• Journal for History of Mathematics
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• v.15 no.1
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• pp.115-134
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• 2002

The increasing use of computers in mathematics and in mathematics education is strongly reflected in the teaching on Euclid geometry, in particular in the use of dynamic graphics software. This development has raised questions about the role of analytic proof in school geometry. One can sometimes find a proof which is rather more explanatory than the one commonly used. Because we, math educators are concerned with tile explanatory power of the proofs, as opposed to mere verification, we should devise ways to use dynamic software in the use of explanatory proofs.

• Honam Mathematical Journal
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• v.21 no.1
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• pp.19-33
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• 1999

In this paper we reproduce results on $Gr{\ddot{o}}bner$ fans of ideals following a paper by T. Mora and L. Robbiano [5], and introduce stable $Gr{\ddot{o}}bner$ fans of ideals introduced by D. Mall [4]. We make minor corrections for the clarification, simplify proofs and provide new proofs. At the end we give a description of $Gr{\ddot{o}}bner$ fans of toric ideals.

• Bulletin of the Korean Mathematical Society
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• v.47 no.1
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• pp.103-111
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• 2010

In the present paper, we give two new proofs for the necessary and sufficient condition $\alpha\leq1$ such that the function $x^{\alpha}[lnx-\psi(x)]$ is completely monotonic on (0,$\infty$).

• Bulletin of the Korean Mathematical Society
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• v.53 no.5
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• pp.1281-1289
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• 2016

Richard Stanley suggested the problem of finding combinatorial proofs of formulas for counting regions of certain hyperplane arrangements defined by hyperplanes of the form $x_i=0$, $x_i=x_j$, and $x_i=2x_j$ that were found using the finite field method. We give such proofs, using embroidered permutations and linear extensions of posets.

• Bulletin of the Korean Mathematical Society
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• v.56 no.2
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• pp.491-499
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• 2019

The problem of finding fast computing methods for Bernoulli numbers has a long and interesting history. In this paper, the author provides new proofs for two lacunary recurrence relations with gaps of length four and eight for the Bernoulli numbers. These proofs invoked the fact that the nth powers of ${\pi}^2$, ${\pi}^4$ and ${\pi}^8$ can be expressed in terms of the nth elementary symmetric functions.

• The Pure and Applied Mathematics
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• v.29 no.1
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• pp.31-35
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• 2022

In the note, by virtue of Abel's theorem and Abel's limit theorem in the theory of power series, the author provides three proofs for a sum of an alternating series involving central binomial numbers.

• Smart Media Journal
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• v.10 no.3
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• pp.48-53
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• 2021

The authentication process is a key step that should be used to verify that a user is legitimate, and it should be used to verify that a user is a legitimate user and grant access only to that user. Recently, two-factor authentication and OTP schemes are used by most applications to add a layer of security to the login process and to address the vulnerability of using only one factor for authentication, but this method also allows access to user accounts without permission. This is a known security vulnerability. In this paper, we propose a Zero Knowledge Proofs (ZKP) personal information authentication scheme based on a Smart Contract of a block chain that authenticates users with minimal personal information exposure conditions. This has the advantage of providing many security technologies to the authentication process based on blockchain technology, and that personal information authentication can be performed more safely than the existing authentication method.

• Communications of Mathematical Education
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• v.24 no.2
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• pp.325-344
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• 2010

The purpose of this study is to investigate what influences students' preferences on empirical and deductive proofs and find their relations. Although empirical and deductive proofs have been seen as a significant aspect of school mathematics, literatures have indicated that students tend to have a preference for empirical proof when they are convinced a mathematical statement. Several studies highlighted students'views about empirical and deductive proof. However, there are few attempts to find the relations of their views about these two proofs. The study was conducted to 47 students in 7~9 grades in the transition from empirical proof to deductive proof according to their mathematics curriculum. The data was collected on the written questionnaire asking students to choose one between empirical and deductive proofs in verifying that the sum of angles in any triangles is $180^{\circ}$. Further, they were asked to provide explanations for their preferences. Students' responses were coded and these codes were categorized to find the relations. As a result, students' responses could be categorized by 3 factors; accuracy of measurement, representative of triangles, and mathematics principles. First, the preferences on empirical proof were derived from considering the measurement as an accurate method, while conceiving the possibility of errors in measurement derived the preferences on deductive proof. Second, a number of students thought that verifying the statement for three different types of triangles -acute, right, obtuse triangles - in empirical proof was enough to convince the statement, while other students regarded these different types of triangles merely as partial examples of triangles and so they preferred deductive proof. Finally, students preferring empirical proof thought that using mathematical principles such as the properties of alternate or corresponding angles made proof more difficult to understand. Students preferring deductive proof, on the other hand, explained roles of these mathematical principles as verification, explanation, and application to other problems. The results indicated that students' preferences were due to their different perceptions of these common factors.

• Journal of the Chungcheong Mathematical Society
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• v.8 no.1
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• pp.79-87
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• 1995

We complete the proofs of integral and integro-differential inequalities presented in Lakshmikantham, Leela and Raos' results.

• Research in Mathematical Education
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• v.19 no.4
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• pp.229-245
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• 2015

In this study, I investigated how pre-service teachers (PSTs) proved three geometric problems by using Geometer's SketchPad (GSP) software. Based on observations in class and results from a test of geometric reasoning, eight PSTs were sorted into four of the five van Hiele levels of geometric reasoning, which were then used to predict the PSTs' levels of reasoning on three tasks involving proofs using GSP. Findings suggested that the ways the PSTs justified their geometric reasoning across the three questions demonstrated their different uses of GSP depending on their van Hiele levels. These findings also led to the insight that the notion of "proof" had somewhat different meanings for students at different van Hiele levels of thought. Implications for the effective integration of technology into pre-service teacher education programs are discussed.