In this paper, we study the emergence of different patterns that are formed on both static and growing domains and their bifurcation structure. One of these is the so-called Turing-Hopf morphogenetic mechanism. The reactive part we consider is of FitzHugh-Nagumo type. The analysis was carried out on a flat square by considering both fixed and growing domain. In both scenarios, sufficient conditions on the parameter values are given for the formation of specific space-time structures or patterns. A series of numerical solutions of the corresponding initial and boundary value problems are obtained, and a comparison between the resulting patterns on the fixed domain and those arising when the domain grows is established. We emphasize the role of growth of the domain in the selection of patterns. The paper ends by listing some open problems in this area.
The present work proposes the design of mathematical problems, which allow the adequate understanding of mathematical concepts of probability for its correct interpretation and later application in the resolution of probabilistic problems. For the development of this work we rely on the theory of didactic situations of Brousseau (1997) and Sadovsky (2005). We believe that new materials and didactic models of this type have great educational potential because they encourage the analysis and understanding of various probability problems (Panizza, 2003). Accurate communication between teachers and students in the approach, interpretation, resolution, and testing of probability problems is of vital importance. The software used for this purpose is the MATHEMATICA program, a tool that facilitates the writing of formulas and calculations, as well as the construction of graphs, through a friendly interface, facilitating the self-taught work of the student and encouraging the development of analysis skills and problem solving. We believe that these materials will contribute to the teaching and learning processes of probability at higher education levels.
En este ensayo se describen algunas regularidades morfológicas que existen en la arquitectura de las plantas. Con el propósito de emprender su modelación matemática, se estudian sus regularidades geométricas basadas en la sucesión de Fibonacci y la sección ́áurea. Se muestra el modelo morfogenético de Alan M. Turing y se presenta un modelo que reproduce la forma de una clase de cactáceas.
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