Exploring the Spectra of Some Classes of Singular Integral Operators with Symbolic Computation

Spectral theory has many applications in several main scientific research areas (structural mechanics, aeronautics, quantum mechanics, ecology, probability theory, electrical engineering, among others) and the importance of its study is globally acknowledged. In recent years, several software applications were made available to the general public with extensive capabilities of symbolic computation. These applications, known as computer algebra systems (CAS), allow to delegate to a computer all, or a significant part, of the symbolic calculations present in many mathematical algorithms. In our work we use the CAS Mathematica to implement for the first time on a computer analytical algorithms developed by us and others within the Operator Theory. The main goal of this paper is to show how the symbolic computation capabilities of Mathematica allow us to explore the spectra of several classes of singular integral operators. For the one-dimensional case, nontrivial rational examples, computed with the automated process called [ASpecPaired-Scalar], are presented. For the matrix case, nontrivial essentially bounded and rational examples, computed with the analytical algorithms [AFact], [SInt], and [ASpecPaired-Matrix], are presented. In both cases, it is possible to check, for each considered paired singular integral operator, if a complex number (chosen arbitrarily) belongs to its spectrum.     

Students' attitude towards computer algebra systems (CAS) and their choice of using CAS in problem-solving

We explore students choice of using computer algebra systems (CAS) in problem-solving relative to their self-reported attitude towards learning mathematics with CAS. Our research design is a case study of nine Norwegian upper-secondary mathematics students with a wide range of attitude towards CAS. Our findings on routine problems indicate that (1) students use CAS whenever students perceive the problem as time-consuming regardless of their attitude towards CAS, and (2) students attitude affects their use of CAS whenever students perceive the problem as non-time-consuming. Norway, among other countries, has implemented CAS as an essential digital resource towards learning mathematics in upper-secondary school. Our discussion focuses on the implications of our findings have on local mathematics educators and national policy-makers.     

Inequalities,assessment and computer algebra

The goal of this paper is to examine single variable real inequalities that arise as tutorial problems and to examine the extent to which current computer algebra systems (CAS) can (1) automatically solve such problems and (2) determine whether students’ own answers to such problems are correct. We review how inequalities arise in contemporary curricula. We consider the formal mathematical processes by which such inequalities are solved, and we consider the notation and syntax through which solutions are expressed. We review the extent to which current CAS can accurately solve these inequalities, and the form given to the solutions by the designers of this software. Finally, we discuss the functionality needed to deal with students’ answers, i.e. to establish equivalence (or otherwise) of expressions representing unions of intervals. We find that while contemporary CAS accurately solve inequalities there is a wide variety of notation used.     

Teaching spline approximation techniques using maple

Using Computer Algebra Systems (CAS)-such as MAPLE-in teaching and learning mathematical concepts is a great challenge both from a didactical and a scientific point of view. We have to rewrite our traditional paper based teaching materials for interactive and living electronic worksheets, Only few statements and principles have to be acquired by the learner and the teacher from the CAS and after they can visualise, make animations modify quickly the program data, perform symbolic and numeric calculations step by step and in the whole, and verify deductions on their own. The author prepared Maple worksheets for teaching different types of function approximating techniques, such as interpolation-, least square-, spline and uniform approximation methods for post-graduate mechanical engineering students. In this paper we want to demonstrate how can we keep and improve the famous problem solving principles and rules given by G. Pólya and R. Descartes (Pólya 1962), when we use the capabilities of CAS. The education principles active learning, motivations and the successive phases are getting new meaning in the CAS. Our examples are always concerning with spline functions. Handling the formulas, calculating values and giving proofs are always in the form of Maple statements.     

Instrumenting Mathematical Activity: Reflections on Key Studies of the Educational Use of Computer Algebra Systems

