The focus of this paper is a penalty-based strategy for preconditioning elliptic saddle point systems. As the starting point, we consider the regularization approach of Axelsson in which a related linear system, differing only in the (2,2) block of the coefficient matrix, is introduced. By choosing this block to be negative definite, the dual unknowns of the related system can be eliminated res...

Bipartite graphs are used to describe the generalized Schur complements of real matrices having no square submatrix with two or more nonzero diagonals. For any matrix A with this property, including any nearly reducible matrix, the sign pattern of each generalized Schur complement is shown to be determined uniquely by the sign pattern of A. Moreover, if A has a normalized LU factorization A = L...

This paper describes a multilevel preconditioning technique for solving sparse symmetric linear systems of equations. This “Multilevel Schur Low-Rank” (MSLR) preconditioner first builds a tree structure T based on a hierarchical decomposition of the matrix and then computes an approximate inverse of the original matrix level by level. Unlike classical direct solvers, the construction of the MSL...

A number of properties of Young raising operators as applied to S-functions and Schur Q-functions are noted. The order of evaluating the action of inverse raising operators is found to require careful specification and the maximum power of the operators bij is determined. The operation of the inverse raising operator on a partition .A is found to be the same as for its conjugate t A new definit...

The vertex operators Hm act on the Hall-Littlewood polynomials HX; t] with the property that HmHX; t] = H (m;;) X; t]. We present a combinatorial rule for computing the action of Hm on the schur function basis which interpolates the Pieri formula for multiplication by hmX] and the schur function vertex operator. This rule suggests the existence of an operator on column strict tableaux of conten...

We demonstrate that the symmetric elliptic polynomials $E_\lambda(x)$ originally discovered in study of generalized Noumi-Shiraishi functions are eigenfunctions Ruijsenaars-Schneider (eRS) Hamiltonians act on mother function variable $y_i$ (substitute Young-diagram $\lambda$). This means they dual eRS system. At same time, their orthogonal complements Schur scalar product, $P_R(x)$ reduction Ko...

Let X be a Hilbert bimodule over a C∗-algebra A and OX = A⋊X Z. Using a finite section method we construct a sequence of completely positive contractions factoring through matrix algebras over A which act on sξs ∗ η as Schur multipliers converging to the identity. This shows immediately that for a finitely generated X the algebra OX inherits any standard approximation property such as nuclearit...

In [LLT] Lascoux, Leclerc and Thibon introduced symmetric functions Gλ which are spin and weight generating functions for ribbon tableaux. This article is aimed at studying these functions in analogy with Schur functions. In particular we will describe: • a Pieri and dual-Pieri formula for ribbon functions, • a ribbon Murnaghan-Nakayama formula, • ribbon Cauchy and dual Cauchy identities, • and...