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The question of which functions acting entrywise preserve positive semidefiniteness has a long history, beginning with the Schur product theorem [Crelle 1911], which implies that absolutely monotonic functions (i.e., power series with nonnegative coefficients) preserve positivity on matrices of all dimensions. A famous result of Schoenberg and of Rudin [Duke Math. J. 1942, 1959] shows the converse: there are no other such functions. Motivated by modern applications, Guillot and Rajaratnam [Trans. Amer. Math. Soc. 2015] classified the entrywise positivity preservers in all dimensions, which act only on the off-diagonal entries. These two results are at "opposite ends", and in both cases the preservers have to be absolutely monotonic. We complete the classification of positivity preservers that act entrywise except on specified "diagonal/principal blocks", in every case other than the two above. (In fact we achieve this in a more general framework.) This yields the first examples of dimension-free entrywise positivity preservers - with certain forbidden principal blocks - that are not absolutely monotonic., Minor edits in exposition, 19 pages. The paper now uses the style file of Trans. AMS (to appear)

In this paper, we estimate the Nekrasov diagonally dominant degree on the Schur complement of Nekrasov matrices. As an application, we offer new bounds of the determinant for several special matrices, which improve the related results in certain case. Further, we give an estimation on the infinity norm bounds for the inverse of Schur complement of Nekrasov matrices. Finally, we introduce new methods called Schur-based super relaxation iteration (SSSOR) method and Schur-based conjugate gradient (SCG) method to solve the linear equation by reducing order. The numerical examples illustrate the effectiveness of the derived result.

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.

The Schur theorem provides the global bounds for spectrum of the Hadamard product of two positive semi-definite matrices. In this paper, we obtain lower and upper estimations for each eigenvalue of the Hadamard product of two Hermitian matrices.

The distribution for eigenvalues of Schur complement of matrices plays an important role in many mathematical problems. In this paper, we firstly present some criteria for H-matrix. Then as application, for two class matrices whose submatrices are γ-diagonally dominant and product γ-diagonally dominant, we show that the eigenvalues of the Schur complement are located in the Geršgorin discs and the Ostrowski discs of the original matrices under certain conditions.

The purpose of this paper is to deduce a Schur type inequality for six variables.

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.

By a result of Schur [J. Reine Angew. Math. 1911], the entrywise product $M \circ N$ of two positive semidefinite matrices $M,N$ is again positive. Vybiral [Adv. Math. 2020] improved on this by showing the uniform lower bound $M \circ \overline{M} \geq E_n / n$ for all $n \times n$ real or complex correlation matrices $M$, where $E_n$ is the all-ones matrix. This was applied to settle a conjecture of Novak [J. Complexity 1999] and to positive definite functions on groups. Vybiral (in his original preprint) asked if one can obtain similar uniform lower bounds for higher entrywise powers of $M$, or for $M \circ N$ when $N \neq M, \overline{M}$. A natural third question is to obtain a tighter lower bound that need not vanish as $n \to \infty$, i.e. over infinite-dimensional Hilbert spaces. In this note, we affirmatively answer all three questions by extending and refining Vybiral's result to lower-bound $M \circ N$, for arbitrary complex positive semidefinite matrices $M, N$. Specifically: we provide tight lower bounds, improving on Vybiral's bounds. Second, our proof is 'conceptual' (and self-contained), providing a natural interpretation of these improved bounds via tracial Cauchy-Schwarz inequalities. Third, we extend our tight lower bounds to Hilbert-Schmidt operators. As an application, we settle Open Problem 1 of Hinrichs-Krieg-Novak-Vybiral [J. Complexity, in press], which yields improvements in the error bounds in certain tensor product (integration) problems., Comment: 14 pages, no figures. Final version, to appear in Proceedings of the American Mathematical Society

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

The study of entrywise powers of matrices was originated by Loewner in the pursuit of the Bieberbach conjecture. Since the work of FitzGerald and Horn (1977), it is known that $A^{\circ \alpha} := (a_{ij}^\alpha)$ is positive semidefinite for every entrywise nonnegative $n \times n$ positive semidefinite matrix $A = (a_{ij})$ if and only if $\alpha$ is a positive integer or $\alpha \geq n-2$. This surprising result naturally extends the Schur product theorem, and demonstrates the existence of a sharp phase transition in preserving positivity. In this paper, we study when entrywise powers preserve positivity for matrices with structure of zeros encoded by graphs. To each graph is associated an invariant called its "critical exponent", beyond which every power preserves positivity. In our main result, we determine the critical exponents of all chordal/decomposable graphs, and relate them to the geometry of the underlying graphs. We then examine the critical exponent of important families of non-chordal graphs such as cycles and bipartite graphs. Surprisingly, large families of dense graphs have small critical exponents that do not depend on the number of vertices of the graphs., Comment: 21 pages; added Proposition 4.15 on correlation matrices. Final version, to appear in Journal of Combinatorial Theory, Series A

In this paper we determine the structure of the total component of the Schur multiplier over an algebraically closed field of some relevant families of groups, such as dihedral groups, dicyclic groups, the infinite cyclic group and the direct product of two finite cyclic groups.

