• Korean Journal of Logic
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• v.9 no.1
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• pp.1-29
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• 2006

This paper investigates algebraic semantics for sKD and its extensions $sKD_\triangle$, $sKD\forall$, and $sKD\forall{_\triangle}$: sKD is a variant of the infinite -valued Kleene- Diense logic KD; $sKD_\triangle$ is the sKD with the Baaz's projection A; and $sKD\forall$ and $sKD\forall{_\triangle}$: are the first order extensions of sKD and $sKD_\triangle$, respectively. I first provide algebraic completeness for each of sKD and $sKD_\triangle$. Next I show that each $sKD\forall$ and $sKD\forall{_\triangle}$: is algebraically complete.

• Journal for History of Mathematics
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• v.15 no.2
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• pp.49-68
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• 2002

One of the characteristics of modem mathematics is to use algebra in every fields of mathematics. But we don't have the exact definition of algebra, and we can't clearly define algebraic thinking. In order to solve this problem, this paper investigate the history of algebra. First, we describe some of the features of proportional Babylonian thinking by analysing some problems. In chapter 4, we consider Greek's analytical method and proportional theory. And in chapter 5, we deal with Diophantus' algebraic method by giving an overview of Arithmetica. Finally we investigate Viete's thinking of algebra through his ‘the analytical art’. By investigating these history of algebra, we reach the following conclusions. 1. The origin of algebra comes from problem solving(various equations). 2. The origin of algebraic thinking is the proportional thinking and the analytical thinking. 3. The thing that plays an important role in transition from arithmetical thinking to algebraic thinking is Babylonian ‘the false value’ idea and Diophantus’ ‘arithmos’ concept.

• Communications of the Korean Mathematical Society
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• v.17 no.4
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• pp.693-708
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• 2002

We generalize algebraic results of Nielsen fixed point theory to Nielsen coincidence theory. We use the algebraic methods of D. Ferrario in Nielsen fixed point theory.

• School Mathematics
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• v.6 no.4
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• pp.411-433
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• 2004

Many researchers have reported that 7th graders have severe difficulties in using letters and algebraic expressions. This study investigated the way Microsoft Excel contributes to student's understanding of letters and algebraic expressions. For six hours through two weeks, four 7th grade students experienced various activities with Excel after school and both before and after the experimentation, the interviews to check their understanding was conducted. The results were as follows; First, after the experimentation, students used various letters to express formulas and recognized that letters represent not only some objects but also changing objects. Also they accepted that same objects could be represented by different letters and different objects could be represented by the same letters. Second, Excel improved students' abilities to discriminate variables and invariables in the problem and to find mathematical relationships among variables. And with Excel students could divide the whole calculation procedure into several steps in order to handle it more easily. Also, Excel made immediate numerical feedback possible and it made students express the calculation in a more formalized way than a paper and pencil environment did.

• School Mathematics
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• v.13 no.4
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• pp.581-598
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• 2011

Given the growing agreement that algebra should be taught in the early stage of the curriculum, considerable studies have been conducted with regard to early algebra in the elementary school. However, there has been lack of research on how to organize mathematic lessons to develop of algebraic reasoning ability of the elementary school students. This research attempted to gain specific and practical information on effective algebraic teaching and learning in the elementary school. An exploratory qualitative case study was conducted to the fourth graders. This paper focused on the associative law of the multiplication. This paper showed what kinds of activities a teacher may organize following three steps: (a) focus on the properties of numbers and operations in specific situations, (b) discovery of the properties of numbers and operations with many examples, and (c) generalization of the properties of numbers and operations in arbitrary situations. Given the steps, this paper included an analysis on how the students developed their algebraic reasoning. This study provides implications on the important factors that lead to the development of algebraic reasoning ability for elementary students.

