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Cloaking and channelling of elastic waves in structured solids
Published in A. B. Movchan, N. V. Movchan, I. S. Jones, D. J. Colquitt, Mathematical Modelling of Waves in Multi-Scale Structured Media, 2017
A. B. Movchan, N. V. Movchan, I. S. Jones, D. J. Colquitt
Whilst cloaking via transformation geometry has been extensively treated in the literature, the sensitivity of the cloaking effect to the boundary conditions is rarely discussed. The cloak is formed by deforming a small region (a point in the case of the classical radial transformation [158]), into a larger finite region. If the region is an inclusion, then the natural interface conditions may be determined following the method outlined in Section 5.2.1.1 and 5.2.1.2. If the cloaked region is a void or rigid inclusion, however, there is some freedom in choosing the boundary condition, subject to the constraints of the physical problem. Figure 5.12 shows the field u(x) for a cloaked void, with Neumann (parts (a) and (b)) and Dirichlet (parts (c) and (d)) conditions applied to the interior of the cloaked region. The corresponding scattering measures are shown in Table 5.4.
Geometry and geospatial data on the web
Published in Pieter Pauwels, Kris McGlinn, Buildings and Semantics, 2023
Anna Wagner, Mathias Bonduel, Jeroen Werbrouck, Kris McGlinn
With OMG and FOG concerning relations between non-geometric objects and their geometry descriptions together with their geometry format, metadata is omitted so far. Therefore, the Geometry Metadata Ontology (GOM)15 has been introduced [44]. GOM aims to extend functionalities of the OMG/FOG methodology to depict additional geometric information regarding coordinate systems and transformation, geometry representation context, represented accuracy and other geometry metadata.
Stresses in Membrane Elements
Published in T. C. Hsu Thomas, Unired Theory of Reinforced Concrete, 2017
The relationship between the rotating 2-1 axes and the stationary l-t axes is shown by the transformation geometry in Figure 4.1c. A positive unit length on the l axis will have projections of cos α and – sin α on the 2 and 1 axis, respectively. A positive unit length on the t axis, however, should give projections of sin α and cos α. Hence, the rotation matrix [R] is
Pre-service mathematics teachers’ beliefs: a quantitative study to investigate the complex relationships in their beliefs
Published in International Journal of Mathematical Education in Science and Technology, 2023
P.M. Labulan, Darma Andreas Ngilawajan, Adi Nur Cahyono, Zetra Hainul Putra, Sadrack Luden Pagiling, Benjamin Rott
All participants are PSTs prepared for teaching mathematics in secondary schools. Although they come from five different universities, only a few subjects in the curricula of the five universities are different. Most subjects are similar. For example, related to mathematics, all five universities programme Basic Algebra, Linear Algebra, Abstract Algebra, Basic Geometry, Analytical Geometry, Transformation Geometry, Trigonometry, Differential and Integral Calculus, Differential Equation, Number Theory, Logica, Linear Programme, Real analyses, and Numerical Method. Related to mathematics education, all five universities also programme many similar subjects, for example, Mathematics Learning Strategies, Mathematics Learning Media, Mathematics Curriculum Reviews, and Student Development. The similarity of the curricula occurs because almost all mathematics education departments in Indonesian universities join the association for mathematics education, namely Indonesian Mathematics Educator Society (I-MES, https://i-mes.org/).
An assessment of geometry teaching supported with augmented reality teaching materials to enhance students’ 3D geometry thinking skills
Published in International Journal of Mathematical Education in Science and Technology, 2020
Emin İbili, Mevlüt Çat, Dmitry Resnyansky, Sami Şahin, Mark Billinghurst
The thinking ability about the manipulation of 3D shape representations by students who participating in the AR-supported learning environment were found to be significantly compared to the control group. According to this result, it can be said that AR-supported geometry teaching is effective in increasing this student skill. AR applications have enabled 3D materials to display content from different perspectives and have enabled users to freely explore the information at their own pace [20]. Quintero et al. [13] emphasized the importance of AR-based visual elements for students’ development of 3D geometric thinking skills and understanding transitions between representations. Bruce and Hawes [50] stated that an activity-based geometry teaching with 3D manipulations can enhance the mental rotation skills of the students. Dixon [51] stated that teaching geometry with virtual manipulations is more effective, especially on reflection and rotation skills, which are subdivisions of transformation geometry, and that visualization skills can be enhanced through experience. Therefore, it can be said that AR-supported geometry teaching helps with better learning by understanding the appearance of 3D objects in different directions and seeing various examples [28,29]. In addition, the combination of virtual objects and real environments allows students to visualize complex spatial relationships and abstract concepts [27].
An instructional unit for prospective teachers’ conceptualization of geometric transformations as functions
Published in International Journal of Mathematical Education in Science and Technology, 2021
To establish the construct validity of the TGQ, two mathematics educators and one mathematician with a doctoral degree examined the first versions of the transformation geometry tasks with respect to the table of specifications. Moreover, they assessed the appropriateness of the tasks to the participants, the representativeness of the content presented in each task, the appropriateness of the test format such as clarity of directions and language, and quality of printing. After revising the TGQ tasks in line with the expert opinions, the TGQ was piloted. Based on the results of the pilot study, several revisions were made, including rewording some of the tasks to clarify their meaning, and dividing the tasks into sub-tasks. For instance, the reflection task (Task 1) was divided into three sub-tasks that ask prospective teachers to (i) define the reflection transformation, (ii) provide an example for reflection, and (iii) illuminate the properties of reflection by using the example that they give. Finally, the revised TGQ was reviewed again for content validity by two mathematics educators and one mathematician who were all holding a doctoral degree in their fields. Their feedback helped in finalizing the revisions. In this study, inter-rater reliability was used as an evidence for external reliability. All of the responses for the tasks in the pilot study were scored using a detailed rubric by the first author and a professor specialized in mathematics education. A few disagreements and discrepancies between the two experts were resolved and the TGQ was put into its final form. Moreover, Cronbach’s alpha reliability coefficient was computed to assess the internal consistency and found it to be 0.79.