• Journal for History of Mathematics
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• v.28 no.2
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• pp.85-102
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• 2015

Elliptic curves are a common theme among various fields of mathematics, such as number theory, algebraic geometry, complex analysis, cryptography, and mathematical physics. In the history of elliptic curves, we can find number theoretic problems on the one hand, and complex function theoretic ones on the other. The elliptic curve theory is a synthesis of those two indeed. As an overview of the history of elliptic curves, we survey the Diophantine equations of 3rd degree and the congruent number problem as some of number theoretic trails of elliptic curves. We discuss elliptic integrals and elliptic functions, from which we get a glimpse of idea where the name 'elliptic curve' came from. We explain how the solution of Diophantine equations of 3rd degree and elliptic functions are related. Finally we outline the BSD conjecture, one of the 7 millennium problems proposed by the Clay Math Institute, as an important problem concerning elliptic curves.

• Structural Engineering and Mechanics
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• v.73 no.6
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• pp.641-649
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• 2020

The determination of midtower longitudinal stiffness has become an essential component in the preliminary design of multi-tower suspension bridges. For a specific multi-tower suspension bridge, the midtower longitudinal stiffness must be controlled within a certain range to meet the requirements of sliding resistance coefficient and deflection-to-span ratio. This study presents a numerical method to divide different types of midtower and determine rational range of longitudinal stiffness for rigid midtower. In this method, influence curves of midtower longitudinal stiffness on sliding resistance coefficient and maximum vertical deflection-to-span ratio are first obtained from the finite element analysis. Then, different types of midtower are divided based on the regression analysis of influence curves. Finally, rational range for longitudinal stiffness of rigid midtower is derived. The Oujiang River North Estuary Bridge which is a three-tower four-span suspension bridge with two main spans of 800m under construction in China is selected as the subject of this study. This will be the first three-tower four-span suspension bridge with steel truss girders and concrete midtower in the world. The proposed method provides an effective and feasible tool for engineers to design midtower of multi-tower suspension bridges.

• ETRI Journal
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• v.28 no.6
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• pp.745-760
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• 2006

In this paper, we particularly deal with no $F_p$-rational two-torsion elliptic curves, where $F_p$ is the prime field of the characteristic p. First we introduce a shift product-based polynomial transform. Then, we show that the parities of (#E - 1)/2 and (#E' - 1)/2 are reciprocal to each other, where #E and #E' are the orders of the two candidate curves obtained at the last step of complex multiplication (CM)-based algorithm. Based on this property, we propose a method to check the parity by using the shift product-based polynomial transform. For a 160 bits prime number as the characteristic, the proposed method carries out the parity check 25 or more times faster than the conventional checking method when 4 divides the characteristic minus 1. Finally, this paper shows that the proposed method can make CM-based algorithm that looks up a table of precomputed class polynomials more than 10 percent faster.

• Communications of the Korean Mathematical Society
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• v.22 no.2
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• pp.207-208
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• 2007

In this work, authors considered a result concerning elliptic curves $y^2=x^3+cx$ over $\mathbb{F}_p$ mod 8, given at [1]. They noticed that there should be a slight change at this result. They give counterexamples and the correct version of the result.

• Korean Journal of Computational Design and Engineering
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• v.3 no.3
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• pp.183-191
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• 1998

There are many available algorithms based on the different approaches to solve the intersection problems between two curves. Among them, the implicitization method is frequently used since it computes precise solutions fast and is robust in lower degrees. However, once the degrees of curves to be intersected are higher than cubics, its computation time increases rapidly and the numerical stability gets worse. From this observation, it is natural to transform the original problem into a set of easier ones. Therefore, curves are subdivided appropriately depending on their geometric behavior and approximated by a set of rational quadratic Bezier cures. Then, the implicitization method is applied to compute the intersections between approximated ones. Since the solutions of the implicitization method are intersections between approximated curves, a numerical process such as Newton-Raphson iteration should be employed to find true intersection points. As the seeds of numerical process are close to a true solution through the mix-and-match process, the experimental results illustrates that the proposed algorithm is superior to other algorithms.

• Journal of the Korea Computer Graphics Society
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• v.17 no.1
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• pp.1-7
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• 2011

We present a new technique for constructing a sweep surface using two-parameter motion. Firstly, a new rational B-spline motion with two parameters is introduced, which is obtained by extending its orientation curve and scaling curve to surface counterparts. A sweep surface is then defined by a single vertex v under the two-parameter motion and allows to represent different u-directional iso-curves depending on parameter ${\upsilon}$. Efficient techniques for modeling and editing the surface are achieved by intuitively controlling the two-parameter motion. We demonstrate the effectiveness of our technique with experimental results on modeling and editing a 3D propeller model.

• 한국지구물리탐사학회:학술대회논문집
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• 2003.11a
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• pp.109-114
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• 2003

It is the common practice to reinforce excessively the secondary tunnel lining due to the lack of rational insights into the ground loosening loads. The main load of the secondary lining for drained-type tunnels is the ground loosening. The main cause of the load for secondary tunnel lining is the deterioration of the primary support members such as shotcrete, steel ribs, and rockbolts. Accordingly, the development of the analysis model to consider the ground-primary supports-secondary lining interaction is very important for the rational design of the secondary tunnel lining. In this paper, the interaction is conceptually described by the simple mass-spring model and the load transfer from the primary supports to the ground and the secondary lining is showed by the characteristic curves including the secondary lining reaction curve for the theoretical solution of a circular tunnel. And also, the application of this model to numerical analysis is verified in order to review the potential tool for practical tunnel problems with the complex conditions like non-circular shaped tunnels, multi-layered ground, sequential excavation and so on.

• Journal of applied mathematics & informatics
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• v.6 no.1
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• pp.203-216
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• 1999

In this paper a rational cubic interpolant spline with linear denominator has been constructed and it is used to constrain interpolation curves to be bounded in the given region. Necessary and sufficient conditions for the interpolant to satisfy the constraint have been developed. The existence conditions are computationally efficient and easy to apply. Finally the approximation properties have been studied.

• Kyungpook Mathematical Journal
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• v.49 no.3
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• pp.419-424
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• 2009

Let J be the Jacobian variety of a hyperelliptic curve over $\mathbb{Q}$. Let M be the field generated by all square roots of rational integers over a finite number field K. Then we prove that the Mordell-Weil group J(M) is the direct sum of a finite torsion group and a free $\mathbb{Z}$-module of infinite rank. In particular, J(M) is not a divisible group. On the other hand, if $\widetilde{M}$ is an extension of M which contains all the torsion points of J over $\widetilde{\mathbb{Q}}$, then $J(\widetilde{M}^{sol})/J(\widetilde{M}^{sol})_{tors}$ is a divisible group of infinite rank, where $\widetilde{M}^{sol}$ is the maximal solvable extension of $\widetilde{M}$.

• Journal of Environmental Health Sciences
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• v.21 no.1
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• pp.29-38
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• 1995

The purpose of this study is to improve the present methodology-for the estimation of optimal water supply from an impounding reservoir. The stochastic reservoir storage model presented in this paper is believed to be rational in that. the probability of reservoir depletion (return period) is to be calculated for the various monthly demands and storage capacities. The monthly flows are used to derive the reservoir storage capacity-monthly demand-probability curves at Dalcheon damsite and Hongcheon damsite in Han river basin.