices) and we claim that our criterion is superior to the existing ones. We then show that the proper subset of the H-matrices determined by the condition preserves the closure property under the Schur complement operation.

The principal object of this paper is to present a new approach simultaneously to both nondegenerate and degenerate cases of the matricial Schur problem. This approach is based on an analysis of the central matrixvalued Schur functions which was started in [24]–[26] and then continued in [27]. In the nondegenerate situation we will see that the parametrization of the solution set obtained here coincides with the well-known formula of D. Z. Arov and M. G. Kreĭn for that case (see [1]). Furthermore, we give some characterizations of the situation that the matricial Schur problem has a unique solution (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

This paper deals with an efficient technique for computing high-quality approximations of Schur complement matrices to be used in various preconditioners for the iterative solution of finite element discretizations of elliptic boundary value problems. The Schur complements are based on a two-by-two block decomposition of the matrix, and their approximations are computed by assembly of local (macroelement) Schur complements. The block partitioning is done by imposing a particular node ordering following the grid refinement hierarchy in the discretization mesh. For the theoretical derivation of condition number bounds, but not for the actual application of the method, we assume that the corresponding differential operator is self-adjoint and positive definite. The numerical efficiency of the proposed Schur complement approximation is illustrated in the framework of block incomplete factorization preconditioners of multilevel type, which require approximations of a sequence of arising Schur complement matrices. The behavior of the proposed approximation is compared with that of the coarse mesh finite element matrix, commonly used as an approximation of the Schur complement in the context of the above preconditioning methods. Moreover, the influence of refining a coarse mesh using a higher refinement number ($m$) than the customary $m=2$ is analyzed and its efficiency is also illustrated by numerical tests.