Numerical Algorithm for Nonlinear Delayed Differential Systems of nnth Order

The purpose of this paper is to propose a semi-analytical technique convenient for numerical approximation of solutions of the initial value problem for $p$-dimensional delayed and neutral differential systems with constant, proportional and time varying delays. The algorithm is based on combination of the method of steps and the differential transformation. Convergence analysis of the presented method is given as well. Applicability of the presented approach is demonstrated in two examples: A system of pantograph type differential equations and a system of neutral functional differential equations with all three types of delays considered. Accuracy of the results is compared to results obtained by the Laplace decomposition algorithm, the residual power series method and Matlab package DDENSD. Comparison of computing time is done too, showing reliability and efficiency of the proposed technique.Comment: arXiv admin note: text overlap with arXiv:1501.00411 Author's reply: the text overlap may be caused by the fact that this article is concerning systems of equations, while the other paper was about single equation

Application of Differential Transform to Multi-Term Fractional Differential Equations with Non-Commensurate Orders

The differential transformation, an approach based on Taylor’s theorem, is proposed as convenient for finding an exact or approximate solution to the initial value problem with multiple Caputo fractional derivatives of generally non-commensurate orders. The multi-term differential equation is first transformed into a multi-order system and then into a system of recurrence relations for coefficients of formal fractional power series. The order of the fractional power series is discussed in relation to orders of derivatives appearing in the original equation. Application of the algorithm to an initial value problem gives a reliable and expected outcome including the phenomenon of symmetry in choice of orders of the differential transformation of the multi-order system

Stability of a Functional Differential System with a Finite Number of Delays

The paper is devoted to the study of asymptotic properties of a real two-dimensional differential system with unbounded nonconstant delays. The sufficient conditions for the stability and asymptotic stability of solutions are given. Used methods are based on the transformation of the considered real system to one equation with complex-valued coefficients. Asymptotic properties are studied by means of Lyapunov-Krasovskii functional. The results generalize some previous ones, where the asymptotic properties for two-dimensional systems with one or more constant delays or one nonconstant delay were studied

Capacity Building in Mathematics and Statistics Learning Support in Norway and the Czech Republic (MSLS Net)

This report describes the final meeting of the project "Capacity Building in Mathematics and Statistics Learning Support in Norway and the Czech Republic (MSLS Net)" held at the Tomas Bata University in Zlín, Czech Republic (June 12-14, 2023). Provision of mathematics and statistics learning support (MSLS) is developing rapidly in many parts of the world and activity in Norway and the Czech Republic has been accelerated significantly through this EEA Grants funded project. Representatives of each of the five partner institutions worked on creating a summary of good practices in tutor training, designing learning resources, and in delivering, monitoring and evaluation of mathematics and statistics support. Provision varied considerably across the institutions and the centres represented demonstrated diverse and innovative ways in which mathematics support is evolving. Outputs from the project include a Handbook on good practice and a booklet concerned with mathematics support centre tutor training, including pedagogic training and learning resources for the development of the tutors as described below. Finally, consideration turned to the value of establishing a professional network to continue this important work. The report will be relevant to other international groups interested in working in university level mathematics and statistics support

Inquiry in university mathematics teaching and learning: The PLATINUM project

This book reports on the work carried out within the Erasmus+ PLATINUM project by eight European universities from seven countries: the University of Agder, in Kristiansand, Norway—the coordinator of the project—the University of Amsterdam in The Netherlands, Masaryk University and Brno University of Technology in Czech Republic, Leibniz University Hannover in Germany, the Complutense University of Madrid in Spain, Loughborough University in the UK, and Borys Grinchenko Kyiv University in Ukraine. In this 21st century, projects aimed at studying and disseminating inquiry-based approaches in the teaching of STEM disciplines in primary and secondary education have proliferated in Europe, benefiting from the impulse of the publication of the Rocard’s report in 2007.1 However, university mathematics teaching has remained mainly traditional, especially in the first university years, crucial for the students’ orientation and retention

Inquiry in University Mathematics Teaching and Learning

The book presents developmental outcomes from an EU Erasmus+ project involving eight partner universities in seven countries in Europe. Its focus is the development of mathematics teaching and learning at university level to enhance the learning of mathematics by university students. Its theoretical focus is inquiry-based teaching and learning. It bases all activity on a three-layer model of inquiry: (1) Inquiry in mathematics and in the learning of mathematics in lecture, tutorial, seminar or workshop, involving students and teachers; (2) Inquiry in mathematics teaching involving teachers exploring and developing their own practices in teaching mathematics; (3) Inquiry as a research process, analysing data from layers (1) and (2) to advance knowledge inthe field. As required by the Erasmus+ programme, it defines Intellectual Outputs (IOs) that will develop in the project. PLATINUM has six IOs: The Inquiry-based developmental model; Inquiry communities in mathematics learning and teaching; Design of mathematics tasks and teaching units; Inquiry-based professional development activity; Modelling as an inquiry process; Evalutation of inquiry activity with students. The project has developed Inquiry Communities, in each of the partner groups, in which mathematicians and educators work together in supportive collegial ways to promote inquiry processes in mathematics learning and teaching. Through involving students in inquiry activities, PLATINUM aims to encourage students` own in-depth engagement with mathematics, so that they develop conceptual understandings which go beyond memorisation and the use of procedures. Indeed the eight partners together have formed an inquiry community, working together to achieve PLATINUM goals within the specific environments of their own institutions and cultures. Together we learn from what we are able to achieve with respect to both common goals and diverse environments, bringing a richness of experience and learning to this important area of education. Inquiry communities enable participants to address the tensions and issues that emerge in developmental processes and to recognise the critical nature of the developmental process. Through engaging in inquiry-based development, partners are enabled and motivated to design activities for their peers, and for newcomers to university teaching of mathematics, to encourage their participation in new forms of teaching, design of teaching, and activities for students. Such professional development design is an important outcome of PLATINUM. One important area of inquiry-based activity is that of “modelling” in mathematics. Partners have worked together across the project to investigate the nature of modelling activities and their use with students. Overall, the project evaluates its activity in these various parts to gain insights to the sucess of inquiry based teaching, learning and development as well as the issues and tensions that are faced in putting into practice its aims and goals