For x x1, x2, . . . , xn ∈ R , the symmetric function φn x, r is defined by φn x, r φn x1, x2, . . . , xn; r ∏ 1≤i1<i2 ···<ir≤n ∑r j 1 xij / 1 xij 1/r , where r 1, 2, . . . , n and i1, i2, . . . , in are positive integers. In this article, the Schur convexity, Schur multiplicative convexity and Schur harmonic convexity of φn x, r are discussed. As applications, some inequalities are established...

Multi-Schur functions are symmetric that generalize the supersymmetric Schur functions, flagged and refined dual Grothendieck which have been intensively studied by Lascoux. In this paper, we give a new free-fermionic presentation of them. The multi-Schur indexed partition two ``tuples tuples'' indeterminates. We construct family linear bases fermionic Fock space such data prove they correspond...

The formula (1) was first used by Schur [22], but the idea of the Schur complement goes back to Sylvester (1851), and the term Schur complement was introduced by E. Haynsworth [16]. In the beginning Schur complements were used in the theory of matrices. M.G. Krein [19] and W.N. Anderson and G.E. Trapp [4] extended the notion of Schur complements of matrices to shorted operators in Hilbert space...

This paper uses the theory of dual equivalence graphs to give explicit Schur expansions to several families of symmetric functions. We begin by giving a combinatorial definition of the modified Macdonald polynomials and modified Hall-Littlewood polynomials indexed by any diagram δ ⊂ Z × Z, written as H̃δ(X; q, t) and P̃δ(X; t), respectively. We then give an explicit Schur expansion of P̃δ(X; t) as...

The Schur--Parlett algorithm, implemented in MATLAB as \textttfunm, evaluates an analytic function $f$ at $n\times n$ matrix argument by using the Schur decomposition and a block recurrence of P...

In this paper we study Schur-Weyl duality between the symplectic group and Brauer’s centralizer algebra over an arbitrary infinite field K. We show that the natural homomorphism from the Brauer’s centralizer algebra Bn(−2m) to the endomorphism algebra of tensor space (K) as a module over the symplectic similitude group GSp2m(K) (or equivalently, as a module over the symplectic group Sp2m(K)) is...

Through the matrix rank method, this paper gives necessary and sufficient conditions for a partitioned matrix to have generalized inverses with Banachiewicz-Schur forms. In addition, this paper investigates the idempotency of generalized Schur complements in a partitioned idempotent matrix.

and Applied Analysis 3 where L x, y , I x, y denote logarithmic mean and identric exponential mean, respectively, G a, b √ ab. The Schur convexity of Sp,q a, b and Gp,q a, b on 0,∞ × 0,∞ with respect to a, b was investigated by Qi et al. 4 , Shi et al. 5 , Li and Shi 6 , and Chu and Zhang 7 . Until now, they have been perfectly solved by Chu and Zhang 7 , Wang and Zhang 8 , respectively. Recent...

in this paper, we consider the boundedness of the libera operator on dirichlet spaces in terms of the schur test. moreover, we get its point spectrum and norm.