• Nonlinear Functional Analysis and Applications
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• v.26 no.1
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• pp.137-155
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• 2021

In this paper, first, we prove some common fixed point theorems for the generalized contraction condition under newly defined modified simulation function which generalize and include many results in the literature. Second, we give two numerical examples with graphical representations for verifying the proposed results. Third, we discuss and study a set of common fixed point theorems for two pairs (finite families) of self-mappings. Finally, we give some applications of our results in discrete and functional fractional economic systems.

• Journal of the Korean Mathematical Society
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• v.50 no.5
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• pp.1129-1163
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• 2013

Based on finite element discretization, two linearization approaches to the defect-correction method for the steady incompressible Navier-Stokes equations are discussed and investigated. By applying $m$ times of Newton and Picard iterations to solve an artificial viscosity stabilized nonlinear Navier-Stokes problem, respectively, and then correcting the solution by solving a linear problem, two linearized defect-correction algorithms are proposed and analyzed. Error estimates with respect to the mesh size $h$, the kinematic viscosity ${\nu}$, the stability factor ${\alpha}$ and the number of nonlinear iterations $m$ for the discrete solution are derived for the linearized one-step defect-correction algorithms. Efficient stopping criteria for the nonlinear iterations are derived. The influence of the linearizations on the accuracy of the approximate solutions are also investigated. Finally, numerical experiments on a problem with known analytical solution, the lid-driven cavity flow, and the flow over a backward-facing step are performed to verify the theoretical results and demonstrate the effectiveness of the proposed defect-correction algorithms.

• Journal for History of Mathematics
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• v.22 no.1
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• pp.41-52
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• 2009

Dutch mathematician Simon Stevin published De Thiende(The Tenth) in 1583. In that book Stevin suggested new numerical notation which could express all numbers. That new notation was decimal fraction system. In this article I will argue that Stevin invented new decimal fraction system with two main purposes. The explicit purpose was to invent a new system which could be used easily by practical mathematicians. The implicit purpose which cannot be found in De Thiende alone but in his other writings was to break the Aristotelian tradition which separated geometry and arithmetic which dealt continuous magnitude and discrete numbers respectively.

• Journal of Elementary Mathematics Education in Korea
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• v.18 no.2
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• pp.319-339
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• 2014

Recently, the discussion about division and partition of fraction increases in Korea's national curriculum documents. There are varieties of assertions arranging from the opinion that both interpretations are unintelligible to the opinion that both interpretations are intelligible. In this paper, we investigated a possibility that division and partition interpretation of fraction become valid. As a result, it is appeared that division and partition interpretation of fraction can be defined reasonably through expansion of interpretation of natural number. Besides, division and partition interpretation of fraction can be work in activity, such as constructing equation from sentence problem, or such as proving algorithm of fraction division.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.18 no.1
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• pp.27-41
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• 2014

In this paper, we present an efficient numerical method for multiphase image segmentation using a multiphase-field model. The method combines the vector-valued Allen-Cahn phase-field equation with initial data fitting terms containing prescribed interface width and fidelity constants. An efficient numerical solution is achieved using the recently developed hybrid operator splitting method for the vector-valued Allen-Cahn phase-field equation. We split the modified vector-valued Allen-Cahn equation into a nonlinear equation and a linear diffusion equation with a source term. The linear diffusion equation is discretized using an implicit scheme and the resulting implicit discrete system of equations is solved by a multigrid method. The nonlinear equation is solved semi-analytically using a closed-form solution. And by treating the source term of the linear diffusion equation explicitly, we solve the modified vector-valued Allen-Cahn equation in a decoupled way. By decoupling the governing equation, we can speed up the segmentation process with multiple phases. We perform some characteristic numerical experiments for multiphase image segmentation.

• The Korean Journal of Applied Statistics
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• v.21 no.4
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• pp.603-613
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• 2008

We often encounter the situation that discrete count data have a large portion of zeros. In this case, it is not appropriate to analyze the data based on standard regression models such as the poisson or negative binomial regression models. In this article, we consider Bayesian analysis for two commonly used models. They are zero-inflated poisson and negative binomial regression models. We use the Bayes factor as a model selection tool and computation is proceeded via Markov chain Monte Carlo methods. Crash count data are analyzed to support theoretical results.

