Computer use in mathematics education – present situation,intentions, and problems in the Federal Republic of Germany†

Computers, and computer‐related thinking structures, are only gradually influencing mathematics education. On the one hand, there is a discrepancy between involved teachers who already have changed their own classroom teaching to a great extent, and a majority of mathematics teachers who have not yet taken notice of the computer for teaching purposes. On the other hand, knowledge of the computer and of algorithms is frequently merely added to the mathematical subject matter. As opposed to that, the authors argue that it is necessary to genuinely integrate such subject matter, and to include general topics such as social impact and changed attitudes toward application. With regard to implementation, they develop concrete ideas which are aligned in a differentiated manner to the specific situation and the opportunities offered in the Federal Republic of Germany. The rationale for that is that only such reference to a specific situation will provide an opportunity for readers abroad to usefully apply approaches and ideas to the situation given in their own cultural environment.

An abbreviated version of this paper for cursory reading or other purposes has been marked by bold lines on the margin.

     

Ontological approach to development of computing with words based systems

Computing with words introduced by Zadeh becomes a very important concept in processing of knowledge represented in the form of propositions. Two aspects of this concept – approximation and personalization – are essential to the process of building intelligent systems for human-centric computing.For the last several years, Artificial Intelligence community has used ontology as a means for representing knowledge. Recently, the development of a new Internet paradigm – the Semantic Web – has led to introduction of another form of ontology. It allows for defining concepts, identifying relationships among these concepts, and representing concrete information. In other words, an ontology has become a very powerful way of representing not only information but also its semantics.The paper proposes an application of ontology, in the sense of the Semantic Web, for development of computing with words based systems capable of performing operations on propositions including their semantics. The ontology-based approach is very flexible and provides a rich environment for expressing different types of information including perceptions. It also provides a simple way of personalization of propositions. An architecture of computing with words based system is proposed. A prototype of such a system is described.     

Performance evaluation of a distributed application

Analysing the performance of an application performed on a distributed system is discussed in this paper. An analogy between a distributed system and a production process is portrayed, particularly for an application running on several computers. Consequently, theories of management of production processes are employed to help analyse and manage distributed systems, specifically, the Theory of Constraints (TOC). Using TOC combined with the cost/utilization model, which was initially developed to evaluate the utilization of a single processor and is extended here to handle a distributed system, it is demonstrated how the performance of a distributed system can be examined. The methodology presented here is based on a simple graphic display aimed to allow managers of information systems to locate constrained resources, to optimize the distribution of the computer application, and to examine and pinpoint improper imbalances and fluctuations in the system workload. The model develops into a management decision support tool that may be applied in areas such as buffer policy, assessment of protective capacity, investment in computer resources, and identification of areas for improvement.     

The role of computer simulations in learning analytic mechanics towards chaos theory: a course experimentation

We present the results of an experimental project, made at the University of Calabria, which uses computer simulations and scientific computational systems in teaching and learning the fundamentals of analytic mechanics, in particular the classical Lagrangian and Hamiltonian formulation of the matter. Starting with examples of classical dynamical systems we also introduced students to appreciate chaos concepts going step by step to more complex settings. The success obtained with the use of computer-based methodologies has been very clear and the final students' tests confirm it. In this paper, in particular, we compare the results obtained by four groups of students: the first two attended such an experimental course and employed computer support for their individual work at home, while the learning environment of the second two groups (the control ones) was based only on traditional classroom lectures with paper and pencil home activities. If on the one hand a better overall performance of the first two groups was evident, it is very interesting (and not so easy) to investigate how much in quantity and quality, the learning of the experimental groups differs from that of the other groups, and why. For example, we illustrate some relevant differences between them in the comprehension of the studied phenomena, in dealing with and solving the proposed problems, and also in the individual development of further personal ideas.     

OSiL: An instance language for optimization

Distributed computing technologies such as Web Services are growing rapidly in importance in today’s computing environment. In the area of mathematical optimization, it is common to separate modeling languages from optimization solvers. In a completely distributed environment, the modeling language software, solver software, and data used to generate a model instance might reside on different machines using different operating systems. Such a distributed environment makes it critical to have an open standard for exchanging model instances. In this paper we present OSiL (Optimization Services instance Language), an XML-based computer language for representing instances of large-scale optimization problems including linear programs, mixed-integer programs, quadratic programs, and very general nonlinear programs. OSiL has two key features that make it much superior to current standard forms for optimization problem instances. First, it uses the object-oriented features of XML schemas to efficiently represent nonlinear expressions. Second, its XML schema maps directly into a corresponding in-memory representation of a problem instance. The in-memory representation provides a robust application program interface for general nonlinear programming, facilitates reading and writing postfix, prefix, and infix formats to and from the nonlinear expression tree, and makes the expression tree readily available for function and derivative evaluations.     

