Maple Transactions最新文献

{"title":"Difference equation satisfied by the Stieltjes transform of a sequence","authors":"V. Pillwein, D. Dominici","doi":"10.5206/mt.v2i1.14445","DOIUrl":"https://doi.org/10.5206/mt.v2i1.14445","url":null,"abstract":"We study a class of generating functions related to the Stieltjes transform of a sequence of moments with respect to the basis of falling factorial polynomials. Given a recurrence relation for the coefficient sequence, it is shown how to compute the difference equation satisified by its generating function w.r.t. this basis. We give several examples from the class of discrete semiclassical orthogonal polynomials.","PeriodicalId":355724,"journal":{"name":"Maple Transactions","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126599809","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"MYMathApps Calculus: Maple Plots","authors":"Philip Yasskin","doi":"10.5206/mt.v2i1.14436","DOIUrl":"https://doi.org/10.5206/mt.v2i1.14436","url":null,"abstract":"I am writing an online Calculus text called MYMathApps Calculus. You can see a sample at           https://mymathapps.com/mymacalc-sample/The text is highly interactive and visual. Nearly all of the graphics have been made with Maple, both 2D and 3D, static and animated. The use of plots and animated plots helps students understand concepts such as \u0000 \u0000the definitions of a derivative as the limit of slopes of secant lines, an integral as limits of Riemann sums, partial derivatives as slopes of traces, curvature and torsion, tangential and normal acceleration, divergence and curl, multiple integrals, curvilinear coordinates and Jacobians. \u0000the proofs of the triangle inequality, the mean value theorem and formulas for applications of integrals. \u0000plotting functions, polar curves, and parametric curves and surfaces. \u0000solving applied problems involving linear approximation, related rates, max/min, area, arc length, surface area, volumes by slicing, volumes of revolution, work, mixing problems, geometric series, Taylor series, directional derivatives, Lagrange multipliers, expansion and circulation. \u0000how to use the right hand rule in Green’s, Stokes’ and Gauss’ theorems. \u0000","PeriodicalId":355724,"journal":{"name":"Maple Transactions","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125187540","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A Maple implementation of a modular algorithm for computing the common zeros of a polynomial and a regular chain","authors":"Juan Pablo Gonzalez Trochez, M. Calder, Marc Moreno Maza, Erik Postma","doi":"10.5206/mt.v2i1.14448","DOIUrl":"https://doi.org/10.5206/mt.v2i1.14448","url":null,"abstract":"One of the core commands in the RegularChains library isTriangularize. The underlying decomposes the solution set of anpolynomial system into geometrically meaningful components representedby regular chains.  This algorithm works by repeatedly calling aprocedure, called Intersect, which computes the common zeros of apolynomial p and a regular chain T.As the number of variables of p and T, as well as their degrees,increase, the call Intersect(p, T) becomes more and morecomputationally expensive. It was observed in (C. Chen an M. MorenoMaza, JSC 2012) that when the input polynomial system iszero-dimensional and T is one-dimensional then this cost can besubstantially reduced. The method proposed by the authors is aprobabilistic algorithm based on evaluation and interpolationtechniques. This is the type of method which is typically challengingto implement in a high-level language like Maple's language, as asharp control of computing resources (in particular memory) is needed.In this paper, we report on a successful Maple implementation of thisalgorithm.  We take advantage of Maple's modp1 function which offersfast arithmetic for univariate polynomials over a prime field.The method avoids unlucky specialization and the probabilistic aspectonly comes from the fact that non-generic solutions are notcomputed. ","PeriodicalId":355724,"journal":{"name":"Maple Transactions","volume":"534 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133110942","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Another Famous Unsolved Problem: Improving Diversity in STEM","authors":"Bryon Thur","doi":"10.5206/mt.v2i1.15198","DOIUrl":"https://doi.org/10.5206/mt.v2i1.15198","url":null,"abstract":"A report on the panel of the same name at the Maple Conference.","PeriodicalId":355724,"journal":{"name":"Maple Transactions","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116727121","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Maple Conference 2021 Mathematical Art and Creative Works Exhibit","authors":"J. May","doi":"10.5206/mt.v2i1.15001","DOIUrl":"https://doi.org/10.5206/mt.v2i1.15001","url":null,"abstract":"The report from the curator on the first Maple Art and Creative Works Exhibit that took place as part of the2021 Maple Conference","PeriodicalId":355724,"journal":{"name":"Maple