Stephen Sugden Copyright (c) 2012 All rights reserved. http://works.bepress.com/stephen_sugden Recent documents in Stephen Sugden en-us Thu, 11 Oct 2012 01:31:07 PDT 3600 Spreadsheets and Bulgarian goats http://works.bepress.com/stephen_sugden/25 http://works.bepress.com/stephen_sugden/25 Tue, 09 Oct 2012 20:30:38 PDT We consider a problem appearing in an Australian Mathematics Challenge in 2003. This article considers whether a spreadsheet might be used to model this problem, thus allowing students to explore its structure within the spreadsheet environment. It then goes on to reflect on some general principles of problem decomposition when the final goal is a successful and lucid spreadsheet implementation.

]]> Stephen Sugden Discrete phase-locked loop systems and spreadsheets http://works.bepress.com/stephen_sugden/24 http://works.bepress.com/stephen_sugden/24 Tue, 09 Oct 2012 20:25:34 PDT This paper demonstrates the use of a spreadsheet in exploring non-linear difference equations that describe digital control systems used in radio engineering, communication and computer architecture. These systems, being the focus of intensive studies of mathematicians and engineers over the last 40 years, may exhibit extremely complicated behavior interpreted in contemporary terms as transition from global asymptotic stability to chaos through period-doubling bifurcations. The authors argue that embedding advanced mathematical ideas in the technological tool enables one to introduce fundamentals of discrete control systems in tertiary curricula without learners having to deal with complex machinery that rigorous mathematical methods of investigation require. In particular, in the appropriately designed spreadsheet environment, one can effectively visualize a qualitative difference in the behavior of systems with different types of non-linear characteristic.

]]> Sergei Abrramovich et al. The slumbarumba jumbuck problem http://works.bepress.com/stephen_sugden/23 http://works.bepress.com/stephen_sugden/23 Tue, 09 Oct 2012 20:25:33 PDT The potential for simple linear relationships arising from a computer game to build student modelling and "word problem" skills is explored. The fundamental capability of the spreadsheet to tabulate and graph possible solutions is used to lay bare the problem structure for the students.

]]> Stephen Sugden Spreadsheet conditional formatting illuminates investigations into modular arithmetic http://works.bepress.com/stephen_sugden/21 http://works.bepress.com/stephen_sugden/21 Thu, 04 Oct 2012 21:55:26 PDT Modular arithmetic has often been regarded as something of a mathematical curiosity, at least by those unfamiliar with its importance to both abstract algebra and number theory, and with its numerous applications. However, with the ubiquity of fast digital computers, and the need for reliable digital security systems such as RSA, this important branch of mathematics is now considered essential knowledge for many professionals. Indeed, computer arithmetic itself is, ipso facto, modular. This chapter describes how the modern graphical spreadsheet may be used to clearly illustrate the basics of modular arithmetic, and to solve certain classes of problems. Students may then gain structural insight and the foundations laid for applications to such areas as hashing, random number generation, and public-key cryptography.

]]> David Miller et al. Basic finance made accessible in Excel 2007: "The Big 5, Plus 2" http://works.bepress.com/stephen_sugden/20 http://works.bepress.com/stephen_sugden/20 Wed, 17 Aug 2011 17:14:47 PDT The basic principles and equations are developed for elementary finance, based on the concept of compound interest. The five quantities of interest in such problems are present value, future value, amount of periodic payment, number of periods and the rate of interest per period. We consider three distinct means of computing each of these five quantities in Excel 2007: (i) use of algebraic equations, (ii) by recursive schedule and the Goal Seek facility, and (iii) use of Excel's intrinsic financial functions. The paper is intended to be used as the basis for a lesson plan and contains many examples and solved problems. Comment is made regarding the relative difficulty of each approach, and a prominent theme is the systematic use of more than one method to increase student understanding and build confidence in the answer obtained.

]]> Stephen Sugden et al. Exploring the fundamental theorem of arithmetic in Excel 2007 http://works.bepress.com/stephen_sugden/19 http://works.bepress.com/stephen_sugden/19 Wed, 17 Aug 2011 17:14:45 PDT This paper discusses how fundamentals of number theory, such as unique prime factorization and greatest common divisor can be made accessible to secondary school students through spreadsheets. In addition, the three basic multiplicative functions of number theory are defined and illustrated through a spreadsheet environment. Primes are defined simply as those natural numbers with just two divisors. One focus of the paper is to show the ease with which spreadsheets can be used to introduce students to some basics of elementary number theory. Complete instructions are given to build a spreadsheet to enable the user to input a positive integer, either with a slider or manually, and see the prime decomposition. The spreadsheet environment allows students to observe patterns, gain structural insight, form and test conjectures, and solve problems in elementary number theory.