This paper examines the process through which students learn to make functional use of computer algebra systems (CAS), and the interaction between that process and the wider mathematical development of students. The result of ‘instrumentalising‘ a device to become a mathematical tool and correspondingly ‘instrumenting’ mathematical activity through use of that tool is not only to extend students' mathematical technique but to shape their sense of the mathematical entities involved. These ideas have been developed within a French programme of research – as reported by Artigue in this issue of the journal – which has explored the integration of CAS – typically in the form of symbolic calculators – into the everyday practice of mathematics classrooms. The French research –influenced by socio-psychological theorisation of the development of conceptual systems- seeks to take account of the cultural and cognitive facets of these issues, noting how mathematical norms – or their absence – shape the mental schemes which students form as they appropriate CAS as tools. Instrumenting graphic and symbolic reasoning through using CAS influences the range and form of the tasks and techniques experienced by students, and so the resources available for more explicit codification and theorisation of such reasoning. This illuminates an influential North American study– conducted by Heid – which French researchers have seen as taking a contrasting view of the part played by technical activity in developing conceptual understanding. Reconsidered from this perspective, it appears that while teaching approaches which ‘resequence skills and concepts’ indeed defer – and diminish –attention to routinised skills, the tasks introduced in their place depend on another –albeit less strongly codified – system of techniques, supporting more extensive and active theorisation. The French research high lights important challenges which arise in instrumenting classroom mathematical activity and correspondingly instrumentalising CAS. In particular, it reveals fundamental constraints on human-machine interaction which may limit the capacity of the present generation of CAS to scaffold the mathematical thinking and learning of students. This revised version was published online in July 2006 with corrections to the Cover Date.     

The Impact of Teacher Privileging on Learning Differentiation with Technology

This study examines how two teachers taught differentiation using a hand held computer algebra system, which made numerical,graphical and symbolic representations of the derivative readily available. The teachers planned the lessons together but taught their Year 11 classes in very different ways. They had fundamentally different conceptions of mathematics with associated teaching practices,innate ‘privileging’ of representations, and of technology use. This study links these instructional differences to the different differentiation competencies that the classes acquired. Students of the teacher who privileged conceptual understanding and student construction of meaning were more able to interpret derivatives. Students of the teacher who privileged performance of routines made better use of the CAS for solving routine problems. Comparison of the results with an earlier study showed that although each teacher's teaching approach was stable over two years, each used technology differently with further experience of CAS. The teacher who stressed understanding moved away from using CAS, whilst the teacher who stressed rules,adopted it more. The study highlights that within similar overall attainment on student tests, there can be substantial variations of what students know. New technologies provide more approaches to teaching and so greater variations between teaching and the consequent learning may become evident. This revised version was published online in July 2006 with corrections to the Cover Date.     

Constraint-based analysis of concurrent probabilistic hybrid systems: An application to networked automation systems

In previous publications, the authors have introduced the notion of stochastic satisfiability modulo theories (SSMT) and the corresponding SiSAT solving algorithm, which provide a symbolic method for the reachability analysis of probabilistic hybrid systems. SSMT extends satisfiability modulo theories (SMT) with randomized (or stochastic), existential, and universal quantification, as known from stochastic propositional satisfiability. In this paper, we extend the SSMT-based procedures to the symbolic analysis of concurrent probabilistic hybrid systems. After formally introducing the computational model, we provide a mechanized translation scheme to encode probabilistic bounded reachability problems of concurrent probabilistic hybrid automata as linearly sized SSMT formulae, which in turn can be solved by the SiSAT tool. We furthermore propose an algorithmic enhancement which tailors SiSAT to probabilistic bounded reachability problems by caching and reusing solutions obtained on bounded reachability problems of smaller depth. An essential part of this article is devoted to a case study from the networked automation systems domain. We explain in detail the formal model in terms of concurrent probabilistic automata, its encoding into the SiSAT modeling language, and finally the automated quantitative analysis.     

Boundary-contact problems for domains with edge singularities

We study boundary-contact problems for elliptic equations (and systems) with interfaces that have edge singularities. Such problems represent continuous operators between weighted edge spaces and subspaces with asymptotics. Ellipticity is formulated in terms of a principal symbolic hierarchy, containing interior, transmission, and edge symbols. We construct parametrices, show regularity with asymptotics of solutions in weighted edge spaces and illustrate the results by boundary-contact problems for the Laplacian with jumping coefficients.     

A Framework for Monitoring Progress and Planning Teaching Towards the Effective Use of Computer Algebra Systems

This article suggests a framework to organise a cluster of variables that are associated with students' effective use of computer algebra systems (CAS) in mathematics learning. Based on a review of the literature and from the authors' own teaching experience, the framework identifies the main characteristics of students' interactions with CAS technology and how these may be used to monitor students' developing use of CAS; from this, the framework may be used to plan teaching in order to gain greater benefit from the availability of CAS. Four case studies describing students' development over a semester are reported. These demonstrate a variety of combinations of technical competencies and personal attributes. They indicate the importance of both the technical and personal aspects but suggest that negative attitudes rather than technical difficulties can limit the effective use of CAS. Finally practical suggestions are given for teaching strategies which may promote effective use of CAS.This revised version was published online in September 2005 with corrections to the Cover Date.     

