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Introduction to Numerical Analysis
Published in James P. Howard, Computational Methods for Numerical Analysis with R, 2017
Applied mathematics is sometimes considered as distinct from pure mathematics. However, applications of mathematics are based on the underlying pure mathematics. Often, numerical methods are straightforward implementations of the related pure mathematical solution for the problem. Because of this relationship, some problems are better understood after applying pure mathematics. Accordingly, when a function generates an error, it can sometimes be resolved through symbolic manipulation.
Interactive theorem provers for university mathematics: an exploratory study of students’ perceptions
Published in International Journal of Mathematical Education in Science and Technology, 2023
Interactive theorem provers have been in use in pure mathematics and computer science research since the 1960s with de Bruijn’s Automath prover (Bruijn’s, 1980), but they have just now started to make their way into the teaching of undergraduate mathematics and the first reports of their use are encouraging. Avigad (2019) reports the experience of a first-year module in logic that included Lean programming in the curriculum and observes how the use of Lean supported the students in the transition to the rigour of mathematics and helped them appreciate both the need for proof and the need for technical language (of mathematics and computing). Buzzard (2020) reports his experience of using Lean both as a research tool and for teaching a pure mathematics module in the first year of undergraduate mathematics. In educational research, Thoma and Iannone (2022) analyse written proofs by students who had or had not engaged with Lean programming and show how learning to use Lean can help students overcome some of the difficulties of their first encounter with proof, even when writing proofs with pen and paper. The analysis on aspects of students’ proof production included using the mathematical technical language, being able to divide a proof into goals and subgoals and generally writing proofs in a style that aligns with what mathematicians expect (for a discussion of mathematicians’ expectations of proof-writing see Lew & Mejía-Ramos, 2019).
Fractional-order SMC controller for mobile robot trajectory tracking under actuator fault
Published in Systems Science & Control Engineering, 2022
Minghuang Qin, Songyi Dian, Bin Guo, Xu Tao, Tao Zhao
However, these traditional sliding mode control methods also have many shortcomings, such as slow convergence speed. Thus, the fractional sliding mode has been developed. As a generalization of the traditional integer-order differentiation and integration to non-integer order, the fractional-order theory has been studied by scholars for 300 years as a pure mathematics theory. It is characterized by attenuating old data and storing new data, the data can be used more discriminatively, hence the fractional-order controller is more stable or at least as stable as the integer-order (J. Huang et al., 2014; F. M. Zaihidee et al., 2019), and has a faster response time. But unfortunately, no one has discovered its actual value until the recent decades and began to be widely applied in science and engineering disciplines.
What is the problem in problem-based learning in higher education mathematics
Published in European Journal of Engineering Education, 2018
The choice of the specific content to be included in the project report was guided by the problem statement. It is evident that the majority of the student reports consisted of mainly formal-axiomatic mathematics belonging to the third world of Tall (2013). This was also the case with the project in applied mathematics (Case A). The proofs were not derived from set-theoretical definitions using epsilon–delta notation but was nevertheless formal proofs using (or attempting to use) correct mathematical syntax. Epsilon–delta notation is not introduced until the third semester. Some of the knowledge presented was also in the embodied symbolic world seen in the many examples and figures. Considering the seven mathematical capabilities (OECD 2013), one also sees that to some extent they are all present in the report. In terms of communication, the groups use the report as a means of communicating in first instance to the examiners, but also to each other, thus preparing themselves for mathematical communication in their later professional life. They also do mathematising as they present a problem in a mathematical way and here create a model. Both reports use mathematical representation, reasoning and argument, and symbolic language during, for instance, proofs and other justification. They both devise strategies based on their problem statement and also use mathematical tools such as Matlab. Case A from applied mathematics can furthermore be argued to show that applied and pure mathematics does not need to be seen as opposites as the students were obviously able to both produce Tall’s (2013) third world of mathematics reasoning as well as apply mathematics. The part with pure mathematics took up the majority of the report as the students were in a mathematics programme. One might argue that engineering students might also embark on solving similar optimisation problems using perhaps more convergent methods not learning or describing the proofs of the algorithm but still able to apply the algorithms.