]]> Stephen Sugden et al. Colour by numbers in Excel 2007: Solving algebraic equations without algebra http://works.bepress.com/stephen_sugden/18 http://works.bepress.com/stephen_sugden/18 Wed, 17 Aug 2011 17:14:43 PDT This is an update of an earlier paper, and is written for Excel 2007. A series of Excel 2007 models is described. The more advanced versions allow solution of f(x)=0 by examining change of sign of function values. The function is graphed and change of sign easily detected by a change of colour. Relevant features of Excel 2007 used are Names, Scatter Chart and Conditional Formatting. Several sample Excel 2007 models are available for download, and the paper is intended to be used as a lesson plan for students having some familiarity with derivatives. For comparison and reference purposes, the paper also presents a brief outline of several common equation-solving strategies as an Appendix.

]]> Stephen Sugden The Number Crunch game: A simple vehicle for building algebraic reasoning skills http://works.bepress.com/stephen_sugden/17 http://works.bepress.com/stephen_sugden/17 Wed, 17 Aug 2011 17:14:41 PDT A newspaper numbers game based on simple arithmetic relationships is discussed. Its potential to give students of elementary algebra practice in semi ad-hoc reasoning and to build general arithmetic reasoning skills is explored.

]]> Stephen Sugden Problem solving with Delphi http://works.bepress.com/stephen_sugden/16 http://works.bepress.com/stephen_sugden/16 Mon, 26 Jul 2010 23:36:53 PDT The purpose of the book is to use Delphi as a vehicle to introduce some fundamental algorithms and to illustrate several mathematical and problem-solving techniques. This book is therefore intended to be more of a reference for problem-solving, with the solution expressed in Delphi. It introduces a somewhat eclectic collection of material, much of which will not be found in a typical book on Pascal or Delphi. Many of the topics have been used by the author over a period of about ten years at Bond University, Australia in various subjects from 1993 to 2003. Much of the work was connected with a data structures subject (second programming course) conducted variously in MODULA-2, Oberon and Delphi, at Bond University, however there is considerable other, more recent material, e.g., a chapter on Sudoku.

]]> Stephen John Sugden Insight into the Fractional Calculus via a Spreadsheet http://works.bepress.com/stephen_sugden/14 http://works.bepress.com/stephen_sugden/14 Mon, 11 Jan 2010 17:08:26 PST Many students of calculus are not aware that the calculus they have learned is a special case (integer order) of fractional calculus. Fractional calculus is the study of arbitrary order derivatives and integrals and their applications. The article begins by stating a naive question from a student in a paper by Larson (1974) and establishes, for polynomials and exponential functions, that they can be deformed into their derivative using the μ-th order fractional derivatives for 0<μ<1. Through the power of Excel we illustrate the continuous deformations dynamically through conditional formatting. Some applications are discussed and a connection made to mathematics education.

]]> David A. Miller et al. Mathematics EASIPLOT A Graphics Tool for Ordinary Differential Equations http://works.bepress.com/stephen_sugden/15 http://works.bepress.com/stephen_sugden/15 Mon, 11 Jan 2010 17:08:26 PST Advanced computer hardware available at Bond University made it possible to aim fairly high when planning our graphics CAL program for support of introductory calculus and differential equations subjects. Having laboratories equipped with IBM PS/2 60 machines having VGA graphics (16 colours at 640x480 resolution), some exciting possibilities emerged. The fast pace at which events happen at Bond University has had a great influence on planning, and while many candidates for plotting on MS-DOS machines were considered, none was considered entirely suitable. EASIPLOT features cartesian, polar, parametric plotting in the (x,y) plane with the facility to plot more than one function simultaneously. Families of curves can be defined and plotted and the integrals of first-order ordinary differential equations can be graphed.

]]> Stephen J. Sugden et al. Computing sequences and series by recurrence http://works.bepress.com/stephen_sugden/13 http://works.bepress.com/stephen_sugden/13 Wed, 19 Aug 2009 17:50:23 PDT Extract:

Many commonly-used mathematical functions may be computed via carefully-constructed recurrence formulas. Sequences are typically defined by giving a formula for the general term. Series is the mathematical name given to partial sums of sequences. In either case we may often take advantage of the great expressive power of recurrence relations to create code which is both lucid and compact. Further, this does not necessarily mean that we must use recursive code. In many instances, iterative code is adequate, and often more efficient.