Transactions","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123676263","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Substituting Units into Multivariate Power Series","authors":"Erik Postma, Marc Moreno Maza","doi":"10.5206/mt.v2i1.14469","DOIUrl":"https://doi.org/10.5206/mt.v2i1.14469","url":null,"abstract":"Substituting non-unit formal power series into formal power series is a well-understood concept. This article describes a sense in which one can define substitution of unit formal power series, implemented in Maple’s MultivariatePowerSeries package.","PeriodicalId":355724,"journal":{"name":"Maple Transactions","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128134283","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Exploring the Mysteries of Babylonian Astronomy with Maple","authors":"Douglas W. MacDougal","doi":"10.5206/mt.v2i1.14357","DOIUrl":"https://doi.org/10.5206/mt.v2i1.14357","url":null,"abstract":"Modern scholarship asserts that the Babylonians were able to determine the synodic periods (discussed below) of the outer planets with exceptional accuracy. We experimented with a simple Maple mathematical software model to see if we could mathematically re-create the synodic cycles of Mars determined by Babylonian astronomers starting from around 400 BC. We sought first to understand whether unique planetary cycles known as Goal-Year periods which the Babylonians gleaned from centuries of observation were mathematically inevitable. We wanted to determine their accuracy quantitatively and find out if other plausible Goal-Year choices were available to them. The Babylonians also invented a method of using Goal-Years in error-canceling combinations to create long-term ‘exact’ periods that markedly improved the accuracy of their planetary predictions for centuries ahead. We tested whether the mathematical method surmised by scholars made sense in terms of our own program and whether we could replicate it. This paper is a report on the success of those efforts.","PeriodicalId":355724,"journal":{"name":"Maple Transactions","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132119089","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Ten Tips for Successful Creation of Contextualized Problems for Secondary School Students with Maple","authors":"M. Sacchet","doi":"10.5206/mt.v2i1.14446","DOIUrl":"https://doi.org/10.5206/mt.v2i1.14446","url":null,"abstract":"Students at all levels of schooling in all countries of the world need to practice mathematical problem solving to develop competencies that they will apply in real-life scenarios. On the other hand, concerning solving, problem posing refers to both the generation of new problems and the re-formulation of given problems. Teaching mathematics from a problem posing and problem-solving perspective entails more than solving non-routine problems or typical textbook types of problems. It is a way for students to exercise all aspects of problem solving: exploring, conjecturing, examining, testing, and generalizing. Tasks should be accessible and extend students’ knowledge. Even students should formulate problems from given situations and create new problems by modifying the conditions of a given problem. The quality of problems submitted to students is an issue that needs to be carefully considered. This work presents different ways to apply good practices when designing a problem-solving activity with students. It is based on the experience of Digital Math Training, a project whose aim is to develop and strengthen Mathematics and Computer Science skills through problem solving activities using the Advanced Computing Environment (ACE) Maple. After initial training in the laboratories of the schools, 3 students per class - the most skilled or motivated ones - participate in online training. They are asked to solve a problem every 10 days and to submit their solution. Meanwhile, students can participate in weekly synchronous tutoring on the use of Maple and collaborate with their colleagues through forum discussions. Students are selected in an intermediate competition and a final one. In this setting it is important to carefully plan and present the activity to the students, the text of the problem should be clear, and concise, with little storytelling to enter the setting of the problem. The problems should not be too theoretical, although they may inspect specific aspects of the related theory. They need to be solved by starting with simpler requests until the most difficult ones, close to the edge of students' knowledge, are reached. The solutions can make use of a calculator, in our case the ACE Maple. Precise design principles are based on both the adoption of suitable practice and the use of Maple. These principles can be adapted to different situations. The paper describes all these features with examples, according to the literature.","PeriodicalId":355724,"journal":{"name":"Maple