• East Asian mathematical journal
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• v.35 no.1
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• pp.41-50
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• 2019

This paper is dealt with one-dimensional elliptic jumping problem with nonlinearities crossing n eigenvalues. We get one theorem which shows multiplicity results for solutions of one-dimensional elliptic boundary value problem with jumping nonlinearities. This theorem is that there exist at least two solutions when nonlinearities crossing odd eigenvalues, at least three solutions when nonlinearities crossing even eigenvalues, exactly one solutions and no solution depending on the source term. We obtain these results by the eigenvalues and the corresponding normalized eigenfunctions of the elliptic eigenvalue problem and Leray-Schauder degree theory.

• Honam Mathematical Journal
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• v.20 no.1
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• pp.89-95
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• 1998

We show by using some properties of an elliptic integral that certain scalar semilinear elliptic problem has a unique solution.

• Korean Journal of Mathematics
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• v.20 no.4
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• pp.533-540
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• 2012

We consider the multiplicity of the solutions for the elliptic boundary value problem with $C^1$ nonlinear term decaying at the origin. We get a theorem which shows the existence of the nontrivial solution for the elliptic problem with $C^1$ nonlinear term decaying at the origin. We obtain this result by reducing the elliptic problem with the $C^1$ nonlinear term to the el-liptic problem with bounded nonlinear term and then approaching the variational method and using the mountain pass geometry for the reduced the elliptic problem with bounded nonlinear term.

• Korean Journal of Mathematics
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• v.20 no.3
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• pp.333-341
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• 2012

We investigate the number of solutions for the superlinear elliptic bifurcation problem with Dirichlet boundary condition. We get a theorem which shows the existence of at least $k$ weak solutions for the superlinear elliptic bifurcation problem with boundary value condition. We obtain this result by using the critical point theory induced from invariant linear subspace and the invariant functional.

• East Asian mathematical journal
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• v.34 no.5
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• pp.549-570
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• 2018

We establish existence and bifurcation of global positive solutions for parametrized nonhomogeneous elliptic equations involving critical Sobolev-Hardy exponents and Hardy terms. The main approach to the problem is the variational method.

• Korean Journal of Mathematics
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• v.23 no.2
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• pp.259-267
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• 2015

This paper is devoted to investigate the existence of the solutions for a class of the noncooperative elliptic system involving critical Sobolev exponents. We show the existence of the negative solution for the problem. We show the existence of the unique negative solution for the system of the linear part of the problem under some conditions, which is also the negative solution of the nonlinear problem. We also consider the eigenvalue problem of the matrix.

• Journal of KIISE:Computer Systems and Theory
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• v.29 no.11
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• pp.622-633
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• 2002

Authentication and Key Agreement using password provide convenience and amenity, but what human can remember has extremely low entropy. To overcome its defects, AMP(Authentiration and key agreement via Memorable Password) which performs authentication and key agreement securely via low entropy password are presented. AMP uses Diffie-Hellman problem that depends on discrete logarithm problem. Otherwise, this thesis applies elliptic curve cryptosystem to AMP for further efficiency That is, this thesis presents EC-AMP(Elliptic Curve-AMP) protocol based on elliptic curve discrete logarithm problem instead of discrete logarithm problem, and shows its high performance through the implementation. EC-AMP secures against various attacks in the random oracle model just as AMP Thus, we nay supply EC-AMP to the network environment that requires authentication and key agreement to get both convenience and security from elliptic curve discrete logarithm problem.

• 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.

• Journal of the Korean Mathematical Society
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• v.39 no.3
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• pp.439-460
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• 2002

We prove the multiplicity of ordered positive radial solutions for a semilinear elliptic problem defined on an exterior domain. The key argument is to prove the existence of the third solution in presence of two known solutions. For this, we obtain some partial results related to three solutions theorem for certain singular boundary value problems. Proof are mainly based on the upper and lower solutions method and degree theory.

• Korean Journal of Mathematics
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• v.26 no.1
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• pp.9-21
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• 2018

We investigate existence and multiplicity of the solutions for elliptic boundary value problem with two singularities. We obtain one theorem which shows that there exists at least one nontrivial weak solution under some conditions on which the corresponding functional of the problem satisfies the Palais-Smale condition. We obtain this result by variational method and critical point theory.