• 한국수학교육학회지시리즈E:수학교육논문집
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• 제22권3호
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• pp.289-310
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• 2008

본 연구는 고등학교 <10-나>의 삼각함수의 성질을 단위원이 아니라 삼각함수 sin(x), cos(x), tan(x)의 그래프 성질을 이용하여 구할 수 있는 방법을 제시하여 삼각함수의 값을 구하는 학습활동과 삼각함수의 그래프에 대한 학습활동과의 연계성을 높이고, 이를 통하여 삼각함수의 성질에 대한 학생들의 이해를 향상시킬 수 있는지를 논의하고자 한다. 본 연구의 결론은 삼각함수 그래프 이용에 의한 삼각 함수값 구하기의 효율성 증대, 일반각에 대한 삼각 함수값 구하기의 정확한 용어 사용, 삼각 함수값 구하기와 삼각함수의 그래프 지도내용 사이의 연계성, 삼각함수의 그래프의 선행 지도 등이며 이에 대한 논의가 제시되어 있다.

• 한국수학교육학회지시리즈A:수학교육
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• 제50권3호
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• pp.355-365
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• 2011

The purpose of this study is to provide mathematical knowledge for supporting the didactical knowledge on circular measure and radian in the high school curriculum. We show that circular measure related to arcs can be mathematically justified as an angular measure and radian is a well defined concept to be able to reconcile the values of trigonometric functions and ones of circular functions, which are real variable functions. Radian has two-fold intrinsic attributes of angular measure and arc measure on the unit circle, in particular, the latter property plays a very important role in simplifying the trigonometric derivatives. To improve students's low academic achievement in trigonometry section, the useful advantage and the background over the introduction of radian should be preferentially taught and recognized to students. We suggest some teaching plans to practice in the class of elementary and middle school for enhancing teachers' and students' understanding of radian.

• 대한수학회보
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• 제48권3호
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• pp.617-625
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• 2011

In [6], it was shown that hyponormality for Toeplitz operators with polynomial symbols can be reduced to classical Schur's algorithm in function theory. In [6], Zhu has also given the explicit values of the Schur's functions ${\Phi}_0$, ${\Phi}_1$ and ${\Phi}_2$. Here we explicitly evaluate the Schur's function ${\Phi}_3$. Using this value we find necessary and sufficient conditions under which the Toeplitz operator $T_{\varphi}$ is hyponormal, where ${\varphi}$ is a trigonometric polynomial given by ${\varphi}(z)$ = ${\sum}^N_{n=-N}a_nz_n(N{\geq}4)$ and satisfies the condition $\bar{a}_N\(\array{a_{-1}\\a_{-2}\\a_{-4}\\{\vdots}\\a_{-N}}\)=a_{-N}\;\(\array{\bar{a}_1\\\bar{a}_2\\\bar{a}_4\\{\vdots}\\\bar{a}_N}\)$. Finally we illustrate the easy applicability of the derived results with a few examples.

• Structural Engineering and Mechanics
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• 제17권1호
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• pp.87-97
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• 2004

In gradient-dependent plasticity theory, the yield strength depends on the Laplacian of an equivalent plastic strain measure (hardening parameter), and the consistency condition results in a differential equation with respect to the plastic multiplier. The plastic multiplier is then discretized in addition to the usual discretization of the displacements, and the consistency condition is solved simultaneously with the equilibrium equations. The disadvantage is that the plastic multiplier requires a Hermitian interpolation that has four degrees of freedom at each node. Instead of using a Hermitian interpolation, in this article, a 3-node incompatible (trigonometric) interpolation is proposed for the plastic multiplier. This incompatible interpolation uses only the function values of each node, but it is continuous across element boundaries and its second-order derivatives exist within the elements. It greatly reduces the degrees of freedom for a problem, and is shown through a numerical example on localization to yield good results.

• 한국지능시스템학회논문지
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• 제19권2호
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• pp.265-272
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• 2009

Fuzzy number intuitionistic fuzzy sets (FNIFSs), each of which is characterized by a membership function and a non-membership function whose values are trigonometric fuzzy number rather than exact numbers, are a very useful means to describe the decision information in the process of decision making. Wang [10] developed some arithmetic aggregation operators, such as the fuzzy number intuitionistic fuzzy weighted averaging (FIFWA) operator, the fuzzy number intuitionistic fuzzy ordered weighted averaging (FIFOWA) operator and the fuzzy number intuitionistic fuzzy hybrid aggregation (FIFHA) operator. In this paper, based on the FIFHA operator and the FIFWA operator, we investigate the group decision making problems in which all the information provided by the decision-makers is presented as fuzzy number intuitionistic fuzzy decision matrices where each of the elements is characterized by fuzzy number intuitionistic fuzzy numbers, and the information about attribute weights is partially known. An example is used to illustrate the applicability of the proposed approach.

