• 대한수학회보
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• 제52권2호
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• pp.649-659
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• 2015

The space $BV^2_{\alpha}(I)$ of all the real functions defined on interval $I=[a,b]{\subset}\mathbb{R}$, which are of bounded second ${\alpha}$-variation (in the sense De la Vall$\acute{e}$ Poussin) on I forms a Banach space. In this space we define an operator of substitution H generated by a function $h:I{\times}\mathbb{R}{\rightarrow}\mathbb{R}$, and prove, in particular, that if H maps $BV^2_{\alpha}(I)$ into itself and is globally Lipschitz or uniformly continuous, then h is an affine function with respect to the second variable.

• Korean Journal of Mathematics
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• 제32권1호
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• pp.43-57
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• 2024

Given a domain X and a collection H of functions h : X → {0, 1}, the Vapnik-Chervonenkis (VC) dimension of H measures its complexity in an appropriate sense. In particular, the fundamental theorem of statistical learning says that a hypothesis class with finite VC-dimension is PAC learnable. Recent work by Fitzpatrick, Wyman, the fourth and seventh named authors studied the VC-dimension of a natural family of functions ℋ'²_t(E) : 𝔽²_q → {0, 1}, corresponding to indicator functions of circles centered at points in a subset E ⊆ 𝔽²_q. They showed that when |E| is large enough, the VC-dimension of ℋ'²_t(E) is the same as in the case that E = 𝔽²_q. We study a related hypothesis class, ℋ^d_t(E), corresponding to intersections of spheres in 𝔽^d_q, and ask how large E ⊆ 𝔽^d_q needs to be to ensure the maximum possible VC-dimension. We resolve this problem in all dimensions, proving that whenever |E| ≥ C_dq^d-1/(d-1) for d ≥ 3, the VC-dimension of ℋ^d_t(E) is as large as possible. We get a slightly stronger result if d = 3: this result holds as long as |E| ≥ C_3q^7/3. Furthermore, when d = 2 the result holds when |E| ≥ C_2q^7/4.

• 대한수학회논문집
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• 제32권2호
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• pp.233-260
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• 2017

Let R be a factorial domain. In this work we investigate the connections between the arithmetic of Int(R) (i.e., the ring of integer-valued polynomials over R) and its monadic submonoids (i.e., monoids of the form {$g{\in}Int(R){\mid}g{\mid}_{Int(R)}f^k$ for some $k{\in}{\mathbb{N}}_0$} for some nonzero $f{\in}Int(R)$). Since every monadic submonoid of Int(R) is a Krull monoid it is possible to describe the arithmetic of these monoids in terms of their divisor-class group. We give an explicit description of these divisor-class groups in several situations and provide a few techniques that can be used to determine them. As an application we show that there are strong connections between Int(R) and its monadic submonoids. If $R={\mathbb{Z}}$ or more generally if R has sufficiently many "nice" atoms, then we prove that the infinitude of the elasticity and the tame degree of Int(R) can be explained by using the structure of monadic submonoids of Int(R).

• 대한수학회보
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• 제46권3호
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• pp.511-519
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• 2009

A matrix $P{\in}\mathbb{C}^{n{\times}n}$ is called a generalized reflection matrix if $P^*$ = P and $P^2$ = I. An $n{\times}n$ complex matrix A is said to be a reflexive (anti-reflexive) matrix with respect to the generalized reflection matrix P if A = PAP (A = -PAP). It is well-known that the reflexive and anti-reflexive matrices with respect to the generalized reflection matrix P have many special properties and widely used in engineering and scientific computations. In this paper, we give new necessary and sufficient conditions for the existence of the reflexive (anti-reflexive) solutions to the linear matrix equation AXB + CY D = E and derive representation of the general reflexive (anti-reflexive) solutions to this matrix equation. By using the obtained results, we investigate the reflexive (anti-reflexive) solutions of some special cases of this matrix equation.

• 대한수학회논문집
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• 제33권4호
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• pp.1141-1158
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• 2018

Hahn difference operator $D_{q,{\omega}}$ which is defined by $$D_{q,{\omega}}g(t)=\{{\frac{g(gt+{\omega})-g(t)}{t(g-1)+{\omega}}},{\hfill{20}}\text{if }t{\neq}{\theta}:={\frac{\omega}{1-q}},\\g^{\prime}({\theta}),{\hfill{83}}\text{if }t={\theta}$$ received a lot of interest from many researchers due to its applications in constructing families of orthogonal polynomials and in some approximation problems. In this paper, we investigate sufficient conditions for stability of the abstract linear Hahn difference equations of the form $$D_{q,{\omega}}x(t)=A(t)x(t)+f(t),\;t{\in}I$$, and $$D^2{q,{\omega}}x(t)+A(t)D_{q,{\omega}}x(t)+R(t)x(t)=f(t),\;t{\in}I$$, where $A,R:I{\rightarrow}{\mathbb{X}}$, and $f:I{\rightarrow}{\mathbb{X}}$. Here ${\mathbb{X}}$ is a Banach algebra with a unit element e and I is an interval of ${\mathbb{R}}$ containing ${\theta}$.