In the recent years, the Schur convexity and Schur geometrically convexity of Sto- larsky's mean values have grabed the focus of many mathematicians and researchers. In this article, the Schur convexity of Stolarsky's extended type mean values are discussed.

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.

The paper is established on Schur complements and block Kronecker product of positive semidefinite matrices. In particular, a formulation for the block Kronecker product of Schur complements of block matrices is improved. Additionally, an new following inequality for block Kronecker product of Schur complements of two matrices and their conjugate transpose is proved, Bu çalışma, yarı pozitif tanımlı blok matrislerin blok kronecker çarpımları ve Schur tamamlayıcıları üzerine kurulmuştur. Özellikle blok matrislerin Schur tamamlayıcılarının blok kronecker çarpımı için bir formülasyon geliştirilmiştir. Bundan başka, iki matris ve eşlenik transpozlarının schur tamamlayıcılarının blok kronecker çarpımları için aşağıdaki yeni eşitsizlik kanıtlanmıştır

We consider the representation dimension, for fixed $n\geq2$, of ordinary and quantised Schur algebras $S(n,r)$ over a field $k$. For $k$ of positive characteristic $p$ we give a lower bound valid for all $p$. We also give an upper bound in the quantum case, when $k$ has characteristic $0$.

[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.

A finite group $G$ is called a Schur group, if any Schur ring over $G$ is the transitivity module of a permutation group on the set $G$ containing the regular subgroup of all right translations. It was proved by R. P\"oschel (1974) that given a prime $p\ge 5$ a $p$-group is Schur if and only if it is cyclic. We prove that a cyclic group of order $n$ is a Schur group if and only if $n$ belongs to one of the following five (partially overlapped) families of integers: $p^k$, $pq^k$, $2pq^k$, $pqr$, $2pqr$ where $p,q,r$ are distinct primes, and $k\ge 0$ is an integer., Comment: the second version; the proof was substantially improved; 29 pages

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

Properties of Schur type for Banach lattices of regular operators and tensor products are analyzed. It is shown that the dual positive Schur property behaves well with respect to Fremlin’s projective tensor product, which allows us to construct new examples of spaces with this property. Similar results concerning the positive Grothendieck property are also presented.

In this note we extend the results of Bekkert and Futorny in [2] and Hemmer, Kujawa and Nakano in [10] and determine the derived representation type of Schur superalgebras.

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.

In this paper, we prove the almost Schur theorem, introduced by De Lellis and Topping, for the Riemannian manifold M of nonnegative Ricci curvature with totally geodesic boundary. Examples are given to show that it is optimal when the dimension of M is at least 5. We also prove that the almost Schur theorem is true when M is a 4-dimensional manifold of nonnegative scalar curvature with totally geodesic boundary. Finally we obtain a generalization of the almost Schur theorem in all dimensions only by assuming the Ricci curvature is bounded below.

In this paper, we study the Schur geometric convexity (concavity) of the ratio of difference of means. Also established some inter related mean inequalities related to the ratio of difference of means.

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.

A diameter graph in ℝd is a graph in which vertices are points of a finite subset of ℝd and two vertices are joined by an edge if the distance between them is equal to the diameter of the vertex set. This paper is devoted to Schur’s conjecture, which asserts that any diameter graph on n vertices in ℝd contains at most n complete subgraphs of size d. It is known that Schur’s conjecture is true in dimensions d ≤ 3. We prove this conjecture for d = 4 and give a simple proof for d = 3.

In this technical note, we demonstrate how the classical Gauss-Lucas theorem can aid in the determination of stabilizing controllers for feedback systems. Moreover, we show how to apply the Gauss-Lucas theorem in the construction of new necessary conditions for Schur and Hurwitz stability. The results are used to derive new bounds on the stability margins and performance of control systems.

The purpose of this paper is to deduce a Schur type inequality for five variables.

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.

An approach based on the Rouche theorem was introduced in the literature to compute the optimum radius for robust stability of Schur polynomials. Later an attempt was made to improve the result, but it was shown to be incorrect. The purpose of this note is to show that an improved optimum radius still can be obtained by modifying the proposed method. The result of this note can be easily extended to the multidimensional cases.

A well known theorem of Schur states that for any group G, if G/Z(G) is finite, then G′ is finite. We give a very short and elementary proof of a further generalization of the converse of Schur’s theorem proved by Niroomand [5] and Sury [7] and also improve the bound for the order of G/Z(G) obtained by Niroomand and Sury.

The recently developed theory of Schur rings over a finite cyclic group is generalized to Schur rings over a ring \(R\) being a product of Galois rings of coprime characteristics. It is proved that if the characteristic of \(R\) is odd, then as in the cyclic group case any pure Schur ring over \(R\) is the tensor product of a pure cyclotomic ring and Schur rings of rank \(2\) over non-fields. Moreover, it is shown that in contrast to the cyclic group case there are non-pure Schur rings over \(R\) that are not generalized wreath products.

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.