• Earthquakes and Structures
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• 제8권6호
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• pp.1463-1480
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• 2015

This research aims to develop an empirical model for simulation of time-varying frequency in earthquake ground motion so as to be used easily in engineering applications. Briefly, 10545 recordings of the Next Generation Attenuation (NGA) global database of accelerograms from shallow crustal earthquakes are selected and binned by magnitude, distance and site condition. Then the wavelet spectrum of each acceleration record is calculated by using one-dimensional continuous wavelet transform, and the frequencies corresponding to the maximum values of the wavelet spectrum at a series of sampling time, named predominant frequencies, are extracted to analyze the variation of frequency content of seismic ground motions in time. And the time-variation of the predominant frequencies of 178 magnitude-distance-site bins for different directions are obtained by calculating the mean square root of predominant frequencies within a bin. The exponential trigonometric function is then use to fit the data, which describes the predominant frequency of ground-motion as a function of time with model parameters given in tables for different magnitude, distance, site conditions and direction. Finally, a practical frequency-dependent amplitude envelope function is developed based on the time-varying frequency derived in this paper, which has clear statistical parameters and can emphasize the effect of low-frequency components on later seismic action. The results illustrate that the time-varying predominant frequency can preferably reflect the non-stationarity of the frequency content in earthquake ground motions and that empirical models given in this paper facilitates the simulation of ground motions.

• 대한수학교육학회지:학교수학
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• 제15권1호
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• pp.159-177
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• 2013

본 연구는 최근 중요성이 부각되고 있는 STEAM 교육의 사례 연구로서 STEAM에 적합한 수업 자료 개발, 교수 학습 방법의 제안, 효과 분석을 통한 STEAM의 가능성 탐색을 목적으로 한다. STEAM 교육의 소재는 수학과 과학이 분리되기 이전의 수학사를 기반으로, 과학(S)은 24절기, 태양의 고도, 천체의 운동, 기술(T)은 그래픽 계산기를 이용한 탐구, 공학(E)은 해시계의 설계 원리 탐구 및 제작, 예술(A)은 수학사 문헌 탐구 및 수학의 심미적 가치에 대한 이해, 수학(M)은 삼각함수와 관련된다. 또한 STEAM의 특성과 문헌 연구를 토대로 프로젝트기반학습이 STEAM 교육에 적합할 것으로 보고 융합적 프로젝트기반학습을 설계하여 STEAM PBL이 융합적 지식 형성과 수학적 가치를 포함하는 수학적 태도와 21세기 역량 신장에 효과적인지에 중점을 두어 분석하였다. 수업을 분석한 결과, 학생들이 팀원 간의 협력, 의사소통과 표현, 비판적 사고를 토대로 문제해결을 하는 모습이 많이 포착되었으며, 수업을 통해 수학적 가치를 포함한 수학적 태도가 긍정적인 방향으로 변화된 것으로 나타났다.

• Journal of Power Electronics
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• 제16권3호
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• pp.1110-1121
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• 2016

The mathematical model of a three phase PWM converter under the stationary αβ reference frame is deduced and constructed based on a Proportional-Resonant (PR) regulator, which can replace trigonometric function calculation, Park transformation, real-time detection of a Phase Locked Loop and feed-forward decoupling with the proposed accurate calculation of the inductance voltage vector. To avoid the parallel resonance of the LCL topology, the active damping method of the proportional capacitor-current feedback is employed. As to current vector error elimination, an optimized PR controller of the inner current loop is proposed with the zero-pole matching (ZPM) and cancellation method to configure the regulator. The impacts on system's characteristics and stability margin caused by the PR controller and control parameter variations in the inner-current loop are analyzed, and the correlations among active damping feedback coefficient, sampling and transport delay, and system robustness have been established. An equivalent model of the inner current loop is studied via the pole-zero locus along with the pole placement method and frequency response characteristics. Then, the parameter values of the control system are chosen according to their decisive roles and performance indicators. Finally, simulation and experimental results obtained while adopting the proposed method illustrated its feasibility and effectiveness, and the inner current loop achieved zero static error tracking with a good dynamic response and steady-state performance.