]]> Stephen J. Sugden Diophantine equations as a context for technology-enhanced training in conjecturing and proving http://works.bepress.com/stephen_sugden/12 http://works.bepress.com/stephen_sugden/12 Tue, 11 Aug 2009 18:45:46 PDT Solving indeterminate algebraic equations in integers is a classic topic in the mathematics curricula across grades. At the undergraduate level, the study of solutions of non-linear equations of this kind can be motivated by the use of technology. This article shows how the unity of geometric contextualization and spreadsheet-based amplification of this topic can provide a discovery experience for prospective secondary teachers and information technology students. Such experience can be extended to include a transition from a computationally driven conjecturing to a formal proof based on a number of simple yet useful techniques.

]]> Sergei Abramovich et al. Bits, Binary, Binomials and Recursion: Helping IT Students Understand Mathematical Induction http://works.bepress.com/stephen_sugden/11 http://works.bepress.com/stephen_sugden/11 Wed, 17 Jun 2009 19:39:17 PDT Discrete Mathematics is a fundamental subject for Information Technology (IT) majors. For students whose primary orientation is not to mathematics, but computing, it is important to relate fundamental mathematical principles to practical computing realities. At Bond University, Microsoft Excel is the primary practical vehicle used to illustrate these connections, and has proved to be an extremely valuable tool for this purpose. No programming or macro creation is required to implement most of the basic principles taught in Discrete Mathematics. In particular, the topic of Mathematical Induction (MI) poses a considerable challenge to many students, and there are a number of ways in which a spreadsheet such as Excel can assist with the learning process.

]]> Stephen J. Sugden An Exact Algorithm for the Constant and Non-linear Cost Function Problem in Economic Synthesis Of Networks http://works.bepress.com/stephen_sugden/10 http://works.bepress.com/stephen_sugden/10 Tue, 03 Mar 2009 22:02:12 PST In this paper we introduce a branch and bound algorithm for finding a tree solution of the minimum cost network flow problem with constant arc cost. We consider different situations such as unit and non-unit traffic demand. The entire searching process is interpreted and the methods used in different situations to prune the search tree are emphasized respectively. Empirical results indicate the efficiency of these techniques. While no algorithm exists for a piecewise-constant cost function problem, we provide a model for transforming the problem to one with constant arc costs. After the transformation, the piecewise constant cost function problem can be solved by most existing algorithms for the linear cost function problem.

]]> Stephen J. Sugden et al. The spreadsheet as a tool for teaching set theory: Part 1 – an Excel lesson plan to help solve Sudokus http://works.bepress.com/stephen_sugden/7 http://works.bepress.com/stephen_sugden/7 Tue, 03 Mar 2009 22:02:11 PST This paper is intended to be used in the classroom. It describes essentially every step of the construction of an Excel model to help solve Sudoku puzzles. For those up to moderate difficulty, it will usually solve the puzzle to completion. For the more difficult ones, it still provides a platform for decision support. The paper may be found useful for a lesson in which students, who, having some basic knowledge of Excel, are learning some of its lesser-known features, such as conditional formatting. It also generates a useful tool for working with Sudoku puzzles, from the very easiest right up to the ones often labelled as fiendish or diabolical. Fundamental mathematical concepts such as set intersection, set partition and reduction of set partition to singletons are very graphically illustrated by the present Excel model for Sudoku. Prominent spreadsheet concepts presented here are conditional formatting, names, COUNTIF, CONCATENATE. The paper is accompanied by a completed Excel model, constructed by using the steps described herein. No VBA code is employed; the whole thing is done with Excel formulas and conditional formatting.

]]> Stephen J. Sugden Network Design With A Genetic Algorithm http://works.bepress.com/stephen_sugden/9 http://works.bepress.com/stephen_sugden/9 Tue, 03 Mar 2009 22:02:11 PST In telecommunications network design, nodes need to be linked in an economical way to handle expected traffic. Capacity constraints, degree constraints and hop limits are to be respected. A genetic algorithm with some novel features is described. The crossover method generates an optimal child solution for the parents selected.

]]> Stephen J. Sugden et al. A Simulated Annealing Approach to Communication Network Design http://works.bepress.com/stephen_sugden/8 http://works.bepress.com/stephen_sugden/8 Tue, 03 Mar 2009 22:02:11 PST This paper explores the use of the meta-heuristic search algorithm Simulated Annealing for solving a minimum cost network synthesis problem. This problem is a common one in the design of telecommunication networks. The formulation we use models a number of practical problems with hop-limit, degree and capacity constraints. Emphasis is placed on a new approach that uses a knapsack polytope to select amongst a number of pre-computed traffic routes in order to synthesise the network. The advantage of this approach is that a subset of the best routes can be used instead of the whole set, thereby making the process of designing large networks practicable. Using simulated annealing, we solve moderately large networks (up to 30 nodes) efficiently.

