Order and chaos: interactive computational activities for the classroom

It has long been believed that typical students learn better through contemporary approaches to questions originated by physics problems that allow experiments. This belief motivated us to develop interactive computational didactic materials about contemporaneous mathematics that can be used both in the classroom and in mathematics clubs in school. Dynamical Systems, the study of how physical systems evolve with time, inspired the activities developed. They share a key goal of understanding the order/chaos relationship in natural phenomena, human behaviour and social systems. Another goal to achieve is to give mathematics an experimental/laboratorial component, which rarely is present. In fact, all the interactive computational didactic materials developed include simulations and the capability to generate wonderful pictures, from which students can enjoy the beauty of mathematics.The first author was supported by the Centre of Research in Mathematics and Applications, University of Évora, through the FCT Pluriannual Funding Program. The second author was supported by FEDER Funds through "Programa Operacional Factores de Competitividade – COMPETE" and by Portuguese Funds through FCT -"Fundação para a Ciência e a Tecnologia", within the Project PEst-C/MAT/UI0013/2011

Basins of attraction for a quadratic coquaternionic map

In this paper we consider the extension, to the algebra of coquaternions, of a complex quadratic map with a real super-attractive 8-cycle. We establish that, in addition to the real cycle, this new map has sets of non-isolated periodic points of period 8, forming four attractive 8-cycles. Here , the cycles are to be interpreted as cycles of sets and an appropriate notion of attractivity is used. Some characteristics of the basins of attraction of the five attracting 8-cycles are discussed and plots revealing the intertwined nature of these basins are shown.Research at CMAT was financed by Portuguese Funds through FCT - Funda¸c˜ao para a Ciˆencia e a Tecnologia, within the Project UID/MAT/00013/2013. Research at NIPE has been carried out within the funding with COMPETE reference number POCI-01- 0145-FEDER-006683, with the FCT/MEC’s (Funda¸c˜ao para a Ciˆencia e a Tecnologia, I.P.) financial support through national funding and by the ERDF through the Operational Programme on “Competitiveness and Internationalization – COMPETE 2020” under the PT2020 Partnership Agreement.info:eu-repo/semantics/publishedVersio

Quaternionic polynomials with multiple zeros: A numerical point of view

In the ring of quaternionic polynomials there is no easy solution to the problem of finding a suitable definition of multiplicity of a zero. In this paper we discuss different notions of multiple zeros available in the literature and add a computational point of view to this problem, by taking into account the behavior of the well known Newton's method in the presence of such roots.COMPETE reference number POCI-01-0145-FEDER-006683; FCT/MEC's financial support through national funding and by the ERDF through the Operational Programme on ``Competitiveness and Internationalization -- COMPETE 2020" under the PT2020 Partnership Agreementinfo:eu-repo/semantics/publishedVersio

Remarks on the zeros of quadratic coquaternionic polynomials

In this paper we focus on the study of monic quadratic polynomials whose coefficients are coquaternions and present several new results concerning the number and nature of its zeros. Examples specially constructed to illustrate the diversity of cases that can occur are also presented.FCT - Fundação para a Ciência e a Tecnologia(UIDB/00013/2020, UIDP/00013/2020 and UIDB/03182/2020

A modified quaternionic Weierstrass method

In this paper we focus on the study of monic polynomials whose coefficients are quaternions located on the left-hand side of the powers, by addressing three fundamental polynomial problems: factor, evaluate and deflate. An algorithm combining a deflaction procedure with a Weierstrass-like quaternionic method is presented. Several examples illustrate the proposed approach.FCT - Fundação para a Ciência e a Tecnologia(UIDB/00013/2020, UIDP/00013/2020 and UIDB/03182/2020

Fixed points for cubic coquaternionic maps

This paper deals with the dynamics of a special two-parameter family of coquaternionic cubic maps. By making use of recent results for the zeros of one-sided coquaternionic polynomials, we analytically determine the fixed points of these maps. Some numerical examples illustrating the theory are also presented. The results obtained show an unexpected richness for the dynamics of cubic coquaternionic maps when compared to the already studied dynamics of quadratic maps.FCT - Fundação para a Ciência e a Tecnologia(UIDB/00013/2020, UIDP/00013/2020, UIDB/03182/2020

The stability of complex dynamics for two families of coquaternionic quadratic polynomials

In this work, we begin by demonstrating that attractors, both periodic and aperiodic, of the one-parameter family of complex quadratic maps x2+ c, where c is a complex number, maintain their stability when we transition from the complex plane C to the coquaternions Hcoq as the map’s phase space. Next, we investigate the same question for a different family of quadratic maps, x2+ bx, and find that this is not the case. In fact, the situation for this family of maps turns out to be quite complicated. Our results show that there are complex attractors that undergo changes in their stability, while others maintain it. However, the most intriguing result is that certain regions of the parameter space, known as bulbs, which correspond to the existence of attracting cycles of some fixed period n, exhibit a mixture of stability behavior when we consider coquaternionic quadratics.FCT -Fundação para a Ciência e a Tecnologia(UIDB/00013/2020

A Two-step Quaternionic Root-finding Method

In this paper we present a new method for determining simultaneously all the simple roots of a quaternionic polynomial. The proposed algorithm is a two-step iterative Weierstrass-like method and has cubic order of convergence. We also illustrate a variation of the method which combines the new scheme with a recently proposed deflation procedure for the case of polynomials with spherical roots.FCT -Fundação para a Ciência e a Tecnologia(UIDB/00013/2020

Mathematica tools for coquaternions

Coquaternions form a four dimensional real algebra generalizing complex numbers and were introduced by James Cockle at about the same time that Hamilton discovered the famous algebra of quaternions. Although not as popular as quaternions, in recent years one can observe an emerging interest among mathematicians and physicists on the study of these numbers. In this work we revisit a Mathematica package for implementing the algebra of coquaternions - Coquaternions - and discuss a set of Mathematica functions - CoqPolynomi al - to deal with coquaternionic polynomials. These two sets of functions provide the basic tools necessary for manipulating coquaternions and unilateral coquaternionic polynomials, reflecting, in its present form, the recent interests of the authors in the area.Supported by FCT - Fundacao para a Ciencia e a Tecnologia, within the Projects UIDB/00013/2020, UIDP/00013/2020 and UIDB/03182/2020

Viagem à Índia para prospecção de tecnologias sobre mamona e pinhão manso.

Viagem á Índia para prospecção de tecnologias sobre mamona e pinhão manso ; Introdução; Roteiro; Detalhamento; Sardar Krushinagar Dantiwada University; Pesquisas com mamona; Pesquisas com pinhão manso; Outras oleaginosas ; Indústria de extração de óleo de mamona ; Mercado de leilões para comercialização de produtos agrícolas ; Mini-usina de biodiesel de pinhão manso; 4º Seminário Internacional sobre semente e óleo de mamona e produtos de valor agregado ; Visita à Secretaria de Agricultura do Estado de Tamil Nadu ; Projeto MGR jatropha biodiesel ; Loco Works - Empresa operadora de ferrovias ; Exposição sobre a mini-usina de biodiesel da Empresa Loco Works ; Considerações finais; Pinhão manso ; Mamona.bitstream/CNPA/18319/1/DOC153.pd