A TWO-STEP EXPLICIT METHOD FOR A CLASS OF LINEAR PERIODIC INITIAL VALUE PROBLEMS

This paper presents a two-step explicit method of order four for solving a class of linear periodic initial value problems. At each computational step, only two right function evaluations and one derivative evaluation are employed. Basing on a special vector operation, the method can be extended to the vector-applicable in multidimensional space.     

Relaxed Steepest Descent and Cauchy-Barzilai-Borwein Method

The negative gradient direction to find local minimizers has been associated with the classical steepest descent method which behaves poorly except for very well conditioned problems. We stress out that the poor behavior of the steepest descent methods is due to the optimal Cauchy choice of steplength and not to the choice of the search direction. We discuss over and under relaxation of the optimal steplength. In fact, we study and extend recent nonmonotone choices of steplength that significantly enhance the behavior of the method. For a new particular case (Cauchy-Barzilai-Borwein method), we present a convergence analysis and encouraging numerical results to illustrate the advantages of using nonmonotone overrelaxations of the gradient method.     

On the Structure of the Augmentation Quotient Group for Some Non-abelian p-groups

In this paper,we study the basis of augmentation ideals and the quotient groups of finite non-abelian p-group which has a cyclic subgroup of index p,where p is an odd prime,and k is greater than or equal to 3.A concrete basis for the augmentation ideal is obtained and then the structure of its quotient groups can be determined.     

Mutation pairs and quotient categories of Abelian categories

The notion of 𝒟-mutation pairs of subcategories in an abelian category is defined in this article. When (𝒵,𝒵) is a 𝒟-mutation pair in an abelian category 𝒜, the quotient category 𝒵∕𝒟 carries naturally a triangulated structure. Moreover, our result generalize the construction of the quotient triangulated category by Happel [10 Happel, D. (1988). Triangulated Categories in the Representation of Finite Dimensional Algebras. London Mathematical Society, LMN, Vol. 119. Cambridge: Cambridge University Press.[Crossref] , [Google Scholar], Theorem 2.6]. Finally, we find a one-to-one correspondence between cotorsion pairs in 𝒜 and cotorsion pairs in the quotient category 𝒵∕𝒟, and study homological finiteness of subcategories in a mutation pair.     

QZ algorithm with two-sided generalized Rayleigh quotient shifts

We generalize the recently proposed two-sided Rayleigh quotient single-shift and the two-sided Grassmann–Rayleigh quotient double-shift used in the QR algorithm and apply the generalized versions to the QZ algorithm. With such shift strategies the QZ algorithm normally has a cubic local convergence rate. Our main focus is on the modified shift strategies and their corresponding truncated versions. Numerical examples are provided to demonstrate the convergence properties and the efficiency of the QZ algorithm equipped with the proposed shifts. For the truncated versions, local convergence analysis is not provided. Numerical examples show they outperform the modified shifts and the standard Rayleigh quotient single-shift and Francis double-shift.     

Weakly left localizable rings

A new class of rings, the class of weakly left localizable rings, is introduced. A ring R is called weakly left localizable if each non-nilpotent element of R is invertible in some left localization S^?1R of the ring R. Explicit criteria are given for a ring to be a weakly left localizable ring provided the ring has only finitely many maximal left denominator sets (eg, this is the case for all left Noetherian rings). It is proved that a ring with finitely many maximal left denominator sets that satisfies some natural conditions is a weakly left localizable ring iff its left quotient ring is a direct product of finitely many local rings such that their radicals are nil ideals.     

Convergence of Nonstationary Iterative Methods for Solving Singular Linear Equations with Index One

In this paper, we investigate the nonstationary iterative scheme for solving consistent singular linear system with index one. We utilize the group inverse to present a sufficient condition for the convergence of the nonstationary iterative method. Our result extends the known results of the stationary iterative scheme. Finally, we present a sufficient condition for the multisplitting algorithm and provide numerical examples to illustrate the advantages of nonstationary method.     

离散变量框架结构的可靠性优化

以框架结构系统的可靠性分析为基础,给出离散变量框架结构可靠性优化的相对差商方法及迭代格式,并用算例证明了该法的有效性及实用性。