Theorems on Schur complement of block diagonally dominant matrices and their application in reducing the order for the solution of large scale linear systems

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.     

On sets of integers with the Schur property

We investigate sets of integers for which Rado and Schur theorems are true from the point of view of their local density. We establish the existence of locally sparse Rado and Schur sets in a strong sense.     

赋范线性空间中的列收敛

本文证明了Ｓｃｈｕｒ空间与有限维Ｂａｎａｃｈ空间的两个特征定理，并对赋范线性空间中的列收敛进行了讨论，得到了一些有趣的结果。     

Schur lemma and limit theorems in lattice groups with respect to filters

Some Schur, Nikodým, Brooks-Jewett and Vitali-Hahn-Saks-type theorems for (?)-group-valued measures are proved in the setting of filter convergence. Finally we pose an open problem.     

Inners and Schur complement

It is shown that for any square matrix having a left triangle of zeros, the determinants of its inners are equal to the leading principal minors of its Schur complement. In particular, if the original matrix has Sylvester-type form, then relationship between zero location theorems for polynomials are recovered.     

Bijective proofs of some classical partition identities

A bijective proof of a general partition theorem is given which has as direct corollaries many classical partition theorems due to Euler, Glaisher, Schur, Andrews, Subbarao, and others. It is shown that the bijective proof specializes to give bijective proofs of these classical results and moreover the bijections which result often coincide with bijections which have occurred in the literature. Also given are some sufficient conditions for when two classes of words omitting certain sequences of words are in bijection.     

(Shifted) Macdonald polynomials: q-Integral representation and combinatorial formula

We extend some results about shifted Schur functions to the general context of shifted Macdonald polynomials. We strengthen some theorems of F. Knop and S. Sahi and give two explicit formulas for these polynomials: a q-integral representation and a combinatorial formula. Our main tool is a q-integral representation for ordinary Macdonald polynomial. We also discuss duality for shifted Macdonald polynomials and Jack degeneration of these polynomials.     

Application of Conformal Mappings to Inequalities for Polynomials

Applications of the geometric theory of functions to inequalities for algebraic polynomials are considered. The main attention is paid to constructing a univalent conformal mapping for a given polynomial and to applying the Lebedev and Nehari theorems to this mapping. A new sharp inequality of Bernshtein type for polynomials with restrictions on the growth on a segment or on a circle, inequalities with restrictions on the zeros of the polynomial, and other inequalities are obtained. In particular, classical inequalities by Markov, Bernshtein, and Schur are strengthened. Bibiography: 13 titles.     

Superfilters, Ramsey theory, and van der Waerden's Theorem

Superfilters are generalizations of ultrafilters, and capture the underlying concept in Ramsey-theoretic theorems such as van der Waerden's Theorem. We establish several properties of superfilters, which generalize both Ramsey's Theorem and its variants for ultrafilters on the natural numbers. We use them to confirm a conjecture of Kočinac and Di Maio, which is a generalization of a Ramsey-theoretic result of Scheepers, concerning selections from open covers. Following Bergelson and Hindman's 1989 Theorem, we present a new simultaneous generalization of the theorems of Ramsey, van der Waerden, Schur, Folkman–Rado–Sanders, Rado, and others, where the colored sets can be much smaller than the full set of natural numbers.     

A new concept for block operator matrices:the quadratic numerical range

In this paper a new concept for 2×2-block operator matrices – the quadratic numerical range – is studied. The main results are a spectral inclusion theorem, an estimate of the resolvent in terms of the quadratic numerical range, factorization theorems for the Schur complements, and a theorem about angular operator representations of spectral invariant subspaces which implies e.g. the existence of solutions of the corresponding Riccati equations and a block diagonalization. All results are new in the operator as well as in the matrix case.     

Bessel sequences as projections of orthogonal systems

We prove generalizations of the Schur and Olevskii theorems on the continuation of systems of functions from an interval I to orthogonal systems on an interval J, I ? J. Only Bessel systems in L ²(I) are projections of orthogonal systems from the wider space L ²(J). This fact allows us to use a certain method for transferring the classical theorems on the almost everywhere convergence of orthogonal series (the Men’shov-Rademacher, Paley-Zygmund, and Garcia theorems) to series in Bessel systems. The projection of a complete orthogonal system from L ²(J) onto L ²)(I) is a tight frame, but not a basis.     

On Some Variants of Theorems of Schur and Baer

A well-known theorem of I. Schur states that if G is a group and G/ζ(G) is finite then G′ is finite. We obtain an analogue of this, and theorems due to R. Baer and P. Hall, for groups G that have subgroups A of Aut(G) such that A/Inn(G) is finite.     