• Journal of the Korea Institute of Information Security & Cryptology
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• v.15 no.1
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• pp.29-40
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• 2005

In this paper we consider the security of recently proposed software-orienred stram cipher HELIX, SCREAM, MUGI, and PANAMA against algebraic attacks. Algebraic attack is a key recovery attack by solving an over-defined system of multi-variate equations with input-output pairs of an algorithm. The attack was firstly applied to block ciphers with some algebraic properties and then it has been mon usefully applied to stream ciphers. However it is difficult to obtain over-defined algebraic equations for a given cryptosystem in general. Here we analyze recently proposed software-oriented stream ciphers by constructing a system of equations for each cipher. furthermore we propose three design considerations of software-oriented stream ciphers.

• Journal of the Korea Institute of Information and Communication Engineering
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• v.26 no.2
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• pp.181-186
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• 2022

Poisson-Boltzmann equation (PBE) is used to model problems arising from various disciplinary including bio-pysics and colloid chemistry. Therefore, to predict a numerical solution of PBE is an important issue. The authors proposed deep learning based methods to solve PBE while the computational time to generate finite element method (FEM) solutions were bottlenecks of the algorithms. In this work, we shorten the generation time of FEM solutions in two directions. First, we experimentally find certain penalty parameter in a bilinear form. Second, we applied algebraic multigrids methods to the algebraic system so that condition number is bounded regardless of the meshsize. In conclusion, we have reduced computation times to solve algebraic systems for PBE. We expect that algebraic multigrids methods can be further employed in various disciplinary to generate deep learning samples.

• The Mathematical Education
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• v.60 no.4
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• pp.409-429
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• 2021

This study investigated instructional methods for fractional division emphasizing algebraic thinking with sixth graders. Specifically, instructional elements for fractional division emphasizing algebraic thinking were derived from literature reviews, and the fractional division instruction was reorganized on the basis of key elements. The instructional elements were as follows: (a) exploring the relationship between a dividend and a divisor; (b) generalizing and representing solution methods; and (c) justifying solution methods. The instruction was analyzed in terms of how the key elements were implemented in the classroom. This paper focused on the fractional division instruction with problem contexts to calculate the quantity of a dividend corresponding to the divisor 1. The students in the study could explore the relationship between the two quantities that make the divisor 1 with different problem contexts: partitive division, determination of a unit rate, and inverse of multiplication. They also could generalize, represent, and justify the solution methods of dividing the dividend by the numerator of the divisor and multiplying it by the denominator. However, some students who did not explore the relationship between the two quantities and used only the algorithm of fraction division had difficulties in generalizing, representing, and justifying the solution methods. This study would provide detailed and substantive understandings in implementing the fractional division instruction emphasizing algebraic thinking and help promote the follow-up studies related to the instruction of fractional operations emphasizing algebraic thinking.

• Journal of Gifted/Talented Education
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• v.26 no.1
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• pp.211-230
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• 2016

The study aimed to investigate the characteristics of algebraic thinking of the mathematically gifted students and search for how to teach algebraic thinking. Research subjects in this study included 93 students who applied for a science gifted education center affiliated with a university in 2015 and previously experienced gifted education. Students' responses on an algebraic item of a creative thinking test in mathematics, which was given as screening process for admission were collected as data. A framework of algebraic thinking factors were extracted from literature review and utilized for data analysis. It was found that students showed difficulty in quantitative reasoning between two quantities and tendency to find solutions regarding equations as problem solving tools. In this process, students tended to concentrate variables on unknown place holders and to had difficulty understanding various meanings of variables. Some of students generated errors about algebraic concepts. In conclusions, it is recommended that functional thinking including such as generalizing and reasoning the relation among changing quantities is extended, procedural as well as structural aspects of algebraic expressions are emphasized, various situations to learn variables are given, and activities constructing variables on their own are strengthened for improving gifted students' learning and teaching algebra.

• Honam Mathematical Journal
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• v.28 no.1
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• pp.83-111
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• 2006

After introducing Tropical Algebraic Varieties, we give a polyhedral description of tropical hypersurfaces. Using TOPCOM and GAP, we show that there exist 59 types of two dimensional tropical quadric surfaces. We also show a criterion for a quadric hypersurface to be non-degenerate in terms of a tropical rank.