• Journal of the Korean Mathematical Society
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• v.48 no.5
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• pp.1065-1081
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• 2011

For any field F, a polynomial f $\in$ F[$x_1,x_2,{\ldots},x_k$] can be associated with a polytope, called its Newton polytope. If the polynomial f has integrally indecomposable Newton polytope, in the sense of Minkowski sum, then it is absolutely irreducible over F, i.e., irreducible over every algebraic extension of F. We present some results giving new integrally indecomposable classes of polygons. Consequently, we have some criteria giving many types of absolutely irreducible bivariate polynomials over arbitrary fields.

• Proceedings of the Computational Structural Engineering Institute Conference
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• 2007.04a
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• pp.513-518
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• 2007

A polygon-wise constant curvature natural element approximation is presented in this paper for the numerical implementation of the abstract Kirchhoff plate model. The strict continuity requirement in the displacement field is relaxed by converting the area integral of the curvatures into the boundary integral along the Voronoi boundary. Curvatures and bending moments are assumed to be constant within each Voronoi polygon, and the Voronoi-polygon-wise constant curvatures are derived in a selective manner for the sake of the imposition of essential boundary conditions. The numerical results illustrating the proposed method are also given.

• Communications of Mathematical Education
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• v.37 no.4
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• pp.635-652
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• 2023

This study conducts a comprehensive analysis of the curricular progression of the concepts and learning sequences of 'lines', specifically, 'line segments', 'straight lines', and 'rays', at the elementary school level. By examining mathematics curricula and textbooks, spanning from 2nd to 7th and 2007, 2009, 2015, and up to 2022 revised version, the study investigates the timing and methods of introducing these essential geometric concepts. It also explores the sequential delivery of instruction and the key focal points of pedagogy. Through the analysis of shifts in the timing and definitions, it becomes evident that these concepts of lines have predominantly been integrated as integral components of two-dimensional plane figures. This includes their role in defining the sides of polygons and the angles formed by lines. This perspective underscores the importance of providing ample opportunities for students to explore these basic geometric entities. Furthermore, the definitions of line segments, straight lines, and rays, their interrelations with points, and the relationships established between different types of lines significantly influence the development of these core concepts. Lastly, the study emphasizes the significance of introducing fundamental mathematical concepts, such as the notion of straight lines as the shortest distance in line segments and the concept of lines extending infinitely (infiniteness) in straight lines and rays. These ideas serve as foundational elements of mathematical thinking, emphasizing the necessity for students to grasp concretely these concepts through visualization and experiences in their daily surroundings. This progression aligns with a shift towards the comprehension of Euclidean geometry. This research suggests a comprehensive reassessment of how line concepts are introduced and taught, with a particular focus on connecting real-life exploratory experiences to the foundational principles of geometry, thereby enhancing the quality of mathematics education.