Using Computer Algebra Packages to Complement the Spreadsheet Construction of Binomial Option Trees: The Example of Mathcad

In this paper we show how the mathematical programming package Mathcad can be used to complement the construction of a binomial option tree in Excel. A binomial option tree is first constructed in Excel using standard spreadsheet 'cut and paste' operations. The same binomial tree is then constructed in Mathcad. We conclude that spreadsheet construction of the tree provides students with a sound concept of the underlying mechanics of the option tree. Additionally, the Mathcad construction reinforces the mathematical notation found in many option pricing texts (e.g. summation signs and indices) and allows for the construction of a more flexible lattice that may be easily altered (e.g. the number of steps). In the process students are provided with an understanding of how to construct option trees in the increasingly important world of computer algebra packages.

The visibility of models: using technology as a bridge between mathematics and engineering

Engineering mathematics is traditionally conceived as a set of unambiguous mathematical tools applied to solving engineering problems, and it would seem that modern mathematical software is making the toolbox metaphor ever more appropriate. We question the validity of this metaphor, and make the case that engineers do in fact use mathematics as more than a set of passive tools—that mathematical models for phenomena depend critically on the settings in which they are used, and the tools with which they are expressed. The perennial debate over whether mathematics should be taught by mathematicians or by engineers looks increasingly anachronistic in the light of technological change, and we think it is more instructive to examine the potential of technology for changing the relationships between mathematicians and engineers, and for connecting their respective knowledge domains in new ways

Оптимальний вибір площин, на яких розміщені томограми, в комп’ютерній томографії

The solution of the problem of reconstructing the internal structure of a three-dimensional body by the known tomograms produced by a computer to-mograph using interflatation of functions and blending approximation is proposed. The known methods ofapproximating functions of one and two variables by interpolation type piecewise constant splines using means and medians are also considered. The paper presents an algorithm for optimizing the choice of the planes in which the tomogramsproduced by a computer tomograph are placed. The case is considered when all the tomograms are parallel to each other. The algorithm developeduses approximations of objects by classical piecewise constant splines. The internal structure of a three-dimensional body (density or absorption coefficient) is assumed to be given by a function of three variables of the form h(x, y, z ) = f (x)g( y, z), where g is an arbitrary function, provided that f is a monotone function on a closed segment. The method of optimal choice of the planes for placing the tomograms is implemented using MathCad computer software.Представлено розв’язок задачі відновлення внутрішньої структури тривимірного тіла за відомими томограмами, що поступають з комп’ютерного томографу, за допомогою інтерфлетації функцій та мішаної апроксимації. Розглянуто також відомі методи наближення функцій однієї та двоx змінних кусково-сталими сплайнами інтерполяційного типу, з використанням середніх та медіан. В статті пропонується алгоритм оптимізації вибору площин, на яких розміщені томограми, що поступають з комп’ютерного томографу. Розглядається випадок, коли всі томограми паралельні одна одній. Запропонований алгоритм використовує наближення об’єктів класичними кусково-сталими сплайнами. При побудові алгоритму істотно використовується припущення про те, що внутрішня структура тривимірного тіла (щільність або коефіцієнт поглинання) є функцією від трьох змінних вигляду h(x, y, z ) = f (x)g( y, z), де g – довільна функція, при умові, що f – монотонна функція на замкненому відрізку. Представлена чисельна реалізація методу оптимального вибору площин, на яких лежать томограми, в системі компʼютерної математики MathCad

Empirical evaluation of accuracy of mathematical software used for availability assessment of fault-tolerant computer systems

Dependability assessment is typically based on complex probabilistic models. Markov and semi-Markov models are widely used to model dependability of complex hardware/software architectures. Solving such models, especially when they are stiff, is not trivial and is usually done using sophisticated mathematical software packages. We report a practical experience of comparing the accuracy of solutions stiff Markov models obtained using well known commercial and research software packages. The study is conducted on a contrived but realistic cases study of computer system with hardware redundancy and diverse software under the assumptions that the rate of failure of software may vary over time, a realistic assumption. We observe that the disagreement between the solutions obtained with the different packages may be very significant. We discuss these findings and directions for future research

