• Journal for History of Mathematics
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• v.32 no.3
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• pp.109-134
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• 2019

In order to improve the educational situation in which the natural number e and the natural logarithm are dealt with somewhat perfunctorily, this study explores the genetic process in which the natural logarithm and its base e occurred, and has an educational discussion based on that analysed process. Specifically, the study inquires into how the natural logarithm happened in relation to the quadrature of the hyperbolic curves through analysis and thought experimentation in mathematics history. Particularly, it sheds light on the role of e and the naturalness of the natural logarithm in terms of the introduction of the real number exponent. Also, this study discusses what the findings suggest educationally.

• Communications of Mathematical Education
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• v.25 no.2
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• pp.323-339
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• 2011

The expansion of exponential law as the law of calculation of integer numbers can be a good material for the students to experience an extended configuration which is based on an algebraic principle of the performance of equivalent forms. While current textbooks described that exponential law can be expanded from natural number to integer, rational number and real number, most teachers force students to accept intuitively that the exponential law is valid although exponent is expanded into real number. However most teachers overlook explaining the value of exponent of rational number or exponent of irrational number so most students have a lot of questions whether this value is a rational number or a irrational number. Related to students' questions, most teacher said that it is out of the current curriculum and students will learn it after going to college instead of detailed answers. In this paper, we will present several examples and the values about irrational exponents of a positive rational and irrational exponents of a positive irrational number, and study the recognition and fallacy of would-be teachers about the cases of irrational exponents of a positive rational and irrational exponents of a positive irrational number at the expansion of exponential law.

• The Mathematical Education
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• v.50 no.1
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• pp.61-67
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• 2011

There may be two methods defining the rational exponent $a^{\frac{m}{n}}$ for any positive real number a. The one which is used in all korean highschool mathematics textbooks is to define it as $\sqrt[n]{a^m}$, that is $(a^m)^{\frac{1}{n}}$. The other is to define it as $(\sqrt[n]a)^m}$, that is $(a^{\frac{1}{n}})^m$. In this paper, we insist that the latter is more appropriate and universal, and that the contents of current textbooks on the definition of the rational exponent should be corrected.

• Asian Journal of Atmospheric Environment
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• v.6 no.4
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• pp.304-313
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• 2012

In this study, the ${\AA}$ngstrom exponent for polydispersed aerosol during dynamic processes was investigated. Log-normal aerosol size distribution was assumed, and a sensitivity analysis of the ${\AA}$ngstrom exponent with regards the coagulation and condensation process was performed. The ${\AA}$ngstrom exponent is expected to decrease because of the particle growth due to coagulation and condensation. However, it is difficult to quantify the degree of change. In order to understand quantitatively the change in the ${\AA}$ngstrom exponent during coagulation and condensation, different real and imaginary parts of the refractive index were considered. The results show that the ${\AA}$ngstrom exponent is sensitive to changes in size distribution and refractive index. The total number concentration decreases and the geometric mean diameter of aerosols increase during coagulation. On the while, the geometric standard deviation approaches monodispersed size distribution during the condensation process, and this change in size distribution affects the ${\AA}$ngstrom exponent. The degree of change in the ${\AA}$ngstrom exponent depends on the refractive index and initial size distribution, and the size parameter changes with the ${\AA}$ngstrom exponent for a given refractive index or chemical composition; this indicates that the size distribution plays an important role in determining the ${\AA}$ngstrom exponent as well as the chemical composition. Subsequently, this study shows how the ${\AA}$ngstrom exponent changes quantitatively during the aerosol dynamics processes for a log-normal aerosol size distribution for different refractive indices; the results showed good agreement with the results for simple analytic size distribution solutions.