Parallel Schur complement method for large-scale systems on distributed memory computers

The Schur complement method, also known as substructuring technique, was widely used in structural mechanics to solve large-scale systems with limited memory computers for more than three decades [J.S. Przemieniecki, AIAA J. 1 (1963) 138]. Currently, due to developments in computer technology, the available on-board memory has increased considerably. Despite the existence of these high-memory systems, the Schur complement method still finds its applications in structural mechanics through parallel computing. When developing a computer program, the Schur method has a significant book-keeping load in comparison to other parallel algorithms used, e.g., Schwarz alternating domain decomposition method [H.A. Schwarz, Gesammelte Mathematiche Abhandlungen, vol. 2, Springer, Berlin, 1890, p. 133]. This results in memory usage. Although parallel systems are used, global coefficient matrices require a large amount of memory. Therefore, significant memory is reserved for the solution of large-scale systems. In this paper, we present an efficient algorithm for the assemblage and solution of interface equations which facilitates the solution of large-scale systems via the Schur complement method on multiple instruction multiple data (MIMD) distributed memory architectures. In this method, we assemble the subdomain and interface coefficient matrices in such a manner that the memory requirements decrease significantly, resulting in the solution of large-scale systems with reasonable memory usage. The computer program is tested on distributed memory architectures with UNIX, WINDOWS NT, and LINUX operating systems.     

Realization of an OpenMath Server for REDUCE

Various approaches and standards for transferring mathematical objects between different computer programs in multimedia environment in the framework of the OpenMath project are described. The use of these standards might revolutionize the exchange of mathematical information caused by the development of computer algebra tools and telecommunication possibilities. The standards make it possible to create databases and easily save and use mathematical objects obtained in different applications, e.g., to exchange the data obtained in different computer algebra systems and place information on the Internet. Bibliography: 9 titles.     

A Knowledge Base on Special Functions

An approach to developing a mathematical knowledge base on special functions is described and an example of such a knowledge base is presented. The problem consists of creating a uniform structural approach to classification of special functions of different types. Such an approach was realized in a series of papers by the authors. Two variants of an experimental version of the SFTools knowledge base developed on the basis of such a classification are presented. The first variant is realized in the Delphi language and works in the environment of the Windows 95 and Windows NT operating systems. The second variant presents an Internet modification of SFTools; it is realized in the Java language. Bibliography: 6 titles.     

Modeling networked systems using the topologically distributed bounded rationality framework

In networked systems research, game theory is increasingly used to model a number of scenarios where distributed decision making takes place in a competitive environment. These scenarios include peer‐to‐peer network formation and routing, computer security level allocation, and TCP congestion control. It has been shown, however, that such modeling has met with limited success in capturing the real‐world behavior of computing systems. One of the main reasons for this drawback is that, whereas classical game theory assumes perfect rationality of players, real world entities in such settings have limited information, and cognitive ability which hinders their decision making. Meanwhile, new bounded rationality models have been proposed in networked game theory which take into account the topology of the network. In this article, we demonstrate that game‐theoretic modeling of computing systems would be much more accurate if a topologically distributed bounded rationality model is used. In particular, we consider (a) link formation on peer‐to‐peer overlay networks (b) assigning security levels to computers in computer networks (c) routing in peer‐to‐peer overlay networks, and show that in each of these scenarios, the accuracy of the modeling improves very significantly when topological models of bounded rationality are applied in the modeling process. Our results indicate that it is possible to use game theory to model competitive scenarios in networked systems in a way that closely reflects real world behavior, topology, and dynamics of such systems. © 2016 Wiley Periodicals, Inc. Complexity 21: 123–137, 2016     

Transparent Rule Based CAS to Support Formalization of Knowledge

The complexity of computer algebra systems hinders many students to develop an adequate mental model of such a system. As a result, they are often suspicious about the results and the didactical benefit is limited. The paper suggests that it is possible to design a transparent version of a computer algebra system that gives students a transparent access to the inner working of such a system. Didactical uses of such a system are discussed.     

Research into Expert Systems to Aid Simulation Model Formulation

The widespread availability of inexpensive computing power and recent developments in machine intelligence now makes it possible to consider computer assistance in the formulation phase of a discrete-event simulation model. The goal is to speed up this process. With this in mind, a series of prototype expert systems (written in LISP and run on an APPLE II) have been produced. The first prototype system was built using the ‘MYCIN’ approach. In the second prototype system a more informal approach was used, and the resulting computer-aided system is a natural-language understanding system. This paper outlines the need for such research and discusses the system development by describing the two main models and explaining why the natural-language approach was adopted.     

