• 한국수학사학회지
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• 제24권2호
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• pp.61-70
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• 2011

본 논문에서는 해석학, 기하학, 정수론, 위상수학, 수리물리학 등 수학의 거의 모든 분야에서 훌륭한 업적을 창출하여 현대의 수학에 가장 큰 영향을 미천 위대한 수학자 중에 하나인 독일의 수학자 리이만(Bernhard Riemann, 1826~1866)의 생애와 그가 이룬 업적을 살펴보고, 리이만 방정식에 대하여 고찰한다.

• 한국수학사학회지
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• 제28권3호
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• pp.133-149
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• 2015

Arrivals of Hilbert and Minkowski at $G\ddot{o}ttingen$ put mathematical science there in full flourish. They further extended its strong mathematical tradition of Gauss and Riemann. Though Riemann envisioned Finsler metric and gave an example of it in his inaugural lecture of 1854, Finsler geometry was officially named after Minkowski's academic grandson Finsler. His tool to generalize Riemannian geometry was the calculus of variations of which his advisor $Carath\acute{e}odory$ was a master. Another $G\ddot{o}ttingen$ graduate Busemann regraded Finsler geometry as a special case of geometry of metric spaces. He was a student of Courant who was a student of Hilbert. These figures all at $G\ddot{o}ttingen$ created and developed Finsler geometry in its early stages. In this paper, we investigate history of works on Finsler geometry contributed by these frontiers.

• Korean Journal of Mathematics
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• 제27권3호
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• pp.743-765
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• 2019

Conceptual and procedural knowledge of integration is necessary not only in calculus but also in real analysis, complex analysis, and differential geometry. However, students show not only focused understanding of procedural knowledge but also limited understanding on conceptual knowledge of integration. So they are good at computation but don't recognize link between several concepts. In particular, Riemann sum is helpful in solving applied problem, but students are poor at understanding structure of Riemann sum. In this study, we try to investigate understanding on conceptual and procedural knowledge of integration and to analyze errors. Conducting experimental class of Riemann sum, we investigate the understanding of Riemann sum structure and so present the implications about improvement of integration teaching.

• 대한수학회지
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• 제56권2호
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• pp.421-437
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• 2019

We establish some Hessian comparison theorems and volume comparison theorems for Riemann-Finsler manifolds under various line radial integral curvature bounds. As their applications, we obtain some results on first eigenvalue, Gromov pre-compactness and generalized Myers theorem for Riemann-Finsler manifolds under suitable line radial integral curvature bounds. Our results are new even in the Riemannian case.

• Journal of applied mathematics & informatics
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• 제9권2호
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• pp.543-559
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• 2002

In this paper we formulate the linear theory for compressible fluids in cylindrical geometry with small perturbation at the material interface. We derive the first order equations in the smooth regions, boundary conditions at the shock fronts and the contact interface by linearizing the Euler equations and Rankine-Hugoniot conditions. The small amplitude solution formulated in this paper will be important for calibration of results from full numerical simulation of compressible fluids in cylindrical geometry.

• 한국수자원학회논문집
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• 제42권4호
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• pp.271-279
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• 2009

본 연구는 단순한 직사각형 하도에서 발생한 댐 붕괴 및 홍수전파 등에서 만족스러운 결과를 보였던 Riemann 근사해법을 이용한 1차원 유한체적기법을 불규칙한 하도형상의 자연하도에 적용하기 위하여 새로운 기법을 개발하는 것이 목적이다. 이를 위하여 자연하천 단면을 등가의 직사각형 단면으로 변환하는 개념을 도입하였으며, 그 결과, 운동량방정식이 수정되었다. 새롭게 개발된 기법을 정확해가 존재하는 삼각형 단면하도의 댐 붕괴 흐름에 적용하고 그 결과를 비교함으로써, 기법의 정확성 및 적용성이 검증되었다. 단면의 형상 및 단면간 거리가 균일하지 않는 자연하도에 적용한 결과는 실측수위와 비교하여 홍수파의 전파 양상, 도수의 발생 위치 및 크기, 그리고 전 구간에서의 최대 수위가 잘 일치함을 나타낸다. 본 연구결과로부터 기존의 균일한 단면을 사용하여 개발된 기법들을 복잡한 수치처리과정 없이 자연하천 단면에 직접 적용할 수 있을 것으로 판단된다.

• 대한수학회지
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• 제40권3호
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• pp.517-561
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• 2003

We establish a link between the study of completely integrable systems of partial differential equations and the study of generic submanifolds in $\mathbb{C}$. Using the recent developments of Cauchy-Riemann geometry we provide the set of symmetries of such a system with a Lie group structure. Finally we determine the precise upper bound of the dimension of this Lie group for some specific systems of partial differential equations.

• 한국수학사학회지
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• 제31권1호
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• pp.37-51
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• 2018

In this paper, our main concern is the historical development of the Finsler geometry in Debrecen, Hungary initiated by L. Berwald. First we look into the research trend in Berwald's days affected by the $G{\ddot{o}}ttingen$ mathematicians from C. Gauss and downward. Then we study how he was motivated to concentrate on the then completely new research area, Finsler geometry. Finally we examine the course of establishing Hungarian Debrecen school of Finsler geometry via the scholars including O. Varga, A. $Rapcs{\acute{a}}k$, L. $Tam{\acute{a}}ssy$ all deeply affected by Berwald after his settlement in Debrecen, Hungary.

• 대한수학회지
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• 제31권4호
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• pp.569-608
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• 1994

The subject matter of this survey has to do with holomorphic maps from a compact Riemann surface to projective space, which are also called algebrac curves; the theory we survey lies at the crossroads of function theory, projective geometry, and commutative algebra (although we should mention that the present survey de-emphasizes the algebraic aspect). Algebraic curves have been vigorously and continuously investigated since the time of Riemann. The reasons for the preoccupation with algebraic curves amongst mathematicians perhaps have to do with-other than the usual usual reason, namely, the herd mentality prompting us to follow the leads of a few great pioneering methematicians in the field-the fact that algebraic curves possess a certain simple unity together with a rich and complex structure. From a differential-topological standpoint algebraic curves are quite simple as they are neatly parameterized by a single discrete invariant, the genus. Even the possible complex structures of a fixed genus curve afford a fairly complete description. Yet there are a multitude of diverse perspectives (algebraic, function theoretic, and geometric) often coalescing to yield a spectacular result.

• 한국수학사학회지
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• 제28권2호
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• pp.103-115
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• 2015

Contemporary differential geometry owes much to the theory of connections on the bundles over manifolds. In this paper, following the work of Gauss on surfaces in 3 dimensional space and the work of Riemann on the curvature tensors on general n dimensional Riemannian manifolds, we will investigate how differential geometry had been developed from mid 19th century to early 20th century through lives and mathematical works of Christoffel, Ricci-Curbastro and Levi-Civita. Christoffel coined the Christoffel symbol and Ricci used the Christoffel symbol to define the notion of covariant derivative. Levi-Civita completed the theory of absolute differential calculus with Ricci and discovered geometric meaning of covariant derivative as parallel transport.