A mathematical model of a 200 MW boiler†

A mathematical model is developed for a 200 MW coal-fired, drum-type natural circulation boiler with reheat. The complete unit is divided, for the convenience of the analysis, into several sections, viz. the economizer, the drum, the downcomer, etc. The dynamic behaviour of each section is described in mathematical terms. The equations are linearized and manipulated into state space form for solution in a digital computer. Results for the more important system variables are presented end discussed. Difficulties found in the modelling are emphasized.     

A mathematical modeling framework for software reliability testing†

Software reliability testing refers to various software testing activities that are driven to achieve a quantitative reliability goal given a priori or lead to a quantitative reliability assessment for the software under test. In this paper we develop a modeling framework for the software reliability testing process, comprising a simplifying model and a generalized model. In both models the software testing action selection process and the defect removal mechanism are explicitly described. Both the discrete-time domain and the continuous-time domain are involved. The generalized model is more accurate or realistic than the simplifying model since the former avoids the assumption that defects are equally detectable and the assumption that defects are removed upon being detected. However simulation examples show that the simplifying model really captures some of essential features of the software testing process after a short initial testing stage. The modeling framework is practically realistic, mathematically rigorous, and quantitatively precise. It demonstrates that the relationship between software testing and delivered software reliability, which was poor understood, can well be formulated and quantified. Rigorous examinations show that several common assumptions adopted in software reliability modeling, including the independence assumption, the exponentiality assumption, and the NHPP assumption, are theoretically false in general. This paper sets a good starting point to further formalize and quantify the software testing process and its relation to delivered software reliability.     

The teacher's responsibility in whole‐class debriefing of mathematical activity

Abstract

Prompted by recent moves in the United Kingdom to guide teachers’ practices in whole‐class, direct interactive teaching, in this article, we offer an opportunity for North American mathematics educators to reflect on possibilities for whole‐class teaching of mathematics. We focus particularly on the plenary aspect of lessons—what might be considered the debriefing of mathematical activity—and specifically on the teacher's responsibility during those sessions, both to his or her students and to the authenticity of the discipline of mathematics. Drawing on data from a Grade‐3 classroom and invoking complexity science as a theoretical lens to explore the classroom as a complex learning system, we present implications for teaching in whole‐class debriefings of mathematical activity.     

A new formalism for mathematical description and verification of component-based systems

In this paper, we introduce a formal approach for composing software components into a distributed system. We describe the system as a hierarchical composition of some components, which can be distributed on a wide variety of hardware platforms and executed in parallel. We represent each component by a mathematical model and specify the abstract communication protocols of the components using Interface Automata (IAs). To model hierarchical systems, besides the basic components’ model, we will present other components, called nodes. A node consists of a set of subnodes interacting under the supervision of a controller. Each subnode, in turn, is a node or discrete event component. By considering a subnode as a node we can make hierarchical nodes/components. The entire system, therefore, forms the root of the hierarchy. A controller, in turn, is a set of subcontrollers/interface automata that specifies interaction protocol of the components inside a node. We have also presented an example demonstrating the model by illustrating nodes, subnodes, controllers, and subcontrollers. To address the state space explosion problem in system verification, we utilize the controller as a contract for independent analysis of the components and their interactions. Therefore, a node will not be analyzed directly; instead, we will analyze the controller.     

Application of mathematical programming in the design of optimal control systems†

The problem of applying the various computational methods of mathematical programming in the design of an optimal control system is discussed. A general case of non-linear, non-autonomous, state equations, subject to inequality constraints on both state and control variables, is considered. Both continuous and discrete time systems are investigated. In case of discrete time systems, the sampling intervals are assumed generally unequal and aperiodic, with inequality constraints imposed upon them.

Systems like these impose considerable computational difficulties when treated by the maximum principle or dynamie programming. Using mathematical programming, one may simplify a wide class of those computational problems.

Several examples of applying mathematical programming to particular control problems are presented.     

Discretization behavior analysis of a switcbing control system from a unified mathematical approach

A useful unified analysis framework is proposed for exploring the intriguing behaviors of a second-order switching control system. Complex discretizafion behaviors of the switching control system are explored in detail, and some intrinsic relationships between the system periodic behaviors and their associated symbolic sequences are studied.     

EMERS: a tree matching–based performance evaluation of mathematical expression recognition systems

Performance evaluation of mathematical expression recognition systems is attempted. The proposed method assumes expressions (input as well as recognition output) are coded following MathML or T_EX/LaT_EX (which also gets converted into MathML) format. Since any MathML representation follows a tree structure, evaluation of performance has been modeled as a tree-matching problem. The tree corresponding to the expression generated by the recognizer is compared with the groundtruthed one by comparing the corresponding Euler strings. The changes required to convert the tree corresponding to the expression generated by the recognizer into the groundtruthed one are noted. The number of changes required to make such a conversion is basically the distance between the trees. This distance gives the performance measure for the system under testing. The proposed algorithm also pinpoints the positions of the changes in the output MathML file. Testing of the proposed evaluation method considers a set of example groundtruthed expressions and their corresponding recognized results produced by an expression recognition system.     

Is students’ computer use at home related to their mathematical performance at school?

Recent research has provided evidence that students’ computer use at home is positively associated with their performance at school. However, most of the previous studies have failed to take into account the multiple determination of school performance and, in addition, to explain why using computers at home should benefit students’ academic achievement. The study described in this article takes a step towards filling this gap by circumventing the theoretical and methodological problems of previous research. First, we analyzed students’ home computer use and their performance in mathematics while accounting for other factors that have been found to affect students’ achievement. Second, we investigated possible differences in students’ school performance depending on how they used the computer. Based on the data from German students who took part in the Programme for International Student Assessment (PISA) in 2003, our results showed that students’ access to a computer was not linked with their performance in mathematics. In addition, it did not matter how often students used a computer at home. A positive effect on mathematical achievement was, however, observed for a small group of students who used the computer in a self-determined way that largely engaged them in problem-solving activities. Overall, the findings indicate that for the majority of students the computer had no substantial influence on their academic achievement. At the same time, the results are suggestive of ways in which performance in mathematics might be enhanced through the use of computers.