An Anthology on the Homology of Simulation with Artificial Intelligence

Simulation and artificial intelligence (AI) can be related in the following ways: the methodological similarities, the actual and potential uses of AI concepts in simulation, expert systems in simulation environments, and the gains for AI when applying ideas from simulation. This paper highlights the personal experiences of the author by discussing, for each of the four above-mentioned areas, a topic or a computer system in which he is directly involved. To highlight the similarities, it will be shown that the three-phase simulation model can be considered as a production-system model. A natural-language understanding system that was used in the definition stage of a simulation project will provide the basis for discussing AI concepts in simulation environments. An expert system used in the development and verification stages will provide an insight into the application of expert systems in simulation modelling. An expert-system development tool which is partially based on simulation techniques will be used to outline the gains for AI.     

云计算环境资源监控系统研究

为保障云计算平台的可靠性,基于Ganglia,采用Eucalyptus开源云平台,提出了云计算平台资源监控系统架构.从总体结构、监控信息处理流程等多个方面对监控系统进行了详细阐述,提出了动态更换虚拟机簇头的算法,并对算法与原型系统进行了验证和性能评测.分析与评测结果表明,所提系统能够适用于云计算环境资源信息的实时监测与预警,具备系统开销较低,用户接口友好的特点,能准确反映系统、虚拟机的负载信息,有助于云计算系统服务可靠性的提升.簇头动态更换算法能及时发现失效的簇头并更换为更健壮的虚拟机,保证了监测系统的正常运行.     

A parallel interior point algorithm for linear programming on a network of transputers

Interior Point algorithms have become a very successful tool for solving large-scale linear programming problems. The Dual Affine algorithm is one of the Interior Point algorithms implemented in the computer program OB1. It is a good candidate for implementation on a parallel computer because it is very computing-intensive. A parallel Dual Affine algorithm is presented which is suitable for a parallel computer with a distributed memory. The algorithm obtains its speedup from parallel sparse linear algebra computations such as Cholesky factorisation, matrix multiplication, and triangular system solving, which form the bulk of the computing work. Efficient algorithms based on the grid distribution of matrices are presented for each of these computations. The algorithm is implemented in occam 2 on a square mesh of transputers. The resulting parallel program is connected to the sequentialFortran 77 program OB1, which performs the preprocessing and the postprocessing. Experimental results on a mesh of 400 transputers are given for a test set of seven realistic planning and scheduling problems from Shell and seven problems from the NETLIB LP collection; the results show a speedup of 88 for the largest problem.     

A market-driven transfer price for distributed products using mathematical programming

A distributed product has its manufacturing activities distributed among many locations. These locations could belong to one or more firms in a manufacturing network. Often, components needed to manufacture a distributed product move through different nodes in the network and sometimes across international borders. Hence, a transfer price is needed for the purpose of estimating duties and drawbacks. Being aware of the fact that transfer price can be used to manipulate taxable profits, many countries have instituted rules concerning transfer price estimation. For example, in the United Sates, the Internal Revenue Service (IRS) says that the right price is the market value. But for many components it is difficult to find a free market. Similar products may exist in the market but they may have different attributes. In such cases, it is important to be able to estimate the market-driven transfer price, given other similar products in the open market.We develop a method using a mathematical programming model and providing companies an opportunity to work proactively with the IRS in a cooperative manner in order to avoid costly audit and litigation. This way, companies avoid penalties and also gain certainty regarding tax liability. An example illustrating the method is presented.     

Concurrent stochastic methods for global optimization

The global optimization problem, finding the lowest minimizer of a nonlinear function of several variables that has multiple local minimizers, appears well suited to concurrent computation. This paper presents a new parallel algorithm for the global optimization problem. The algorithm is a stochastic method related to the multi-level single-linkage methods of Rinnooy Kan and Timmer for sequential computers. Concurrency is achieved by partitioning the work of each of the three main parts of the algorithm, sampling, local minimization start point selection, and multiple local minimizations, among the processors. This parallelism is of a coarse grain type and is especially well suited to a local memory multiprocessing environment. The paper presents test results of a distributed implementation of this algorithm on a local area network of computer workstations. It also summarizes the theoretical properties of the algorithm.Research supported by AFOSR grant AFOSR-85-0251, ARO contract DAAG 29-84-K-0140, NSF grant DCR-8403483, and NSF cooperative agreement DCR-8420944.     

On the GI/M/1/N queue with multiple working vacations—analytic analysis and computation

We consider finite buffer single server GI/M/1 queue with exhaustive service discipline and multiple working vacations. Service times during a service period, service times during a vacation period and vacation times are exponentially distributed random variables. System size distributions at pre-arrival and arbitrary epoch with some important performance measures such as, probability of blocking, mean waiting time in the system etc. have been obtained. The model has potential application in the area of communication network, computer systems etc. where a single channel is allotted for more than one source.