• Bulletin of the Korean Mathematical Society
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• v.54 no.3
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• pp.799-816
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• 2017

In this paper, we investigate the Bonnesen-style Aleksandrov-Fenchel inequalities in ${\mathbb{R}}^n$, which are the generalization of known Bonnesen-style inequalities. We first define the i-th symmetric mixed homothetic deficit ${\Delta}_i(K,L)$ and its special case, the i-th Aleksandrov-Fenchel isoperimetric deficit ${\Delta}_i(K)$. Secondly, we obtain some lower bounds of (n - 1)-th Aleksandrov Fenchel isoperimetric deficit ${\Delta}_{n-1}(K)$. Theorem 4 strengthens Groemer's result. As direct consequences, the stronger isoperimetric inequalities are established when n = 2 and n = 3. Finally, the reverse Bonnesen-style Aleksandrov-Fenchel inequalities are obtained. As a consequence, the new reverse Bonnesen-style inequality is obtained.

• Bulletin of the Korean Mathematical Society
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• v.59 no.5
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• pp.1215-1235
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• 2022

In this paper we study Szegö kernel projections for Hardy spaces in quaternionic Clifford analysis. At first we introduce the matrix Szegö projection operator for the Hardy space of quaternionic Hermitean monogenic functions by the characterization of the matrix Hilbert transform in the quaternionic Clifford analysis. Then we establish the Kerzman-Stein formula which closely connects the matrix Szegö projection operator with the Hardy projection operator onto the Hardy space, and we get the matrix Szegö projection operator in terms of the Hardy projection operator and its adjoint. At last, we construct the explicit matrix Szegö kernel function for the Hardy space on the sphere as an example, and get the solution to a Diriclet boundary value problem for matrix functions.

• The Pure and Applied Mathematics
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• v.31 no.1
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• pp.83-102
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• 2024

In this paper, first we establish a unique common fixed point theorem satisfying new contractive condition on partially ordered non-Archimedean fuzzy metric spaces and give an example to support our result. By using the result established in the first section of the manuscript, we formulate a unique common coupled fixed point theorem and also give an example to validate our result. In the end, we study the existence of solution of integral equation to verify our hypothesis. These results generalize, improve and fuzzify several well-known results in the existing literature.

• Communications of the Korean Mathematical Society
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• v.34 no.1
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• pp.185-202
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• 2019

In this paper, we investigate many types of stability, like (uniform stability, exponential stability and h-stability) of the first order dynamic equations of the form $$\{u^{\Delta}(t)=Au(t)+f(t),\;\;t{\in}{\mathbb{T}},\;t>t_0\\u(t_0)=x{\in}D(A),$$ and $$\{u^{\Delta}(t)=Au(t)+f(t,u),\;\;t{\in}{\mathbb{T}},\;t>t_0\\u(t_0)=x{\in}D(A),$$ in terms of the stability of the homogeneous equation $$\{u^{\Delta}(t)=Au(t),\;\;t{\in}{\mathbb{T}},\;t>t_0\\u(t_0)=x{\in}D(A),$$ where f is rd-continuous in $t{\in}{\mathbb{T}}$ and with values in a Banach space X, with f(t, 0) = 0, and A is the generator of a $C_0$-semigroup $\{T(t):t{\in}{\mathbb{T}}\}{\subset}L(X)$, the space of all bounded linear operators from X into itself. Here D(A) is the domain of A and ${\mathbb{T}}{\subseteq}{\mathbb{R}}^{{\geq}0}$ is a time scale which is an additive semigroup with property that $a-b{\in}{\mathbb{T}}$ for any $a,b{\in}{\mathbb{T}}$ such that a > b. Finally, we give illustrative examples.

