• 대한수학교육학회지:학교수학
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• 제18권1호
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• pp.159-173
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• 2016

실제 교육 현장에서는 구체적 맥락에서 추상화하는 과정과 반대로 추상화된 개념을 먼저 가르치고 구체적인 문제 상황을 도입하는 경우도 있다. 즉, 추상적 지식을 구체화 해야 하는 경우가 있는 것이다. Freudenthal은 이런 상황을 반교수학적인 전도라고 표현하며 부정적인 견해를 나타낸 바 있지만 모든 수업상황이 구체적 상황이나 구체물에서 출발하는 추상화로 진행될 수 있는지는 의문의 여지가 있다. 본 연구에서는 구체물을 추상화하여 추상적 개념을 형성하는 과정과 추상적 개념을 구체적인 상황으로 구체화하는 과정에서 나타나는 수학적 사고의 차이점을 비교 분석하여 그 교육적 시사점을 살펴보고자 한다. 이를위해 AiC의 분석틀을 활용하여 구체물의 추상화 과정에서의 수학적 사고를 분석하였고, AiC의 분석틀을 토대로 연구자가 구안한 방식으로 추상적 개념의 구체화 과정에서의 수학적 사고를 분석하였다. 두 과정을 비교 분석한 결과 구체물의 추상화 과정만큼이나 추상적 개념의 구체화 과정에서도 유의미한 수학적 사고를 유도할 수 있음을 확인할 수 있었다.

• 대한수학회보
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• 제21권2호
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• pp.99-106
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• 1984

J. Yeh has recently introduced the concept of conditional Wiener integrals which are meant specifically the conditional expectation E$^{w}$ (Z vertical bar X) of a real or complex valued Wiener integrable functional Z conditioned by the Wiener measurable functional X on the Wiener measure space (A precise definition of the conditional Wiener integral and a brief discussion of the Wiener measure space are given in Section 2). In [3] and [4] he derived some inversion formulae for conditional Wiener integrals and evaluated some conditional Wiener integrals E$^{w}$ (Z vertical bar X) conditioned by X(x)=x(t) for a fixed t>0 and x in Wiener space. Thus E$^{w}$ (Z vertical bar X) is a real or complex valued function on R$^{1}$. In this paper we shall be concerned with the random vector X given by X(x) = (x(s$_{1}$),..,x(s$_{n}$ )) for every x in Wiener space where 0=s$_{0}$ $_{1}$<..$_{n}$ =t. In Section 3 we will evaluate some conditional Wiener integrals E$^{w}$ (Z vertical bar X) which are real or complex valued functions on the n-dimensional Euclidean space R$^{n}$ . Thus we extend Yeh's results [4] for the random variable X given by X(x)=x(t) to the random vector X given by X(x)=(x(s$_{1}$).., x(s$_{n}$ )).

• 대한수학회지
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• 제55권1호
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• pp.101-129
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• 2018

From an analytical perspective, we introduce a sequence of Cartier operators that act on the field of formal Laurent series in one variable with coefficients in a field of positive characteristic p. In this work, we discover the binomial inversion formula between Hasse derivatives and Cartier operators, implying that Cartier operators can play a prominent role in various objects of study in function field arithmetic, as a suitable substitute for higher derivatives. For an applicable object, the Wronskian criteria associated with Cartier operators are introduced. These results stem from a careful study of two types of Cartier operators on the power series ring ${\mathbf{F}}_q$[[T]] in one variable T over a finite field ${\mathbf{F}}_q$ of q elements. Accordingly, we show that two sequences of Cartier operators are an orthonormal basis of the space of continuous ${\mathbf{F}}_q$-linear functions on ${\mathbf{F}}_q$[[T]]. According to the digit principle, every continuous function on ${\mathbf{F}}_q$[[T]] is uniquely written in terms of a q-adic extension of Cartier operators, with a closed-form of expansion coefficients for each of the two cases. Moreover, the p-adic analogues of Cartier operators are discussed as orthonormal bases for the space of continuous functions on ${\mathbf{Z}}_p$.

