• The Mathematical Education
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• v.45 no.1 s.112
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• pp.61-74
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• 2006

The ability of understanding the number and number systems, grasping the properties of number systems, and manipulating number systems is the foundation to understand algebra. It is useful to deepen students' mathematical understanding of number systems and operations. The authentic understanding of numbers and operations can make it possible for the students to manipulate algebraic symbols, to represent relationship among sets of numbers, and to use variables to investigate the properties of sets of numbers. The high school students need to understand the number systems from more abstract perspective. The purpose of this study is to study on understanding and application ability of eleventh graders of basic properties of operations with real numbers.

• Journal of the Semiconductor & Display Technology
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• v.13 no.1
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• pp.57-62
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• 2014

This paper proposes a new optical recognition method of credit card numbers. Firstly, the proposed method segments numbers from the input image of a credit card. It uses the significant differences of standard deviations between the foreground numbers and the background. Secondly, the method extracts gradient features from the segmented numbers. The gradient features are defined as four directions of grayscale pixels for 16 regions of an input number. Finally, it utilizes an artificial neural network classifier that uses an error back-propagation algorithm. The proposed method is implemented using C language in an embedded Linux system for a high-speed real-time image processing. Experiments were conducted by using real credit card images. The results show that the proposed algorithm is quite successful for most credit cards. However, the method fails in some credit cards with strong background patterns.

• Bulletin of the Korean Mathematical Society
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• v.58 no.5
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• pp.1149-1161
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• 2021

In this paper, we present new bounds on the fundamental units of real quadratic fields ${\mathbb{Q}}({\sqrt{d}})$ using the continued fraction expansion of the integral basis element of the field. Furthermore, we apply these bounds to Dirichlet's class number formula. Consequently, we provide computational advantages to estimate the class numbers of such fields. We also give some numerical examples.

• Journal of applied mathematics & informatics
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• v.20 no.1_2
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• pp.409-420
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• 2006

In this paper we consider the set of all n + 1-tuples of real numbers, not necessarily all distinct, in the decreasing order from the unit interval under the usual ordering of real numbers, always including 1. Such n + 1-tuples inherently arise as the membership values of fuzzy subsets and are called keychains. An natural equivalence relation is introduced on this set and the equivalence classes of keychains are studied here. The number of such keychains is finite and the set of all keychains is a lattice under the coordinate-wise ordering. Thus keychains are subchains of a finite chain of real numbers in the unit interval. We study some of their properties and give some applications to counting fuzzy subsets of finite sets.

• Journal for History of Mathematics
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• v.15 no.3
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• pp.17-24
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• 2002

It will be described the discovery of fundamental algebras such as complex numbers and the quaternions. Cardano(1539) was the first to introduce special types of complex numbers such as 5$\pm$$\sqrt{-15}$. Girald called the number a$\pm$$\sqrt{-b}$ solutions impossible. The term imaginary numbers was introduced by Descartes(1629) in “Discours la methode, La geometrie.” Euler knew the geometrical representation of complex numbers by points in a plane. Geometrical definitions of the addition and multiplication of complex numbers conceiving as directed line segments in a plane were given by Gauss in 1831. The expression “complex numbers” seems to be Gauss. Hamilton(1843) defined the complex numbers as paire of real numbers subject to conventional rules of addition and multiplication. Cauchy(1874) interpreted the complex numbers as residue classes of polynomials in R[x] modulo $x^2$＋1. Sophus Lie(1880) introduced commutators [a, b] by the way expressing infinitesimal transformation as differential operations. In this paper, it will be studied general quaternion algebras to finding of algebraic structure in Algebras.

• Transactions of the Korean Society of Mechanical Engineers B
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• v.28 no.4
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• pp.493-499
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• 2004

Variations of temperature and velocity fields in a Hele-Shaw convection cell (HSC) were investigated using a holographic interferometry and 2-D PIV system with varying Rayleigh number. To measure quasi-steady variation of temperature field, two different measurement methods of holographic interferometry, double-exposure method and real-time method, were employed. In the double-exposure method, unwanted waves were eliminated effectively using a digital image processing technique. The reconstructed images are clear, but transient flow cannot be reconstructed clearly. On the other hand, transient convective flow can be reconstructed well using the real-time method. However, the fringe patterns reconstructed by the real-time method contain more noises, compared with the double-exposure method. Experimental results show a steady flow pattern at low Rayleigh numbers and a time-dependent periodic flow structure at high Rayleigh numbers. The periodic flow pattern at high Rayleigh numbers obtained by the real-time holographic interferometer method is in a good agreement with the PIV results.

• East Asian mathematical journal
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• v.28 no.2
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• pp.159-172
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• 2012

In all Mathematics I Textbooks(Kim, S. H., 2010; Kim, H. K., 2010; Yang, S. K., 2010; Woo, M. H., 2010; Woo, J. H., 2010; You, H. C., 2010; Youn, J. H., 2010; Lee, K. S., 2010; Lee, D. W., 2010; Lee, M. K., 2010; Lee, J. Y., 2010; Jung, S. K., 2010; Choi, Y. J., 2010; Huang, S. K., 2010; Huang, S. W., 2010) in high schools in Korea these days, it is written and taught that for a positive real number $a$, $a^{\frac{m}{n}}$ is defined as $a^{\frac{m}{n}}={^n}\sqrt{a^m}$, where $m{\in}\mathbb{Z}$ and $n{\in}\mathbb{N}$ have common prime factors. For that situation, the author shows his opinion that the definition is not well-defined and $a^{\frac{m}{n}}$ must be defined as $a^{\frac{m}{n}}=({^n}\sqrt{a})^m$, whenever $^n\sqrt{a}$ is defined, based on the field axiom of the real number system including rational number system and natural number system. And he shows that the following laws of exponents for reals: $$\{a^{r+s}=a^r{\cdot}a^s\\(a^r)^s=a^{rs}\\(ab)^r=a^rb^r$$ for $a$, $b$>0 and $s{\in}\mathbb{R}$ hold by the completeness axiom of the real number system and the laws of exponents for natural numbers, integers, rational numbers and real numbers are logically equivalent.

• Honam Mathematical Journal
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• v.32 no.4
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• pp.563-579
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• 2010

In this paper we introduce the new analogs of Genocchi numbers and polynomials. Finally, we investigate the zeros of the twisted (h,q)-extension of Genocchi polynomials ${G_{n,q,w}^{(h)}}$(x).

• The Pure and Applied Mathematics
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• v.12 no.2 s.28
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• pp.125-131
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• 2005

In this paper, we introduce the concept of derivative of the function f : $\mathbb{Q}p{\to} R$ where $\mathbb{Q}p$ is the field of the p-adic numbers and R is the set of real numbers. And some basic theorems on derivatives are given.

• The Pure and Applied Mathematics
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• v.2 no.1
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• pp.9-16
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• 1995

D. Dubois and H. Prade introduced the notions of fuzzy numbers and defined its basic operations [3]. R. Goetschel, W. Voxman, A. Kaufmann, M. Gupta and G. Zhang [4,5,6,9] have done much work about fuzzy numbers. Let $\mathbb{R}$ the set of all real numbers and $F^{*}(\mathbb{R})$ all fuzzy subsets defined on $\mathbb{R}$. G. Zhang [8] defined the fuzzy number $\tilde{a}\;\in\;F^{*}(\mathbb{R})$ as follows : (omitted)