• The Journal of Korean Physical Therapy
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• v.13 no.3
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• pp.509-527
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• 2001

The Sun's ray is composed of Infrared(49%), Visible light(40%) and Ultra violet(11%), however the ray getting to the earth is FIR(Far infrared; 60%), IR(Infrared; 20%), and UV(Ultra Violet; 20%). Human beings has utilized FIR already from time immemorial. Hershel found out Infrared for the first time. in the Industrial Revolution the Infrared and FIR had been begun to use making products. In these days, with contemporary science FIR would be begun to clear up the implication in the human body and organic compound. IR classified by wavelength three parts NlR, MIR, FIR. There is FIR which is radiated from healthy human body the wave length is 8-l4m. The human body is composed of proteins which get easily changed by a thermal factor (about 42 $^{\circ}$C over). FIR with low temperature can deeply penetrate on the human body composed things without troublesomes, since FIR has effectively operated on the human body at low temperature (35-40 $^{\circ}$C). When FlR penetrated on the human body. it would inhibit the abnormal genes and cells expression, and then information of DNA and RNA would be reexpressed for arranging DNA and RNA abnormal state. As FlR's receptors in the body, it could be presumed that N-glycosyl linkage of purine and deoxyribose, RNA splicing process, and Heat shock protein. To take the FIR which was a optimized wavewlength and strength, at first, we induced the characteristic algorithm and the computerized programing. Then we formed that the formular of optimized FIR with physical, mathematical logic and theory. especially, Plank, Kirchhoff, Wien, Stefan-Boltzmann's logic and law. In the long run, the formular was induced with integration mathematical, since we had to know the molecular wavelength. Based on the induced formular as above, we programmed the optimized FlR radiating computerized program. In this research, we designed the eletronic circuit f3r interfacing with human body to diagnosis and treatment with FIR sensor which radiated FIR wavelength optimized.

• The Pure and Applied Mathematics
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• v.2 no.2
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• pp.149-156
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• 1995

In stability theory of polysystems two concepts that playa very important role are the limit set and the prolongational limit set. For the above two concepts, A.Bacciotti and N.Kalouptsidis studied their properties in a locally compact metric space [2]. In this paper we investigate their results in c-first countable space which is more a general space than a metric space.(omitted)

• The Pure and Applied Mathematics
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• v.28 no.2
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• pp.119-142
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• 2021

In this paper, we first introduce a new L_p analytic Fourier-Feynman transform with respect to subordinate Brownian motion (AFFTSB), which extends the Fourier-Feynman transform in the Wiener space. We next examine several relationships involving the L_p-AFFTSB, the convolution product, and the gradient operator for several types of functionals.

• The Pure and Applied Mathematics
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• v.30 no.3
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• pp.249-267
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• 2023

This manuscript is divided into three segments. In the first segment, we prove a unique common fixed point theorem satisfying generalized weak contraction on partially ordered metric spaces and also give an example to support our results presented here. In the second segment of the article, some common coupled fixed point results are derived from our main results. In the last segment, we investigate the solution of integral equation as an application. Our results generalize, extend and improve several well-known results of the existing literature.

• The Mathematical Education
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• v.44 no.3 s.110
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• pp.337-360
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• 2005

In this paper we develop textbook 'Modem Algebra' for training mathematics teacher of secondary schools. In order to understand mathematics teacher's viewpoint about desirable textbook 'Modem Algebra' we created a Questionnaire related with curriculum and textbook for training mathematics teacher of secondary schools. We analyze the result of the questionnaire along with recent studies on teacher education and come up with basic principles of developing textbook 'Modern Algebra'. The first version of 'Modern Algebra for Mathematics Teachers' that we have developed based on our study can be found in website 'www.teacheredu.co.kr'.

• Research in Mathematical Education
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• v.12 no.1
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• pp.67-83
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• 2008

Promoting students' mathematics understanding is an important research theme in mathematics education. According to general theories of learning, mathematics understanding is close to active learning or significant learning. Thus, if a teacher wants to promote his/her students' mathematics understanding, he/she should pay attention to the students so that the students' thinking is in active situation. In the first part of this paper, some mathematics teachers' ideas about paying attention to their students in Chinese high school are given by questionnaire and interview. In the second part of this paper, we give some teaching episodes about how experienced mathematics teachers promote their students' mathematics understanding based on paying attention on them.