Computer Algebra System Maxima Research Articles

Solving plane stress problems using EsfPlano2D educational software

An educational computational application for analyzing plane stress problems oriented to engineering students called EsfPlano2D is presented to motivate students to study the subject and strengthen didactic strategies in the classroom. The application assists the step-by-step calculation of plane stress problems, which is a fundamental topic in teaching the subject of Mechanics of Materials in engineering careers. The application is a computer program developed in Fortran, linked to the free Computer Algebra System MAXIMA to plot Mohr’s circle of stresses. The developed tool is user-friendly and completely solves the plane stress problem, the associated deformations, and the strain energy by drawing Mohr’s circle of plane stresses. A validation case is presented with a typical example of the mechanics of materials. The results show the feasibility of using custom-designed tools to strengthen classroom learning and autonomous work.

Using a constraint-based expert model to provide step-level feedback for user-inputted mathematics equations

Studies across a variety of educational fields have shown the efficacy of feedback on student performance and learning. Web-based homework is a common feature of secondary and collegiate mathematics courses to provide such feedback. While web-based homework provides often instantaneous feedback to students as they complete assignments, the feedback is limited to the correctness of the final answer or to a limited number of pre-programmed common mistakes. Intelligent Tutoring Systems have the ability to provide step-by-step feedback and guidance tailored to individual students, however authoring content for such systems is generally considered a huge barrier to their wider use. This paper presents an alternative system, called SANYMS (Show and Name Your Math Steps), which combines the wide-scale usability of web-based homework with a Constraint-Based Intelligent Tutoring System Expert Model based on the open-source computer algebra system Maxima. With the integration of Maxima, SANYMS can provide step-by-step feedback to students constructing their own solutions to a wide variety of algebraic equations of their choice without the need for extensive content authoring inherent in many Intelligent Tutoring Systems.

Regularized Integral Representations of the Reciprocal Gamma Function

This paper establishes a real integral representation of the reciprocal Gamma function in terms of a regularized hypersingular integral along the real line. A regularized complex representation along the Hankel path is derived. The equivalence with the Heine’s complex representation is demonstrated. For both real and complex integrals, the regularized representation can be expressed in terms of the two-parameter Mittag-Leffler function. Reference numerical implementations in the Computer Algebra System Maxima are provided.

Rozrahunok dovgyh truboprovodiv u veb-interfejsi systemy kompjuternoi algebry MAXIMA

Веб-інтерфейс системи CAS MAXIMA й мобільний Інтернет дозволяють практично без обмежень використовувати весь спектр математичного апарату в учбовому процесі. Методика розрахунків типового учбово-го завдання базується на використанні аналітичних й чисельних алгоритмів із елементами програмування в системі CAS MAXIMA. Задіяна стандартна практика розрахунку тестового завдання за стандарт-ними формулами для перевірки результатів розрахунку вищеозначеними методами. Система CAS MAXIMA дозволяє на практичних заняттях вивільнити час для впровадження елементів моделювання й графічного представлення результатів [1 − 4]. Поява сучасних смартфонів та планше-тів у яких присутні системи комп’ютерної алгебри, дає змогу швидше і простіше розв’язати будь-яку задачу. Для цього необхідно лише знати формули розрахунку і вміти задавати їх в електронному вигляді. Розглянемо розрахунок на прикладі ста-ндартної задачі гідродинаміки-розрахунок довгих трубопроводів. Стандартний спосіб розрахунку полягає у розв’язанні системи рівнянь і її вирішенні. В системі комп’ютерної алгебри CAS MAXIMA система алгебраїчних рівнянь вирішується однією командою [5]. Система CAS MAXIMA дозволяє представити результати розв’язання задачі у графічному вигляді й провести візуалізацію варіантів вирішення задачі із елементами моделювання [6]. Висновок: вирішення задач в системі комп’ютерної алгебри CAS MAXIMA дозволяє розширити способи розв’язання учбових задач [7].

Clifford Algebra Implementations in Maxima

This tutorial focuses on the packages $\texttt{clifford}$ and $\texttt{cliffordan}$ for the computer algebra system Maxima. Maxima is the open source descendant of the first computer algebra system and features a rich functionality from a large number of shared packages. The Maxima language is based on the ideas of functional programming, which is particularly well suited for transformations of formal mathematical expressions. While $\texttt{clifford}$ implements Clifford algebras $C\ell_{p,q,r}$ of arbitrary signatures and order based on the elementary construction of Macdonald, $\texttt{cliffordan}$ features geometric calculus functionality. Using $\texttt{clifford}$ expressions containing geometric, outer and inner products can be simplified. Applications of $\texttt{clifford}$ and $\texttt{cliffordan}$ in linear algebra and calculus are demonstrated.

