Inverse homogenization using the topological derivative

PurposeThe purpose of this study is to solve the inverse homogenization problem, or so-called material design problem, using the topological derivative concept.Design/methodology/approachThe optimal topology is obtained through a relaxed formulation of the problem by replacing the characteristic function with a continuous design variable, so-called density variable. The constitutive tensor is then parametrized with the density variable through an analytical interpolation scheme that is based on the topological derivative concept. The intermediate values that may appear in the optimal topologies are removed by penalizing the perimeter functional.FindingsThe optimization process benefits from the intermediate values that provide the proposed method reaching to solutions that the topological derivative had not been able to find before. In addition, the presented theory opens the path to propose a new framework of research where the topological derivative uses classical optimization algorithms.Originality/valueThe proposed methodology allows us to use the topological derivative concept for solving the inverse homogenization problem and to fulfil the optimality conditions of the problem with the use of classical optimization algorithms. The authors solved several material design examples through a projected gradient algorithm to show the advantages of the proposed method.

Pre-service Mathematics Teacher's Mathematical Connections in the Context of Problem-Solving About the Derivative

The mathematical connections made by five Pre-Service Mathematics Teachers (PSMTs) when solving problems on the derivative were analyzed. The conceptual framework used was the typology of intra-mathematical connections. For data collection, semi-structured interviews were conducted, and a questionnaire was designed, which included three tasks about of the derivative, which were analyzed through the thematic analysis method. The results showed that PSMTs made mathematical connections: meaning, different representations, procedural, part-whole, implication and feature. We identified that the PSMT's difficulties in establishing connections are caused by the meaning they have on the derivative concept, acquired in their received teaching and, if they are not attended, they can be reproduced in their future practice as in-service teachers.

Teaching Derivative Concept Using 6 Questions Cognitive Model

Derivative is one of the basic concepts of calculus, has an extremely rich practical background and wide application. However, "low exploration" of the limit thought in teaching and learning activities have caused difficulties in future advanced mathematics learning. The purpose of this study is to create a learning model that helps students understand calculus. The method in this research is using research and development method, designing a learning model using 6 questions cognitive model on derivative material. Based on the results of developing a new learning model in this study using the six questions cognitive model to explore mathematics from the six dimensions of "from where", "what ", "why", "how", "what if it changed" and "think about it", The teaching and learning activities when teaching derivative under the cultural background reflects the continuity, naturalness and order of teaching. The results of this study indicate that 6 question cognitive models can help students learn the basic concept of calculus. The results of this study can be implemented by teachers in schools to improve the quality of high school student learning

On an Interpreter’s Game with the Legal Text and Extratextual Factors in the Derivative Concept of Legal Interpretation

The aim of this paper is to show the derivative concept of legal interpretation from the point of view of the structure of limits of the juridical power. This structure includes the politico-legal culture, the legal text, the juridical culture, and personal factors, such as ethical and aesthetic judgements. These days, the derivative concept is the most influential Polish theory of legal interpretation. According to this concept the process of interpreting the law is a kind of a game between the legal text and extratextual factors, which are treated as extratextual limits of juridical power. On the one hand, the legal text does not determine the full meaning of the law, although it has great importance for it. On the other hand, the derivative concept precisely identifies certain others factors that are relevant for the content of law.

Students’ Understanding of the Derivative Concept in the Context of Mathematics for Economics

The derivative concept plays a major role in economics. However, its use in economics is very heterogeneous, sometimes inconsistent, and contradicts students’ prior knowledge from school. This applies in particular to the common economic interpretation of the derivative as the amount of change while increasing the production by one unit. Hence, in calculus courses for economics students, learners should acquire an understanding of the derivative that is mathematically acceptable and connected to their prior knowledge, but which also takes into account its practical use in economics. In this paper we first develop a theoretical model describing such an understanding of the derivative for economics students. We then present an exploratory study investigating the extent to which economics students have such an understanding after their calculus course. The results indicate that many of them might not have acquired this kind of understanding, in particular concerning the common economic interpretation of the derivative. The study furthermore yields possible gaps in students’ understanding and possible misconceptions.