• The Mathematical Education
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• v.50 no.3
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• pp.309-327
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• 2011

In this paper, we extended the division algorithm for the integers to the finite decimals. Though the remainder for the finite decimals is able to be defined as various ways, the remainder could be defined as 'the remained amount' which is the result of the division and as "the remainder" only if 'the remained amount' is decided uniquely by certain conditions. From the definition of "the remainder" for the finite decimal, it could be inferred that 'the division by equal part' and 'the division into equal parts' are proper for the division of the finite decimal concerned with the definition of "the remainder". The finite decimal, based on the unit of measure, seemed to make it possible for us to think "the remainder" both ways: 1" in the division by equal part when the quotient is the discrete amount, and 2" in the division into equal parts when the quotient is not only the discrete amount but also the continuous amount. In this division context, it could be said that the remainder for finite decimal must have the meaning of the justice and the completeness as well. The theorem of the division algorithm for the finite decimal could be accomplished, based on both the unit of measure of "the remainder", and those of the divisor and the dividend. In this paper, the meaning of the division algorithm for the finite decimal was investigated, it is concluded that this theory make it easy to find the remainder in the usual unit as well as in the unusual unit of measure.

• Journal of Educational Research in Mathematics
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• v.14 no.3
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• pp.221-237
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• 2004

This research analyzed mathematical discourse of the students in an RME-based differential equations course at a university in order to investigate the social transformation of the students' conceptual model of differential equations. The analysis focused on the change in the students' use of conceptual metaphor for differential equations and pedagogical factors promoting the change. The analysis shows that discrete and quantitative conceptual model was prevalent in the beginning of the semester However, continuous and qualitative conceptual model emerged through the negotiation of mathematical meaning based on the inquiry of context problems. The participation in the project class has a positive impact on the extension of the students' conceptual model of differential equations and increases the fluency of the students' problem solving in differential equations. Moreover, this paper provides a discussion to identify the pedagogical factors Involved with the transformation of the students' conceptual model. The discussion highlights the sociocultural aspect of teaching and learning of mathematics and provides implications to improve teaching of mathematics in school.

• Structural Engineering and Mechanics
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• v.72 no.5
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• pp.597-615
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• 2019

The paper studies the dispersion of the axisymmetric longitudinal wave propagating in the "hollow cylinder + surrounding medium" system with inhomogeneous initial stresses caused by the uniformly distributed radial compressional forces acting at infinity. Up to now in the world literature, there exist only a few investigations related to the wave dispersion in a hollow cylinder with inhomogeneous initial stresses. Therefore, this paper is one of the first attempts in this field in the sense of the development of investigations for the case where the cylinder is surrounded with an infinite medium. The three-dimensional linearized theory of elastic waves is used for describing the considered wave propagation problem and, for a solution to the corresponding mathematical problem, the discrete-analytical solution method is developed and employed. The corresponding dispersion equation is obtained and this equation is solved numerically and, as a result of this solution, the dispersion curves are constructed for the first and second modes. By analyzing these curves, the character of the influence of the inhomogeneous initial stresses on the dispersion curves is established. In particular, it is established that as a result of the inhomogeneity of the initial stresses both new dispersion curves and the "band gap" for the wave frequencies can appear.

• Bulletin of the Korean Mathematical Society
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• v.53 no.4
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• pp.1051-1069
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• 2016

Combining the concept of self-certified public key and message recovery, Li-Zhang-Zhu (LZZ) gives the proxy signature scheme with message recovery using self-certified public key. The security of the proposed scheme is based on the discrete logarithm problem (DLP) and one-way hash function (OWHF). Their scheme accomplishes the tasks of public key verification, proxy signature verification, and message recovery in a logically single step. In addition, their scheme satisfies all properties of strong proxy signature and does not use secure channel in the communication between the original signer and the proxy signer. In this paper, it is shown that in their signature scheme a malicious signer can cheat the system authority (SA), by obtaining a proxy signature key without the permission of the original signer. At the same time malicious original signer can also cheat the SA, he can also obtain a proxy signature key without the permission of the proxy signer. An improved signature scheme is being proposed, which involves the remedial measures to get rid of security flaws of the LZZ et al.'s. The security and performance analysis shows that the proposed signature scheme is maintaining higher level of security, with little bit of computational complexity.