Bounds for Systems of Maximal Differential Polynomials without Mixed Derivatives

Mathematical Notes -     

Matrix extensions and eigenvalue completions, the generic case

In this paper we provide new necessary and sufficient conditions for the so-called eigenvalue completion problem.

     

Non-cooperative implementation: A survey of recent results

Strategic decentralization of collective decision-making is considered when the agents' behaviour is non-cooperative. Specifically they use dominant, maximin risk-avert or sophisticated strategies. The existing results applying these equilibrium concepts are reviewed in three familiar economically meaningful contexts: ordinal (as in voting), random (as in bargaining) and quasilinear (as in public goods problems).     

Fast algorithms for placing large entries along the diagonal of a sparse matrix

Solving a sparse system of linear equations Ax=b is one of the most fundamental operations inside any circuit simulator. The equations/rows in the matrix A are often rearranged/permuted before factorization and applying direct or iterative methods to obtain the solution. Permuting the rows of the matrix A so that the entries with large absolute values lie on the diagonal has several advantages like better numerical stability for direct methods (e.g., Gaussian elimination) and faster convergence for indirect methods (such as the Jacobi method). Duff (2009) [3] has formulated this as a weighted bipartite matching problem (the MC64 algorithm). In this paper we improve the performance of the MC64 algorithm with a new labeling technique which improves the asymptotic complexity of updating dual variables from O(|V|+|E|) to O(|V|), where |V| is the order of the matrix A and |E| is the number of non-zeros. Experimental results from using the new algorithm, when benchmarked with both industry benchmarks and UFL sparse matrix collection, are very promising. Our algorithm is more than 60 times faster (than Duff’s algorithm) for sparse matrices with at least a million non-zeros.     

The DJL Conjecture for CP Matrices over Special Inclines

Drew, Johnson, and Loewy conjectured that for n ≥ 4, the CP-rank of every n × n completely positive real matrix is at most [n²/4]. While this conjecture has recently been disproved for completely positive real matrices, we show that this conjecture is true for n × n completely positive matrices over certain special types of inclines. In addition, we prove an incline version of Markham's theorems which gives sufficient conditions for completely positive matrices over special inclines to have triangular factorizations.     

病态线性方程组新的Jacobi迭代解法

给出了解病态线性方程组的一种新的Jacobi迭代算法,并证明了算法的收敛性;通过具体算例说明了算法的实用性和有效性.     

Updating Insights into the Catalytic Domain Properties of Plant Cellulose synthase (CesA) and Cellulose synthase-like (Csl) Proteins

The wall is the last frontier of a plant cell involved in modulating growth, development and defense against biotic stresses. Cellulose and additional polysaccharides of plant cell walls are the most abundant biopolymers on earth, having increased in economic value and thereby attracted significant interest in biotechnology. Cellulose biosynthesis constitutes a highly complicated process relying on the formation of cellulose synthase complexes. Cellulose synthase (CesA) and Cellulose synthase-like (Csl) genes encode enzymes that synthesize cellulose and most hemicellulosic polysaccharides. Arabidopsis and rice are invaluable genetic models and reliable representatives of land plants to comprehend cell wall synthesis. During the past two decades, enormous research progress has been made to understand the mechanisms of cellulose synthesis and construction of the plant cell wall. A plethora of cesa and csl mutants have been characterized, providing functional insights into individual protein isoforms. Recent structural studies have uncovered the mode of CesA assembly and the dynamics of cellulose production. Genetics and structural biology have generated new knowledge and have accelerated the pace of discovery in this field, ultimately opening perspectives towards cellulose synthesis manipulation. This review provides an overview of the major breakthroughs gathering previous and recent genetic and structural advancements, focusing on the function of CesA and Csl catalytic domain in plants.     

一类非奇异H-矩阵判定的新条件

非奇异H-矩阵是在许多领域具有广泛应用的重要矩阵类,但实际判定一个非奇异H-矩阵是十分困难的.在本文中,我们给出了一类关于非奇异H-矩阵新的判定条件,改进了近期的相关结果,并用数值例子说明了文中结果判定范围的更广泛性.     

On diagonal-Schur complements of block diagonally dominant matrices

It is known that the diagonal-Schur complements of strictly diagonally dominant matrices are strictly diagonally dominant matrices [J.Z. Liu, Y.Q. Huang, Some properties on Schur complements of H-matrices and diagonally dominant matrices, Linear Algebra Appl. 389 (2004) 365-380], and the same is true for nonsingular H-matrices [J.Z. Liu, J.C. Li, Z.T. Huang, X. Kong, Some properties of Schur complements and diagonal-Schur complements of diagonally dominant matrices, Linear Algebra Appl. 428 (2008) 1009-1030]. In this paper, we research the properties on diagonal-Schur complements of block diagonally dominant matrices and prove that the diagonal-Schur complements of block strictly diagonally dominant matrices are block strictly diagonally dominant matrices, and the same holds for generalized block strictly diagonally dominant matrices.     

On modified block SSOR iteration methods for linear systems from steady incompressible viscous flow problems

In order to solve the large sparse systems of linear equations arising from numerical solutions of two-dimensional steady incompressible viscous flow problems in primitive variable formulation, we present block SSOR and modified block SSOR iteration methods based on the special structures of the coefficient matrices. In each step of the block SSOR iteration, we employ the block LU factorization to solve the sub-systems of linear equations. We show that the block LU factorization is existent and stable when the coefficient matrices are block diagonally dominant of type-II by columns. Under suitable conditions, we establish convergence theorems for both block SSOR and modified block SSOR iteration methods. In addition, the block SSOR iteration and AF-ADI method are considered as preconditioners for the nonsymmetric systems of linear equations. Numerical experiments show that both block SSOR and modified block SSOR iterations are feasible iterative solvers and they are also effective for preconditioning Krylov subspace methods such as GMRES and BiCGSTAB when used to solve this class of systems of linear equations.