The Lee‐Yang and Pólya‐Schur programs. II. Theory of stable polynomials and applications

In the first part of this series we characterized all linear operators on spaces of multivariate polynomials preserving the property of being nonvanishing in products of open circular domains. For such sets this completes the multivariate generalization of the classification program initiated by Pólya and Schur for univariate real polynomials. We build on these classification theorems to develop here a theory of multivariate stable polynomials. Applications and examples show that this theory provides a natural framework for dealing in a uniform way with Lee‐Yang type problems in statistical mechanics, combinatorics, and geometric function theory in one or several variables. In particular, we answer a question of Hinkkanen on multivariate apolarity. © 2009 Wiley Periodicals, Inc.     

Schur algebras of Brauer algebras I

Schur algebras of Brauer algebras are defined as endomorphism algebras of certain direct sums of ‘permutation modules’ over Brauer algebras. Explicit combinatorial bases of these new Schur algebras are given; in particular, these Schur algebras are defined integrally. The new Schur algebras are related to the Brauer algebra by Schur–Weyl dualities on the above sums of permutation modules. Moreover, they are shown to be quasi-hereditary. Over fields of characteristic different from two or three, the new Schur algebras are quasi-hereditary 1-covers of Brauer algebras, and hence the unique ‘canonical’ Schur algebras of Brauer algebras.     

Primitive Idempotents of Schur Rings

In this paper, we explore the nature of central idempotents of Schur rings over finite groups. We introduce the concept of a lattice Schur ring and explore properties of these kinds of Schur rings. In particular, the primitive, central idempotents of lattice Schur rings are completely determined. For a general Schur ring S, S contains a maximal lattice Schur ring, whose central, primitive idempotents form a system of pairwise orthogonal, central idempotents in S. We show that if S is a Schur ring with rational coefficients over a cyclic group, then these idempotents are always primitive and are spanned by the normal subgroups contained in S. Furthermore, a Wedderburn decomposition of Schur rings over cyclic groups is given. Some examples of Schur rings over non-cyclic groups will also be explored.     

N元指数和对数平均的凸性及几何凸性

讨论了n元指数平均和对数平均的凸性、S-凸性、几何凸性及S-几何凸性，证明了：（1）n元指数平均是S-凹的和S-几何凸的；（2）n元第一对数平均是S-凹的；（3）n元第二对数平均是凹的和几何凸的．最后提出了二个悬而未决的问题．     

N 元指数和对数平均的凸性及几何凸性

讨论了n 元指数平均和对数平均的凸性、S - 凸性、几何凸性及S - 几何凸性, 证明了:(1) n 元指数平均是S - 凹的和S - 几何凸的; (2) n 元第一对数平均是S - 凹的; (3) n 元第二对数平均是凹的和几何凸的. 最后提出了二个悬而未决的问题.     

Permanents and determinants with generic noncommuting entries

This paper studies some basic combinatorial properties of matrix functions of generic matrices. A generic matrix is one with entries from a free associative algebra, over a field, and on a finite set of non-commuting variables (i.e. a tensor algebra). The principal tools are shuffle products. Generic column and row permanents are defined and analogs of the Laplace and Cauchy-Binet theorems are derived in terms of shuffles. In this setting, the generic permanents include as special cases all of the classical matrix functions: Schur matrix functions, determinants, and permanents. 1980 Mathematics Classification 05, 15. Keywords: Shuffle product, generic matrix functions, minor expansions, Laplace Expansion Theorem, Cauchy-Binet Theorem, permanents, determinants, tensor algebra, matrices with non-commuting entries.     

Inverse NO-matrices

Results similar to those of Thomas L. Markham concerning inverse M-matrices are shown to hold for inverse N_Omatrices. A more general result and an alternate method of proof using Schur complements are given for one of Markham's theorems. Conditions are given to transform a nonpositive matrix to an N_O-amtrix by multiplication by Householder transformations. Also, an LU factorization of an inverse N_O-matrix is given.     

Classifying the near-equality of ribbon Schur functions

We consider the problem of determining when the difference of two ribbon Schur functions is a single Schur function. We fully classify the five infinite families of pairs of ribbon Schur functions whose difference is a single Schur function with corresponding partition having at most two parts at least 2. We also prove an identity for differences of ribbon Schur functions and we determine some necessary conditions for such a difference to be Schur-positive, depending on the distribution of 1’s and the end row lengths.