• Communications of Mathematical Education
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• v.29 no.3
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• pp.465-489
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• 2015

The purpose of this study is to suggest implications for the development and revision of future teaching materials for mathematically gifted students by using network text analysis of secondary mathematics materials. Subjects of the analysis were learning goals of 110 teaching materials in a gifted education center in science attached to a university from 2002 to 2014. In analysing the frequency of the texts that appeared in the learning goals, key words were selected. A co-occurrence matrix of the key words was established, and a basic information of network, centrality, centralization, component, and k-core were deducted. For the analysis, KrKwic, KrTitle, and NetMiner4.0 programs were used, respectively. The results of this study were as follows. First, there was a pivot of the network formed with core hubs including 'diversity', 'understanding' 'concept' 'method', 'application', 'connection' 'problem solving', 'basic', 'real life', and 'thinking ability' in the whole network from 2002 to 2014. In addition, knowledge aspects were well reflected in teaching materials based on the centralization analysis. Second, network text analysis based on the three periods of the Mater Plan for the promotion of gifted education was conducted. As a result, a network was built up with 'understanding', and there were strong ties among 'question', 'answer', and 'problem solving' regardless of the periods. On the contrary, the centrality analysis showed that 'communication', 'discovery', and 'proof' only appeared in the first, second, and third period of Master Plan, respectively. Therefore, the results of this study suggest that affective aspects and activities with high cognitive process should be accompanied, and learning goals' mannerism and ahistoricism be prevented in developing and revising teaching materials.

• Communications of Mathematical Education
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• v.37 no.1
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• pp.65-84
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• 2023

The method of Lagrange multipliers, one of the most fundamental algorithms for solving equality constrained optimization problems, has been widely used in basic mathematics for artificial intelligence (AI), linear algebra, optimization theory, and control theory. This method is an important tool that connects calculus and linear algebra. It is actively used in artificial intelligence algorithms including principal component analysis (PCA). Therefore, it is desired that instructors motivate students who first encounter this method in college calculus. In this paper, we provide an integrated perspective for instructors to teach the method of Lagrange multipliers effectively. First, we provide visualization materials and Python-based code, helping to understand the principle of this method. Second, we give a full explanation on the relation between Lagrange multiplier and eigenvalues of a matrix. Third, we give the proof of the first-order optimality condition, which is a fundamental of the method of Lagrange multipliers, and briefly introduce the generalized version of it in optimization. Finally, we give an example of PCA analysis on a real data. These materials can be utilized in class for teaching of the method of Lagrange multipliers.

• Ecology and Resilient Infrastructure
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• v.3 no.3
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• pp.207-211
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• 2016

This study focuses on the categorization of the phenomenon of vegetative recruitment on riparian channels, so called, the phenomenon from "white river" to "green river", and proposes for the corresponding research direction. According to the literature review and research outputs obtained from the authors' previous research performed in Korea within a limited scope, the necessary and sufficient conditions for the recruitment and retrogression of riparian vegetation may be the mechanical disturbance (riverbed tractive stress), soil moisture (groundwater level, topography, composition of riverbed material, precipitation etc.), period of submergence, extreme weather, and nutrient inflow. In this study, two categories, one for the reduction in spring flood due to the change in spring precipitation pattern in unregulated rivers and the other for the increase in nutrient inflow into streams, both of which were partially proved, have been added in the categorization of the vegetative recruitment and retrogression on the riparian channels. In order to scientifically investigate further the phenomenon of the riparian vegetative recruitment and retrogression and develop the working riparian vegetative models, it is necessary to conduct a systematic nationwide survey on the "white to green" rivers, establishment of the categorization of the vegetation recruitment and retrogression based on the proof of those hypotheses and detailed categorization, development of the working mathematical models for the dynamic riparian vegetative recruitment and retrogression, and adaptive management for the river changes.

• Journal of the Korean School Mathematics Society
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• v.14 no.3
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• pp.355-375
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• 2011

This paper was designed for the purpose of helping the functional comprehension on the concept of a circumcenter and an incenter of triangle and offering the help for teaching-learning process on their definitions. We analysed the characteristic of the definition on a circumcenter and an incenter of triangle and studied the context, mean and purpose on the definition. The definition focusing on the construction is the definition stressed on the consistency of the concept through the fact that it is possible to draw figure of the concept. And this definition is the thing that consider the extend of the concept from triangle to polygon. Meanwhile this definition can be confused because the concept is not connected with the terminology. The definition focusing on the meaning is easy to memorize the concept because the concept is connected with the terminology but is difficult to search for the concept truth. And this definition is the thing that has the grounds on the occurrence but is taught in a made-knowledge. The definition focusing on both the construction and meaning is the definition that the starting point is vague in the logical proof process. We hope that the results are used to improve the understanding the concept of a circumcenter and an incenter of triangle in the field of mathematical education.

• Kyungpook Mathematical Journal
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• v.61 no.4
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• pp.679-710
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• 2021

We study the tensor product algebra P_k(x₁) ⊗ P_k(x₂) ⊗ ⋯ ⊗ P_k(x_m), where P_k(x) is the partition algebra defined by Jones and Martin. We discuss the centralizer of this algebra and corresponding Schur-Weyl dualities and also index the inequivalent irreducible representations of the algebra P_k(x₁) ⊗ P_k(x₂) ⊗ ⋯ ⊗ P_k(x_m) and compute their dimensions in the semisimple case. In addition, we describe the Bratteli diagrams and branching rules. Along with that, we have also constructed the RS correspondence for the tensor product of m-partition algebras which gives the bijection between the set of tensor product of m-partition diagram of P_k(n₁) ⊗ P_k(n₂) ⊗ ⋯ ⊗ P_k(n_m) and the pairs of m-vacillating tableaux of shape [λ] ∈ Γ_k^m, Γ_k^m = {[λ] = (λ₁, λ₂, …, λ_m)|λ_i ∈ Γ_k, i ∈ {1, 2, …, m}} where Γ_k = {λ_i ⊢ t|0 ≤ t ≤ k}. Also, we provide proof of the identity $(n_1n_2{\cdots}n_m)^k={\sum}_{[{\lambda}]{\in}{\Lambda}^k_{{n_1},{n_2},{\ldots},{n_m}}}$ f^[λ]m_k^[λ] where m_k^[λ] is the multiplicity of the irreducible representation of $S{_{n_1}}{\times}S{_{n_2}}{\times}....{\times}S{_{n_m}}$ module indexed by ${[{\lambda}]{\in}{\Lambda}^k_{{n_1},{n_2},{\ldots},{n_m}}}$, where f^[λ] is the degree of the corresponding representation indexed by ${[{\lambda}]{\in}{\Lambda}^k_{{n_1},{n_2},{\ldots},{n_m}}}$ and ${[{\lambda}]{\in}{\Lambda}^k_{{n_1},{n_2},{\ldots},{n_m}}}=\{[{\lambda}]=({\lambda}_1,{\lambda}_2,{\ldots},{\lambda}_m){\mid}{\lambda}_i{\in}{\Lambda}^k_{n_i},i{\in}\{1,2,{\ldots},m\}\}$ where ${\Lambda}^k_{n_i}=\{{\mu}=({\mu}_1,{\mu}_2,{\ldots},{\mu}_t){\vdash}n_i{\mid}n_i-{\mu}_1{\leq}k\}$.