Euklides dynamic geometry

The spreading of computers is becoming more and more inevitable in education as well. Promising pathfinders are the computerised algebraic systems (CAS) and the dynamic geometrical systems (DGS). These tools are very important for students to be able to play, experiment, recognise the problems and ask the questions themselves. If we would give way to more experimenting activities, we could bring those number of students already derailed by the educational system back to the joy of gaining knowledge on their own.This article introduces the Euklides Dynamic Geometry software, comparing to concurrent softwares and suggesting conceptional questions of dynamic geometry.Important special abilities are animation, track of a point (locus) and tools for examination of structure.     

Inversion of general tridiagonal matrices

In the current work, the authors present a symbolic algorithm for finding the inverse of any general nonsingular tridiagonal matrix. The algorithm is mainly based on the work presented in [Y. Huang, W.F. McColl, Analytic inversion of general tridiagonal matrices, J. Phys. A 30 (1997) 7919–7933] and [M.E.A. El-Mikkawy, A fast algorithm for evaluating nth order tridiagonal determinants, J. Comput. Appl. Math. 166 (2004) 581–584]. It removes all cases where the numeric algorithm in [Y. Huang, W.F. McColl, Analytic inversion of general tridiagonal matrices, J. Phys. A 30 (1997) 7919–7933] fails. The symbolic algorithm is suited for implementation using Computer Algebra Systems (CAS) such as MACSYMA, MAPLE and MATHEMATICA. An illustrative example is given.     

Boundary-contact problems for domains with conical singularities

We study boundary-contact problems for elliptic equations (and systems) with interfaces that have conical singularities. Such problems represent continuous operators between weighted Sobolev spaces and subspaces with asymptotics. Ellipticity is formulated in terms of extra transmission conditions along the interfaces with a control of the conormal symbolic structure near conical singularities. We show regularity and asymptotics of solutions in weighted spaces, and we construct parametrices. The result will be illustrated by a number of explicit examples.     

A problem structuring method for ecosystem-based management: The DPSIR modelling process

The purpose of this paper is to learn from Complex Adaptive Systems (CAS) theory to inform the development of Problem Structuring Methods (PSMs) both in general and in the specific context of marine management. The focus on marine management is important because it is concerned with a CAS (formed through the interconnection between natural systems, designed systems and social systems) which exemplifies their particularly ‘wicked’ nature. Recognition of this compels us to take seriously the need to develop tools for knowledge elicitation and structuring which meet the demands of CAS. In marine management, chief among those tools is the DPSIR (Drivers – Pressures – State Changes – Impacts – Responses) model and, although widely applied, the extent to which it is appropriate for dealing with the demands of a CAS is questionable. Such questioning is particularly pertinent in the context of the marine environment where there is a need to not only recognise a broad range of stakeholders (a question of boundary critique) but also to manage competing knowledge (economic, local and scientific) and value claims. Hence this paper emphasises how a CAS perspective might add impetus to the development of a critical perspective on DPSIR and PSM theory and practice to promote a more systemic view of decision-making and policy development.     

CAS in general mathematics education

A rational discussion of the use of Computer algebra systems (CAS) in mathematics teaching in general education needs an explicit image of (general) mathematics education, an explication of global perspectives and goals on mathematics teaching focusing on general education (chapter 1). The conception of general education according to the «ability of communication with experts» described in chapter 2 can be such an orientation for analysing, considering, classifying and assessing the didactical possibilities of using CAS. CAS are materialised mathematics allowing for more or less exhaustive outsourcing of operative (also symbolically) knowledge and skills to the machine. This frees up space of time as well as mental space for the development of those competences being in our view relevant for general mathematics education. In chapter 3 the idea of outsourcing and the role of CAS for it is discussed more detailed as well as consequences being possible for the CAS-supported teaching of mathematics. Beyond, CAS can be didactically used and reflected as a model of communication between (mathematical) experts and lay-persons (chapter 4). Chapter 5 outlines some research perspectives.     