New mathematical model for analysing three-phase controlled rectifier using switching functions

This paper is a postprint of a paper submitted to and accepted for publication in IET Power Electronics and is subject to Institution of Engineering and Technology Copyright. The copy of record is available at IET Digital Library.The aim of this study is to present a set of closed-form analytical equations in order to enable the computation of the three-phase bridge rectifier steady-state performance estimation. The proposed method presented in this study is a fast, accurate and effective mathematical model for analysing three-phase full-wave controlled rectifiers. The steady-state mathematical model is based on the derivation of an appropriate set of switching functions using the general switching matrix circuit (GSMC) techniques. Once the switching functions are derived, the output current, input current and output dc voltage can all be easily derived and generated from the application of this technique. The effect of overlap is accurately modelled and the distortion (notches), frequency content on the input (voltage and current) and output voltage distortion are derived. The proposed mathematical model, unlike conventional analytical methods, can be integrated in the design of active filters. Furthermore, the output voltage reduction, the rms, average and peak values of voltages and currents for the thyristors and any other semiconductor devices used are readily available for the designer by direct substitution into closed-form equations without any need for the waste of time for worst-case scenario simulations. This method can also be applied to other types of converters, specifically to all voltage fed power converters

Using the Mathcad Solver to Teach Portfolio Optimisation

This paper introduces a Mathcad program for teaching optimisation. The program, which involves a portfolio of equity shares, is less code oriented than GAMS and reinforces mathematical notation to a greater extent than the Excel solver. Also, the program adjusts automatically for alternative portfolios (other than the one presented) without the need for further programming. Thus, students can concentrate on examining alternative portfolios with little need to change parameters consistently.

Застосування MathCAD для визначення параметрів складної нелінійної моделі

Проведено дослідження щодо використання програми MathCAD для визначення параметрів складної нелінійної моделі. Розглянуто числовий приклад та здобуті результати обчислення параметрів моделі. Знайдені результати показали доцільність використання MathCAD для дослідження складних нелінійних моделей об’єктів до яких відносяться харчові продукти.The researches for application of the software MathCAD for determination of parameters of nonlinear composite model are conducted. A numerical example and the results of calculation of model parameters are got. The got results of the use of MathCAD for research of composite nonlinear models which food products behave to are reasonable

Design and finite element mode analysis of noncircular gear

The noncircular gear transmission is an important branch of the gear transmission, it is characterized by its compact structure, good dynamic equilibration and other advantages, and can be used in the automobile, engineering machine, ship, machine tool, aviation and spaceflight field etc. Studying on the dynamics feature of noncircular gear transmission can improve the ability to carry loads of, reduce the vibration and noise of, increase the life of the noncircular gear transmission machine, provides guidance for the design of the noncircular gear, and has significant theories and practical meanings. In this paper, the gear transmission technique is used to studied the design method of the noncircular gear, which contains distribution of teeth on the pitch curve, designs of the tooth tip curve and the tooth root curve, design of the tooth profile curve, the gear system dynamics principle is introduced to establish dynamics model for the noncircular gear; basic theory of finite element and mode analysis method are applied, finite element model for the noncircular gear is established, natural vibration characteristic of the noncircular gear is studied. And the oval gear is taken as an example, the mathematics software MathCAD, the 3D modeling software UG and the finite element software ABAQUS are used to realize precise 3D model of the oval gear. The finite element method is used, the natural vibration characteristic of the oval gear is studied, the main vibration types and natural frequencies of the oval gear and that of the equivalent cylindrical gears are analyzed and compared, the conclusions received reflect the dynamics performance of the oval gear, and solid foundation is laid for dynamics research and engineering application of the oval gear transmission