• Geomechanics and Engineering
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• 제31권1호
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• pp.99-112
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• 2022

The present study examines the natural frequencies (NFs) of perfect/imperfect functionally graded sandwich beams (P/IP-FGSBs), which are composed of a porous core constructed of functionally graded materials (FGMs) and a homogenous isotropic metal and ceramic face sheets resting on elastic foundations. To accomplish this, the material properties of the FGSBs are assumed to vary continuously along the thickness direction as a function of the volume fraction of constituents expressed by the modified rule of the mixture, which includes porosity volume fraction represented using four distinct types of porosity distribution models. Additionally, to characterize the reaction of the two-parameter elastic foundation to the Perfect/Imperfect (P/IP) FGSBs, the medium is assumed to be linear, homogeneous, and isotropic, and it is described using the Winkler-Pasternak model. Furthermore, the kinematic relationship of the P/IP-FGSBs resting on the Winkler-Pasternak elastic foundations (WPEFs) is described using trigonometric shear deformation theory (TrSDT), and the equations of motion are constructed using Hamilton's principle. A closed-form solution is developed for the free vibration analysis of P/IP-FGSBs resting on the WPEFs under four distinct boundary conditions (BCs). To validate the new formulation, extensive comparisons with existing data are made. A detailed investigation is carried out for the effects of the foundation coefficients, mode numbers (MNs), porosity volume fraction, power-law index, span to depth ratio, porosity distribution patterns (PDPs), skin core skin thickness ratios (SCSTR), and BCs on the values of the NFs of the P/IP-FGSBs.

• 자원환경지질
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• 제12권2호
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• pp.95-104
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• 1979

The position of any point on the earth's surface can be. represented in the spherical coordinates by surface spherical harmonics. Since geomagnetic field is a function of position on the earth, it can be also expressed by spherical harmonic analysis as spherical harmonics of trigonometric series of $a_m({\theta})$ cos $m{\phi}$ and $b_m({\theta})$ sin $m{\phi}$. Coefficients of surface spherical harmonics, $a_m({\theta})$ and $b_m({\theta})$, can be drawn from the components of the geomagnetic field, declination and inclination, and vice versa. In this paper, components of geomagnetic field, declination and inclination in the Korean peninsula are obtained by spherical harmonic analysis using the Gauss coefficients calculated from the world-wide magnetic charts of 1960. These components correspond to the values of normal geomagnetic field having no disturbances of subsurface mass, structure, and so on. The vertical and total components offer the zero level for the interpretation of geomagnetic data obtained by magnetic measurement in the Korean peninsula. Using this zero level, magnetic anomaly map is obtained from the data of airborne magnetic. prospecting carried out during 1958 to 1960. The conclusions of this study are as follows; (1) The intensity of horizontal component of normal geomagnetic field in Korean peninsula ranges from $2{\times}10^4$ gammas to $2.45{\times}10^4$ gammas. It decreases about 500 with the increment of $1^{\circ}$ in latitude. Along the same. latitude, it increases 250 gammas with the increment of $1^{\circ}$ in longitude. (2) Intensity of vertical component ranges from $3.85{\times}10^4$ gammas to $5.15{\times}10^4$ gammas. It increases. about 1000 gammas with the increment of $1^{\circ}$ in latitude. Along the same latitude, it decreases. 150~240 gammas with the increment of $1^{\circ}$ in longitude. Decreasing rate is considerably larger in higher latitude than in lower latitude. (3) Total intensity ranges from $4.55{\times}10^4$ gammas to $5.15{\times}10^4$ gammas. It increases 600~700 gammas with the increament of $1^{\circ}$ in latitude. Along the same latitude, it decreases 10~90 gammas. with the increment of $1^{\circ}$ in longitude. Decreasing rate is considerably larger in higher latitude as the case of vertical component. (4) The declination ranges from $-3.8^{\circ}$ to $-11.5^{\circ}$. It increases $0.6^{\circ}$ with the increment of $1^{\circ}$ in latitude. Along the same latutude, it increases $0.6^{\circ}$ with the increment of l O in longitude. Unlike the cases of vertical and total component, the rate of change is considerably larger in lower latitude than in higher latitude. (5) The inclination ranges from $57.8^{\circ}$ to $66.8^{\circ}$. It increases about $1^{\circ}$ with 'the increment of $1^{\circ}$ in latitude Along the same latitude, it dereases $0.4^{\circ}$ with the increment of $1^{\circ}$ in longitude. (6) The Boundaries of 5 anomaly zones classified on the basis of the trend and shape of anomaly curves correspond to the geologic boundaries. (7) The trend of anomaly curves in each anomaly zone is closely related to the geologic structure developed in the corresponding zone. That is, it relates to the fault in the 3rd zone, the intrusion. of granite in the 1st and 5th zones, and mountains in the 2nd and 4th zones.