Relationships between variables in learning and modes of presenting mathematics concepts

The purpose of this study was to investigate relationships between figural and symbolic aptitudes and figural and symbolic modes of presenting mathematics concepts to secondary school students. One hundred and sixty students were measured on 11 aptitudes (five figural, five symbolic, and one semantic) from Guilford's structure‐of‐intellect cube and were randomly assigned to either a figural or symbolic instructional mode for learning the mathematical concept of function. Subjects studied the function concept using one of two sets (figural or symbolic) of programmed instructional materials during three consecutive mathematics classes. Immediately following instruction a learning test was given, which was followed by a retention test 1 week later. Data analysis showed that females scored significantly higher than males on all dependent measures, and their scores were independent of instructional mode. For male students figural instruction was superior to the symbolic mode. Significant relationships were found between instructional mode and the figural aptitude divergent production of figural systems. The symbolic aptitude cognition of symbolic systems was a predictor of success for subjects studying symbolic materials. Cognition of semantic systems was a good predictor of success for students receiving the figural instructional treatment.     

Symbolic computing

It is shown that symbolic computing can be effectively applied to derive actuarial formulae and used to prove theorems. Illustrating examples are given through applying the computer package MACSYMA to problems in deriving prob- ability functions for fractional ages, deriving the power series of an annuity coefficient, calculating annuities-certain, invert- ing a matrix containing a variable in graduation, and providing Jensen's inequality. Using symbolic computing effectively is a first step to building expert systems in actuarial science.     

Instructional Supports for Representational Fluency in Solving Linear Equations with Computer Algebra Systems and Paper‐and‐Pencil

This research addresses the issue of how to support students' representational fluency—the ability to create, move within, translate across, and derive meaning from external representations of mathematical ideas. The context of solving linear equations in a combined computer algebra system (CAS) and paper‐and‐pencil classroom environment is targeted as a rich and pressing context to study this issue. We report results of a collaborative teaching experiment in which we designed for and tested a functions approach to solving equations with ninth‐grade algebra students, and link to results of semi‐structured interviews with students before and after the experiment. Results of analyzing the five‐week experiment include instructional supports for students' representational fluency in solving linear equations: (a) sequencing the use of graphs, tables, and CAS feedback prior to formal symbolic transpositions, (b) connecting solutions to equations across representations, and (c) encouraging understanding of equations as equivalence relations that are sometimes, always, or never true. While some students' change in sophistication of representational fluency helps substantiate the productive nature of these supports, other students' persistent struggles raise questions of how to address the diverse needs of learners in complex learning environments involving multiple tool‐based representations.     

Nonlinear transformations for the simplification of unconstrained nonlinear optimization problems

Formalization decisions in mathematical programming could significantly influence the complexity of the problem, and so the efficiency of the applied solver methods. This widely accepted statement induced investigations for the reformulation of optimization problems in the hope of getting easier to solve problem forms, e.g. in integer programming. These transformations usually go hand in hand with relaxation of some constraints and with the increase in the number of the variables. However, the quick evolution and the widespread use of computer algebra systems in the last few years motivated us to use symbolic computation techniques also in the field of global optimization. We are interested in potential simplifications generated by symbolic transformations in global optimization, and especially in automatic mechanisms producing equivalent expressions that possibly decrease the dimension of the problem. As it was pointed out by Csendes and Rapcsák (J Glob Optim 3(2):213–221, 1993), it is possible in some cases to simplify the unconstrained nonlinear objective function by nonlinear coordinate transformations. That means mostly symbolic replacement of redundant subexpressions expecting less computation, while the simplified task remains equivalent to the original in the sense that a conversion between the solutions of the two forms is possible. We present a proper implementation of the referred theoretical algorithm in a modern symbolic programming environment, and testing on some examples both from the original publications and from the set of standard global optimization test problems to illustrate the capabilities of the method.     

Simplification of symbolic polynomials on non-commutative variables

We address the issue of simplifying symbolic polynomials on non-commutative variables. The problem is motivated by applications in optimization and various problems in systems and control. We develop theory for polynomials which are linear in a subset of the variables and develop algorithms to produce representations which have the minimal possible number of terms. The results can handle polynomial matrices as well as block-matrix variables.     

On the expressiveness and decidability of o-minimal hybrid systems

This paper is driven by a general motto: bisimulate a hybrid system by a finite symbolic dynamical system. In the case of o-minimal hybrid systems, the continuous and discrete components can be decoupled, and hence, the problem reduces in building a finite symbolic dynamical system for the continuous dynamics of each location. We show that this can be done for a quite general class of hybrid systems defined on o-minimal structures. In particular, we recover the main result of a paper by G. Lafferriere, G.J. Pappas, and S. Sastry, on o-minimal hybrid systems. We also provide an analysis and extension of results on decidability and complexity of problems and constructions related to o-minimal hybrid systems.