On inclusion theorems for absolute matrix summability methods, corrections: [J. Math. Anal. Appl. 238 (1999) 82-90]

We modify some of the conditions of the theorem to make it more applicable.     

Inner bounds for the spectrum of quasinormal operators

A linear operator in a separable Hilbert space is called a quasinormal one if it is a sum of a normal operator and a compact one. In the paper, bounds for the spectrum of quasinormal operators are established. In addition, the lower estimate for the spectral radius is derived. Under some restrictions, that estimate improves the well-known results. Applications to integral operators and matrices are discussed. Our results are new even in the finite-dimensional case.

     

Variations on the Theorem of Birkhoff – von Neumann and Extensions

 The theorem of Birkhoff – von Neumann concerns bistochastic matrices (i.e., matrices with nonnegative real entries such that all row sums and all column sums are equal to one). We consider here real matrices with entries unrestricted in sign and we extend the notion of permutation matrices (integral bistochastic matrices); some generalizations of the theorem are derived by using elementary properties of graph theory. Received: October 10, 2000 Final version received: April 11, 2002     

Self-Converse Directed BIBDs with Block Size Four

A directed balanced incomplete block design (or D B(k,;v)) (X,) is called self-converse if there is an isomorphic mapping f from (X,) to (X,^–1), where ^–1={B ^–1:B} and B ^–1=(x _k ,x _{k –1},,x ₂,x ₁) for B=(x ₁,x ₂,,x _{k –1},x _k ). In this paper, we give the existence spectrum for self-converse D B(4,1;v). AMS Classification:05BResearch supported in part by NSFC Grant 10071002 and SRFDP under No. 20010004001     

A Comparative Study on Dynamic and Static Sparsity Patterns in Parallel Sparse Approximate Inverse Preconditioning

Sparse approximate inverse (SAI) techniques have recently emerged as a new class of parallel preconditioning techniques for solving large sparse linear systems on high performance computers. The choice of the sparsity pattern of the SAI matrix is probably the most important step in constructing an SAI preconditioner. Both dynamic and static sparsity pattern selection approaches have been proposed by researchers. Through a few numerical experiments, we conduct a comparable study on the properties and performance of the SAI preconditioners using the different sparsity patterns for solving some sparse linear systems. This revised version was published online in July 2006 with corrections to the Cover Date.     

The Structure of the (Outer) Automorphism Group of a Bieberbach Group

Referring to Tits alternative, we develop a necessary and sufficient condition to decide whether the normalizer of a finite group of integral matrices is polycyclic-by-finite or is containing a non-Abelian free group. This result is of fundamental importance to conclude whether the (outer) automorphism group of a Bieberbach group is polycyclic-by-finite or has a non-cyclic free subgroup.     

Prefiltration technique via aggregation for constructing low‐density high‐quality factorized sparse approximate inverse preconditionings

This paper considers a new approach to a priori sparsification of the sparsity pattern of the factorized approximate inverses (FSAI) preconditioner using the so‐called vector aggregation technique. The suggested approach consists in construction of the FSAI preconditioner to the aggregated matrix with a prescribed sparsity pattern. Then small entries of the computed ‘aggregated’ FSAI preconditioning matrix are dropped, and the resulting pointwise sparsity pattern is used to construct the low‐density block sparsity pattern of the FSAI preconditioning matrix to the original matrix. This approach allows to minimize (sometimes significantly) the construction costs of low‐density high‐quality FSAI preconditioners. Numerical results with sample matrices from structural mechanics and thin shell problems are presented. Copyright © 2002 John Wiley & Sons, Ltd.     

A fast modified sine transform for solving block-tridiagonal systems with Toeplitz blocks

In this report we consider block-tridiagonal systems with Toeplitz blocks. Each block is of sizen×n consisting ofn _c×n _c matrices as entries, and there arem×m blocks in the system. The solution of those systems consists of 2n _c m modified sine transforms and an intermediate solution ofn block-tridiagonal systems. Symmetries in the data vectors are exploited such that one modified sine transform can be computed in terms of one Fourier transform of half the length of the original one, hence requiringO(2.5nlog₂ n) operations. Similarly, we only have to solve (n+1)/2 of the intermediate systems due to symmetry.This work was supported by the Swedish National Board for Industrial and Technical Development, NUTEK, under contract No. 89-02539 P.     

Cyclic reduction and FACR methods for piecewise hermite bicubic orthogonal spline collocation

Cyclic reduction and Fourier analysis-cyclic reduction (FACR) methods are presented for the solution of the linear systems which arise when orthogonal spline collocation with piecewise Hermite bicubics is applied to boundary value problems for certain separable partial differential equations on a rectangle. On anN×N uniform partition, the cyclic reduction and Fourier analysis-cyclic reduction methods requireO(N ²log₂ N) andO(N ²log₂log₂ N) arithmetic operations, respectively.     

Some sufficient conditions for the E- and MV-optimality of block designs having blocks of unequal size

Summary In this paper we consider experimental situations in which ν treatments are to be tested inb blocks whereb _i blocks containk _i experimental units,i=1,...,p, k ₁<k ₂<...<k _p . The idea of a group divisible (GD) design is extended to that of a group divisible design with unequal block sizes (GDUB design) and then a number of results concerning the E- and MV-optimality of GD designs are generalized to the case of GDUB designs.