• Journal of Information Technology Applications and Management
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• v.12 no.4
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• pp.25-31
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• 2005

We know that the Hurst Exponent (HE) is a real number in [0, 1] which denotes randomness of time series. in this research, we suggest non-linear analysis of human biological signals through HE. The feasibility of human biological signals using inductive incitement provides Some diagnosis for active treatment. In our experiment, we measured the heartbeat through the MCG, 29 healthy and 34 abnormal subjects ostensibly. The raw data of acupuncture incitement are supported by opinions of gross examination and pathological diagnosis. The mean values of HE are 0.345, 0.755 and 0.805 for the periods of before, during and after acupuncture treatment, respectively in case of abnormal subjects. On the other hand, the mean values, 0.808, 0.797 and 0.785 are for normal cases, correspondingly. From this data, we show that HE is very significant in abnormal controls according to an acupuncture incitement, and the incitement effect is evidently extracted in abnormal subjects. But, in normal subjects, the incitement effect is meaningless.

• Journal of Institute of Control, Robotics and Systems
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• v.17 no.3
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• pp.236-240
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• 2011

The terminal sliding mode control schemes have been studied a lot since they can guarantee that the state error gets to zero in a finite time. However, the conventional terminal sliding surfaces have been designed using power function whose exponent is a rational number between 0 and 1, and whose numerator and denominator should be odd integers. It is clearly restrictive. Thus, in this paper, we propose a novel terminal sliding surface using power function whose exponent can be a real number between 0 and 1.

• Journal of KIEE
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• v.10 no.1
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• pp.7-15
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• 2000

Source separation is a statistical method, the goal of which is to separate the linear instantaneous mixtures of statistically independent sources without resorting to any prior knowledge. This paper addresses a source separation algorithm which is able to separate the mixtures of sub- and super-Gaussian sources. The nonlinear function in the proposed algorithm is derived from the generalized Gaussian distribution that is a set of distributions parameterized by a real positive number (Gaussian exponent). Based on the relationship between the kurtosis and the Gaussian exponent, we present a simple and efficient way of selecting proper nonlinear functions for source separation. Useful behavior of the proposed method is demonstrated by computer simulations.

• Journal of the Korean Mathematical Society
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• v.56 no.2
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• pp.475-484
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• 2019

We study the set of critical exponents of discrete groups acting on regular trees. We prove that for every real number ${\delta}$ between 0 and ${\frac{1}{2}}\;{\log}\;q$, there is a discrete subgroup ${\Gamma}$ acting without inversion on a (q+1)-regular tree whose critical exponent is equal to ${\delta}$. Explicit construction of edge-indexed graphs corresponding to a quotient graph of groups are given.

• Journal of Institute of Control, Robotics and Systems
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• v.15 no.11
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• pp.1137-1143
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• 2009

This paper presents the FPGA implementation of efficient algorithms for approximating exponential function based on floating point format data. The Taylor-Maclaurin expansion as a conventional approximation method becomes inefficient since high order expansion is required for the large number to satisfy the approximation error. A format converter is designed to convert fixed data format to floating data format, and then the real number is separated into two fields, an integer field and an exponent field to separately perform mathematic operations. A new assembly command is designed and added to previously developed command set to refer the math table. To test the proposed algorithm, assembly program has been developed. The program is downloaded into the Altera DSP KIT W/STRATIX II EP2S180N Board. Performances of the proposed method are compared with those of the Taylor-Maclaurin expansion.

• Nuclear Engineering and Technology
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• v.55 no.7
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• pp.2697-2711
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• 2023

In this study, the validity of reducing the number of gas turbine stages designed for a nitrogen Brayton cycle coupled to a sodium-cooled fast reactor was assessed. The turbine performance was evaluated through computational fluid dynamics (CFD) simulations under different off-design conditions controlled by a reduced flow rate and reduced rotational speed. Two different multistage gas turbines designed to extract almost the same specific work were selected: two- and three-stage turbines (mid-span stage loading coefficient: 1.23 and 1.0, respectively). Real gas properties were considered in the CFD simulation in accordance with the Peng-Robinson's equation of state. According to the CFD results, the off-design performance of the two-stage turbine is comparable to that of the three-stage turbine. Moreover, compared to the three-stage turbine, the two-stage turbine generates less entropy across the shock wave. The results indicate that under both design and off-design conditions, increasing the stage loading coefficient for a fewer number of turbine stages is effective in terms of performance and size. Furthermore, the Ellipse law can be used to assess off-design performance and increasing exponent of the expansion ratio term better predicts the off-design performance with a few stages (two or three).