]]> Stephen J. Sugden et al. Stochastic Recurrences of Jackpot Keno http://works.bepress.com/stephen_sugden/5 http://works.bepress.com/stephen_sugden/5 Tue, 03 Mar 2009 22:02:10 PST We describe a mathematical model and simulation study for Jackpot Keno, as implemented by Jupiters Network Gaming (JNG) in the Australian state of Queensland, and as controlled by the Queensland Office of Gaming Regulation (QOGR) qogr. The recurrences for the house net hold are derived and it is seen that these are piecewise linear with a ternary domain split, and further, the split points are stochastic in nature. Since this structure is intractable brockett, estimation of house net hold obtained through an appropriately designed simulator using a random number generator with desirable properties is described.

Since the model and simulation naturally derives hold given payscale, but JNG and QOGR require payscale given hold, an inverse problem was required to be solved. This required development of a special algorithm, which may be described as a stochastic binary search.

Experimental results are presented, in which the simulator is used to determine jackpot payscales so as to satisfy legal requirements of approximately 75% of net revenue returned to the players, i.e., 25% net hold for the house (JNG). Details of the algorithm use to solve this problem are presented here, and notwithstanding the stochastic nature of the simulation, convergence to a specified hold for the inverse problem has been achieved to within 0.1% in all cases of interest to date.

]]> Stephen J. Sugden et al. Computer Mathematics using Pascal, 2nd Edition http://works.bepress.com/stephen_sugden/6 http://works.bepress.com/stephen_sugden/6 Tue, 03 Mar 2009 22:02:10 PST This book introduces elementary programming in Pascal as a tool to solve introductory problems in numerical mathematics at post-compulsory level, i.e. secondary schools, colleges of TAFE, senior colleges, and first-year tertiary. In particular, it covers all the material in the Computer Mathematics (Unit V) course in Queensland high schools. A modern, structured approach to programming has been used throughout the book, and emphasis is placed on problem-solving strategies and algorithm development. Students should eventually appreciate that, in general, the sooner you start coding the solution to a problem, the longer it will take to solve successfully, and that the computer is simply a tool to help solve certain classes of problems.

The textbook has the following goals:

1. To introduce many aspects of programming and problem-solving processes including problem specification and organization, algorithm development, coding, testing and debugging, and documentation of method and results.

2. To demonstrate what is currently considered good programming style.

3. To teach the syntax and semantics of a reasonable subset of a modern dialect of the Pascal language.

4. To show how elegantly some mathematical problems can be expressed in a precise algorithmic language, and also finally solved on a digital computer.

5. To allow students to use an experimental approach to computer mathematics by providing examples for each major algorithm both in the text and on diskette. The supplied programs can be run as is, or modified to illustrate different points or examples. Indeed, a number of the exercises require students to extend or change a program in the text.

The emphasis throughout is on mathematics, and not programming, however, it will be appreciated by teachers that a proper treatment of mathematical material in a computing context requires the same attention to detail and accuracy that they would customarily use in other mathematics units. With this in mind, we have included four chapters on the nature of computers and information in general, with some treatment of the Pascal language. The general approach of the book is to draw on topics from many areas of mathematics. Some brief revision of key concepts is presented, but in general, no attempt has been made to 're-teach' basic material. In a number of cases, we have endeavoured also to use the computer to give further insight into some important theoretical concepts. On the whole, however, it has been assumed that students are familiar with some of the basic principles of trigonometry, polynomials, differential calculus, functional notation, and coordinate geometry.

The Pascal language was chosen in preference to the ubiquitous but archaic BASIC because of the very good teaching tools such as Turbo Pascal that are now becoming available on IBM and compatible machines at very reasonable prices. These products serve as excellent vehicles for the illustration of principles of both computer mathematics and sound programming techniques. Further, the recent release of powerful document processing products such as Lotus Manuscript (on which this work was developed), and mathematical typesetting software (such as Donald Knuth's TEX), have made possible the creation of mathematical equations on a computer. Only those parts of Pascal which are necessary to properly develop the programs relevant to the syllabus are covered in this text. Of course, teachers may extend the coverage of the language as they see fit; however our aim has been to present a minimal, but reasonably thorough treatment of the Pascal language which is adequate for the development and coding of mathematical algorithms. No peculiarities of the Turbo Pascal dialect are used, however some graphics support and other routines provided in Turbo Pascal have been used to advantage. For the most part, students should be able to run the programs supplied on the optional diskette with little or no changes. We have planned the book with a view to minimising the tedium that is so often associated with teaching a subject involving some elements of practical computing. In order to save the time needed to key the programs, these are supplied on the optional examples diskette, and solutions to selected exercises (marked * in the text) also available.

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