• Coupled systems mechanics
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• 제9권4호
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• pp.343-358
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• 2020

The present research deals with the time-harmonic deformation in transversely isotropic magneto thermoelastic solid with two temperature (2T), rotation due to inclined load and laser pulse. Generalized theory of thermoelasticity has been formulated for this mathematical model. The entire thermo-elastic medium is rotating with uniform angular velocity and subjected to thermally insulated and isothermal boundaries. The inclined load is supposed to be a linear combination of a normal load and a tangential load. The Fourier transform techniques have been used to find the solution to the problem. The displacement components, stress components, and conductive temperature distribution with the horizontal distance are computed in the transformed domain and further calculated in the physical domain using numerical inversion techniques. The effect of angle of inclination of normal and tangential load for Green Lindsay Model and time-harmonic source for Lord Shulman model is depicted graphically on the resulting quantities.

• Coupled systems mechanics
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• 제9권4호
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• pp.325-342
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• 2020

A mathematical model of electro-thermoelasticity subjected to memory-dependent derivative (MDD) heat conduction law is applied to a one-dimensional problem of a thermoelectric spherical cavity exposed to a warm stun that is an element of time in the presence of a uniform magnetic field. Utilizing Laplace transform as an instrument, the issue has been fathomed logically within the changed space. Numerical inversion of the Laplace transform is carried for the considered distributions and represented graphically. Some comparisons are shown in the figures to estimate the effects of MDD parameters and thermoelectric properties on the behavior of all considered fields.

• Korean Journal of Mathematics
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• 제24권2호
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• pp.235-271
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• 2016

By using the Heckman-Opdam theory on ${\mathbb{R}}^d$ given in [20], we define and study in this paper, the generalized wavelets on ${\mathbb{R}}^d$ and the generalized wavelet transform on ${\mathbb{R}}^d$, and we establish their properties. Next, we prove for the generalized wavelet transform Plancherel and inversion formulas.

• International Journal of CAD/CAM
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• 제2권1호
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• pp.45-54
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• 2002

Given a set of three generator circles in a plane, we want to find a circumcircle of these generators. This problem is a part of well-known Apollonius' $10^{th}$ Problem and is frequently encountered in various geometric computations such as the Voronoi diagram for circles. It turns out that this seemingly trivial problem is not at all easy to solve in a general setting. In addition, there can be several degenerate configurations of the generators. For example, there may not exist any circumcircle, or there could be one or two circumcircle(s) depending on the generator configuration. Sometimes, a circumcircle itself may degenerate to a line. We show that the problem can be reduced to a point location problem among the regions bounded by two lines and two transformed circles via $M{\ddot{o}}bius$ transformations in a complex space. The presented algorithm is simple and the required computation is negligible. In addition, several degenerate cases are all incorporated into a unified framework.

• 한국소음진동공학회:학술대회논문집
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• 한국소음진동공학회 2005년도 추계학술대회논문집
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• pp.213-216
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• 2005

This paper presents force feedback control performance of a 3DOF haptic device that can be used for minimally invasive surgery (MIS). As a first step, a 3DOF electrorheological (ER) joint is designed using a spherical mechanism. And it is optimized based on the mathematical torque modeling. Subsequently, the master haptic device is manufactured by the spherical joint. In order to achieve desired force trajectories, model based compensation strategy is adopted for the ER master. Therefore, Preisach model fur the PMA-based ER fluid is identified using experimental first order descending (FOD) curves. A compensation strategy is then formulated through the model inversion to achieve desired force at the ER master. Tracking control performances for sinusoidal force trajectory are presented, and their tracking errors are evaluated.

• Advances in materials Research
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• 제8권2호
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• pp.83-102
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• 2019

The present research deals with the time harmonic deformation in transversely isotropic magneto thermoelastic solid with two temperature (2T), rotation and without energy dissipation due to inclined load. Lord-Shulman theory has been formulated for this mathematical model. The entire thermo-elastic medium is rotating with a uniform angular velocity. The Fourier transform techniques have been used to find the solution to the problem. The displacement components, stress components and conductive temperature distribution with the horizontal distance are computed in the transformed domain and further calculated in the physical domain using numerical inversion techniques. The effect of time harmonic source and rotation is depicted graphically on the resulting quantities.

• Structural Engineering and Mechanics
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• 제70권2호
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• pp.245-255
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• 2019

In present research, we have considered transversely isotropic magneto thermoelastic solid with two temperature and without energy dissipation due to inclined load. The mathematical model has been formulated using Lord-Shulman theory. The Laplace and Fourier transform techniques have been used to find the solution to the problem. The displacement components, stress components and conductive temperature distribution with the horizontal distance are computed in the transformed domain and further calculated in the physical domain using numerical inversion techniques. The effect of rotation and angle of inclination of inclined load is depicted graphically on the resulting quantities.