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In this paper, we extend Guo and Xia’s necessary condition which has been presented by Guo and Xia (Fuzzy optimizat Decis Mak 5: 33–47, 2006) in order to study the finitely many constraints of fuzzy relation inequalities and optimize a linear objective function on this region which is defined by the fuzzy max–min operator. The new condition provides a means for removing the unnecessary paths resulting from Guo and Xia’s paths. Also, an algorithm and two numerical examples are offered to abbreviate and illustrate the steps of the resolution process of the problem. [ABSTRACT FROM AUTHOR]

We consider quasihomogeneous strings with piecewise constant density and derive the quasihomogeneity condition. For a small number of string components, we present explicit formulas for reconstructing a string from its part. For an arbitrary number of string components, we construct a theory of matrices of special form (that we call distinguished) and present an algorithm for string reconstruction from its part. [ABSTRACT FROM AUTHOR]

Conventional schedulers schedule operations in dependence order and never revisit or undo a scheduling decision on any operation. In contrast, backtracking schedulers may unschedule operations and can often generate better schedules. This paper develops and evaluates the backtracking approach to fill branch delay slots. We first present the structure of a generic backtracking scheduling algorithm and prove that it terminates. We then describe two more aggressive backtracking schedulers and evaluate their effectiveness. We conclude that aggressive backtracking-based instruction schedulers can effectively improve schedule quality by eliminating branch delay slots with a small amount of additional computation. [ABSTRACT FROM AUTHOR]

In this paper, a new parallel algorithm for solving parabolic equations is proposed. The new algorithm includes two domain decomposition methods, each method is applied to compute the values at (n+1)st time level by use of known numerical solutions at nth time level, respectively. Then the average of two above values is chosen to be the numerical solutions at (n+1)st time level. The new algorithm obtains satisfactory accuracy while maintaining parallelism and unconditional stability. This algorithm can be extended to solve two-dimensional parabolic equations by alternating direction implicit (ADI) technique. Both error analysis and numerical experiments illustrate the accuracy and efficiency of the new algorithm. [ABSTRACT FROM AUTHOR]

A comprehensive linear stability analysis of splitting methods is carried out by means of a 2×2 matrix K( x) with polynomial entries (the stability matrix) and the stability polynomial p( x) (the trace of K( x) divided by two). An algorithm is provided for determining the coefficients of all possible time-reversible splitting schemes for a prescribed stability polynomial. It is shown that p( x) carries essentially all the information needed to construct processed splitting methods for numerically approximating the evolution of linear systems. By conveniently selecting the stability polynomial, new integrators with processing for linear equations are built which are orders of magnitude more efficient than other algorithms previously available. [ABSTRACT FROM AUTHOR]

In this article, we take a rapid journey through the history of algebra, noting the important developments and reflecting on the importance of this history in the teaching of algebra in secondary school or university. Frequently, algebra is considered to have three stages in its historical development: the rhetorical stage, the syncopated stage, and the symbolic stage. But besides these three stages of expressing algebraic ideas, there are four more conceptual stages which have happened along side of these changes in expressions. These stages are the geometric stage, where most of the concepts of algebra are geometric ones; the static equation-solving stage, where the goal is to find numbers satisfying certain relationships; the dynamic function stage, where motion seems to be an underlying idea, and finally, the abstract stage, where mathematical structure plays the central role. The stages of algebra are, of course not entirely disjoint from one another; there is always some overlap. We discuss here high points of the development of these stages and reflect on the use of these historical stages in the teaching of algebra. [ABSTRACT FROM AUTHOR]

We construct an algorithm that tests a system of recurrence equations on a tree for the existence of a nontrivial solution and computes it. [ABSTRACT FROM AUTHOR]

The structural theory of short-term damage is generalized to the case where the undamaged components of a granular composite deform nonlinearly. The basis for this generalization is the stochastic elasticity equations for a granular composite with porous components whose skeletons deform nonlinearly. Microvolumes of the composite components meet the Huber–Mises failure criterion. Damaged microvolume balance equations are derived for the physically nonlinear materials of the components. Together with the equations relating macrostresses and macrostrains of a granular composite with porous nonlinear components, they constitute a closed-form system. The system describes the coupled processes of physically nonlinear deformation and microdamage. Algorithms for calculating the microdamage–macrostrain relationship and plotting deformation diagrams are proposed. Uniaxial tension curves are plotted for the case where microdamages occur in the linearly hardened matrix and do not in the inclusions, which are linearly elastic [ABSTRACT FROM AUTHOR]

The problem of solving an infinite system of linear equations finitely expressed is addressed. Modifications of the Gauss–Seidel method are presented, especially suitable for the implementation on SMP machines with a small number of processors. One of the proposed parallel algorithms, which concentrates the computational efforts where they are most needed, results to be more efficient than the sequential algorithm, even from the point of view of the total number of operations. [ABSTRACT FROM AUTHOR]