• International Journal of Fuzzy Logic and Intelligent Systems
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• 제5권2호
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• pp.175-178
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• 2005

In the present paper, we observe a relation between fuzzy norms and induced crisp norms on a linear space. We first prove that if $\rho_1,\;\rho_2$ are equivalent fuzzy norms on a linear space, then for every $\varepsilon\in(0.1)$, the induced crisp norms $P_\varepsilon^1,\;and\;P_\varepsilon^2$, respectively are equivalent. Since the converse does not hold, we prove it under some strict conditions. And consider the following theorem proved in [8]: Let $\rho$ be a lower semicontinuous fuzzy norm on a normed linear space X, and have the bounded support. Then $\rho$ is equivalent to the fuzzy norm $\chi_B$ where B is the closed unit ball of X. The lower semi-continuity of $\rho$ is an essential condition which guarantees the continuity of $P_\varepsilon$, where 0 < e < 1. As the last result, we prove that : if $\rho$ is a fuzzy norm on a finite dimensional vector space, then $\rho$ is equivalent to $\chi_B$ if and only if the support of $\rho$ is bounded.

• 한국안광학회지
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• 제6권2호
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• pp.11-16
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• 2001

렌즈에 가한 응력을 측정하기 위한 편광기는 1/4 파장판-편광판으로 구성하였고, 각 단계별 광파의 $E_1$, $E_2$의 성분을 분석하였다. 주 응력차 ${\sigma}_1-{\sigma}_2$는 무늬 차수(N)가 시료 각 점에서 분석되고, 시료의 광축은 주 응력 방향과 일치하는 것으로부터 2 차원 모델을 결정할 수 있다. 광파에 작용하는 복굴절성과 위상지연은 주 응력차 (${\sigma}_1-{\sigma}_2$)에 비례하고, 최종 광파의 세기는 $sin^2({\Delta}/2)$에 비례하고, ${\Delta}/2=n^{\pi}$ (n=0, 1, 2, ...)일 때 소멸 무늬인 흡광이 일어난다. 실험적 결과로는 렌즈의 외부 응력의 크기에 따라 흡광 band가 이동한다.

• 정보학연구
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• 제8권2호
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• pp.1-16
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• 2005

In high education, knowledge is the very important resource and output. Knowledge management is the art of creating additional value from academic knowledge assets. The e-community is the network community, newly created by information communication technology, and the critical place that share and transfer people's the various knowledge and thinking and the effective space that can apply knowledge management. If we construct a convenient and useful e-community in our university, we will be able to increase the student's intellectual needs, and create new knowledge. Web-portal will promote the activity of e-community providing easy to access and personalized web circumstance. The emergence of Internet and Web access to all university services will force institutions to rethik everything. We should reconfigure our academic environment in order to adapt information technology and educational change, and must do change management based on their structure and culture.

• 충청수학회지
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• 제35권3호
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• pp.227-234
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• 2022

Let 𝔽_q be a finite field with odd q elements. In this article, we prove that if E ⊆ 𝔽^d_q, d ≥ 2, and |E| ≥ q, then there exists a set Y ⊆ 𝔽^d_q with |Y| ~ q^d such that for all y ∈ Y, the number of distances between the point y and the set E is ~ q. As a corollary, we obtain that for each set E ⊆ 𝔽^d_q with |E| ≥ q, there exists a set Y ⊆ 𝔽^d_q with |Y| ~ q^d so that any set E ∪ {y} with y ∈ Y determines a positive proportion of all possible distances. The averaging argument and the pigeonhole principle play a crucial role in proving our results.

• 해양환경안전학회지
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• 제22권2호
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• pp.167-173
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• 2016

본 연구에서는 정박지 해상교통 환경을 재현한 시뮬레이션을 통하여 정박지의 적정 면적을 결정하기 위한 방법론을 제시하고, 이를 울산항 E정박지에 적용하여 울산항 정박지 면적의 적정성을 비교 분석하기 위한 목적이 있다. 이를 위해 정박지 사용현황에 따른 시간별 정박지 점유율과 필요 정박지 면적 계산식을 제시하였다. 시간별 정박지 점유율 분석을 위해 시뮬레이션 알고리즘을 구성하고, MATLAB 프로그램을 이용하여 모델링하였으며, 이를 울산항 E 정박지에 적용하여 시간별 정박지 점유율과 필요 정박지 면적을 도출하고 현재 울산항 E정박지 면적과 비교 분석하였다. 그 결과 E1 정박지의 필요 정박지 면적은 현재 정박지 면적의 1.41배, E2와 E3 정박지는 0.90배, 0.96배로 분석되었다. 본 연구에서 제시한 정박지 면적의 적정성 분석 결과를 정박지 설계기준에 반영할 경우 정박지를 이용하는 선박의 안전에 도움을 줄 수 있을 것으로 판단된다.