Shortest Path Problem on a Network with Imprecise Edge Weight

A network with its arc lengths as imprecise number, instead of a real number, namely, interval number and triangular fuzzy number is considered here. Existing ideas on addition and comparison between two imprecise numbers of same type are introduced. To obtain a fuzzy shortest path from a source vertex to all other vertices, a common algorithm is developed which works well on both types of imprecise numbers under consideration. In the proposed algorithm, a decision-maker is to negotiate with the obtained fuzzy shortest paths according to his/her view only when the means are same but the widths are different of the obtained paths. Otherwise, a fuzzy optimal path is obtained to which the decision-maker always satisfies with different grades of satisfaction. All pairs fuzzy shortest paths can be found by repeated use of the proposed algorithm.     

Curriculum developers and problem solving: the case of Israeli elementary school projects

Problem solving has been a main focus in mathematics education for several decades, yet it seems that its definition and classroom implementation are far from being consensual. We explore the views and approaches of a small community: the project leaders of five elementary mathematics curriculum development projects in Israel, working within a centralized system, which dictates the syllabus. We describe and analyze their views along six categories: What are problems? What are not problems? Classification of problems, problem solving and individual differences, the ratio of problem solving tasks to other tasks in the project, and the role of heuristics and metacognition in teaching problem solving. We describe, exemplify, interpret and discuss the (few) points of convergence and the many different approaches. Finally, we reflect on the possible role of research in settling those differences. We speculate that our analysis and results go beyond the local and the idiosyncratic.     

On the Second Iterate for Critically Diffusive Active Scalar Equations

We consider an iterative resolution scheme for a class of active scalar equations with a fractional power γ of the Laplacian and focus our attention on the second iterate. In the case of critical diffusivity, we extract information relevant to Well-posedness questions in scale-invariant spaces. Our results are Two-fold: we prove continuity of the bilinear operator in ${\dot{B}^{0}_{\infty,1}}$ ; for equations with an even symbol we show that the ${B^{-1/2}_{\infty,q}}$ -regularity, where q > 2, is in a sense a minimal necessary requirement on the solution.     

Analysis of the nonlinear vibration of a two-mass–spring system with linear and nonlinear stiffness

An analytical approach is developed for areas of nonlinear science such as the nonlinear free vibration of a conservative, two-degree-of-freedom mass–spring system having linear and nonlinear stiffnesses. The main contribution of this research is twofold. First, it introduces the transformation of two nonlinear differential equations for a two-mass system using suitable intermediate variables into a single nonlinear differential equation and, more significantly, the treatment of a nonlinear differential system by linearization coupled with Newton’s method. Secondly, the major section is the solving of the governing nonlinear differential equation where the displacement of the two-mass system can be obtained directly from the linear second-order differential equation using a first-order variational approach. The aforementioned approach proposed by J.H. He, who actually developed the method, is exactly He’s variational method. This approach is an explicit method with high validity for resolving strong nonlinear oscillation system problems. Two examples of nonlinear two-degree-of-freedom mass–spring systems are analyzed, and verified with published results and exact solutions. The method can be easily extended to other nonlinear oscillations and so could be widely applicable in engineering and science.     

Augmented Lagrangians with Adaptive Precision Control for Quadratic Programming with Equality Constraints

In this paper we introduce an augmented Lagrangian type algorithm for strictly convex quadratic programming problems with equality constraints. The new feature of the proposed algorithm is the adaptive precision control of the solution of auxiliary problems in the inner loop of the basic algorithm. Global convergence and boundedness of the penalty parameter are proved and an error estimate is given that does not have any term that accounts for the inexact solution of the auxiliary problems. Numerical experiments illustrate efficiency of the algorithm presented