Transactions","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131150938","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Numerical investigation of structural minimality for structures of uncontrolled linear switching systems with Maple","authors":"J. Whyte","doi":"10.5206/mt.v2i1.14385","DOIUrl":"https://doi.org/10.5206/mt.v2i1.14385","url":null,"abstract":"One path to understanding a physical system is to represent it by a model structure (collection of related models). Suppose our system is not subject to external influences, and depends on unobservable state variables (x), and observables (y). Then, a suitable uncontrolled, state-space model structure S is defined by relationships between x and y, involving parameters θ ∈ Θ. That is, each parameter vector in parameter space Θ is associated with a particular model in S. \u0000Before using S for prediction, we require system observations for parameter estimation. This process aims to determine θ values for which predictions “best” approximate the data (according to some objective function). The result is some number of estimates of the true parameter vector, θ*. Multiple parameter estimates are problematic when these cause S to produce dissimilar predictions beyond our data's range. This can render us unable to confidently make predictions, resulting in an uninformative study. \u0000Non-uniqueness of parameter estimates follows when S lacks the property of structural global identifiability (SGI). Fortunately, we may test S for SGI prior to data collection. The absence of SGI encourages us to rethink our experimental design or model structure. \u0000Before testing S for SGI we should check that it is structurally minimal. If so, we cannot replace S by a structure of fewer state variables which produces the same output. Most testing methodology is applicable to structures which employ the same equations for all time. These methods are not appropriate when, for example, a process has an abrupt change in its dynamics. For such a situation, a structure of linear switching systems (LSSs) may be suitable. Any system in the structure has a collection of linear time-invariant state-space systems, and a switching function which determines the system in effect at each instant. As such, we face a novel challenge in testing an LSS structure for SGI. \u0000We will consider the case of an uncontrolled LSS structure of one switching event (a ULSS-1 structure). In this setting, we may approach the structural minimality problem via the Laplace transform of the output function on each time interval. Each rational function yields conditions for pole-zero cancellation. If these conditions are not satisfied for almost all θ ∈ Θ, then S is structurally minimal. \u0000Analytical approaches can be quite laborious. However, we may expect a numerical approach to provide useful insights quickly. For example, if pole-zero cancellation occurs for almost all of a sufficiently large number of parameter values sampled from Θ, then structural minimality is possible. This result may encourage us to prove the existence of structural minimality. \u0000We shall use Maple 2020-2 to conduct a numerical investigation of structural minimality for a test case ULSS-1 structure applicable to flow-cell biosensor experiments used to monitor biochemical interactions, which include the popular Biacore-branded units.","PeriodicalId":355724,"journal":{"name":"Maple Transactions","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133677317","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Symbolic Analysis of Linear Amplifiers with Multi-Loop Feedbacks in Interacting Programs Maple and FASTMEAN","authors":"Yurova Valentina, Filin Vladimir","doi":"10.5206/mt.v2i1.14418","DOIUrl":"https://doi.org/10.5206/mt.v2i1.14418","url":null,"abstract":"A technique of interaction of computer programs for symbolic analysis of complex electronic circuits with amplifying elements is proposed. FASTMEAN simulation program, used in the universities of telecommunications in Russia, has a symbolic analysis module and is capable of generating analytical expressions for Laplace images of a circuit determinant, currents and voltages in complex electronic circuits. However, the obtained expressions have a nested (folded) structure, which makes it difficult to analyze the influence of elements on the properties of a circuit with amplifiers and feedbacks, in particular on its stability. It is proposed to transfer the obtained expressions to  Maplе program for their structural transformation and mathematical processing.                       .          Amplifiers with local, common and crossed feedbacks are considered. The analysis of such circuits in Maple shows that an expression for the circuit determinant in the form of the products of the loop gain functions is a sign of the presence of several feedback loops in the circuit.","PeriodicalId":355724,"journal":{"name":"Maple Transactions","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125079344","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}