On finding and using identifiable parameter combinations in nonlinear dynamic systems biology models and COMBOS: a novel web implementation.

Parameter identifiability problems can plague biomodelers when they reach the quantification stage of development, even for relatively simple models. Structural identifiability (SI) is the primary question, usually understood as knowing which of P unknown biomodel parameters p 1,…, pi,…, pP are-and which are not-quantifiable in principle from particular input-output (I-O) biodata. It is not widely appreciated that the same database also can provide quantitative information about the structurally unidentifiable (not quantifiable) subset, in the form of explicit algebraic relationships among unidentifiable pi. Importantly, this is a first step toward finding what else is needed to quantify particular unidentifiable parameters of interest from new I–O experiments. We further develop, implement and exemplify novel algorithms that address and solve the SI problem for a practical class of ordinary differential equation (ODE) systems biology models, as a user-friendly and universally-accessible web application (app)–COMBOS. Users provide the structural ODE and output measurement models in one of two standard forms to a remote server via their web browser. COMBOS provides a list of uniquely and non-uniquely SI model parameters, and–importantly-the combinations of parameters not individually SI. If non-uniquely SI, it also provides the maximum number of different solutions, with important practical implications. The behind-the-scenes symbolic differential algebra algorithms are based on computing Gröbner bases of model attributes established after some algebraic transformations, using the computer-algebra system Maxima. COMBOS was developed for facile instructional and research use as well as modeling. We use it in the classroom to illustrate SI analysis; and have simplified complex models of tumor suppressor p53 and hormone regulation, based on explicit computation of parameter combinations. It’s illustrated and validated here for models of moderate complexity, with and without initial conditions. Built-in examples include unidentifiable 2 to 4-compartment and HIV dynamics models.

Teaching meaningful mathematics with the Computer Algebra System MAXIMA using the example of inequalities

The paper was originally motivated by the request to accentuate the meaningful contribution of inequalities in Mathematics Education. Additionally nationwide approved competences such as estimating come to the fore when organizing mathematical contents along some central Big Ideas. Not least the integration of computers enriches the reasonable discussion of inequalities by modern well accepted methodological principles. The freeware MAXIMA is used as Computer Algebra System (CAS) representatively.

Modelling the landing of a plane in a calculus lab

We exhibit a simple model of a plane landing that involves only basic concepts of differential calculus, so it is suitable for a first-year calculus lab. We use the computer algebra system Maxima and the interactive geometry software GeoGebra to do the computations and graphics.

107 数式処理システムMaximaを用いた効率の良い技術教育(技術教育・工学教育(II))

107 数式処理システムMaximaを用いた効率の良い技術教育(技術教育・工学教育(II))

Teaching reliability theory with the Computer Algebra System Maxima

The use of the Computer Algebra System Maxima as a teaching aid in an MSc module in Reliability Theory is described here. Extracts from student handouts are used to show how the ideas in Reliability Theory are developed and how they are intertwined with their applications implemented in Maxima. Three themes from the lectures are used to illustrate this: (1) Normal Approximations, (2) Markov Modelling, (3) Laplace Transform Techniques. It is argued that Maxima is a good tool for the task, since: it is fairly easy to learn & use; it is well documented; it has extensive facilities; it is available for any operating system; and, finally, it can be freely downloaded from the Web. Maxima proves to be a useful tool even for Reliability research for certain tasks. This latter feature provides a seamless link from teaching to research – an important feature in postgraduate education.

Assessing elementary algebra with STACK

This paper concerns computer aided assessment (CAA) of mathematics in which a computer algebra system (CAS) is used to help assess students' responses to elementary algebra questions. Using a methodology of documentary analysis, we examine what is taught in elementary algebra. The STACK CAA system, http://www.stack.bham.ac.uk/, which uses the CAS Maxima, is taken as a case study with which to test the implementation of the ideas developed in this paper. The general characteristics needed from a CAS for the application of computer aided assessment, or computer based learning is discussed. In particular the need for consistently implemented noun forms of elementary arithmetic operations, together with their traditional verb forms is identified.

Numerical analysis of higher order singularities in chemical process models

In this contribution, a tool is presented that allows the continuation of singularities of higher codimension for complex chemical process models. The tool is an extension of the process-modelling tool ProMoT. It allows creating augmented systems for simple zero eigenvalue bifurcation points. Parameter continuation of such points allows to produce different varieties for hysteresis, isolas, pitchforks or winged cusps singularities. Required higher order directional derivatives are obtained analytically via an interface to the computer algebra system Maxima.