• Bulletin of the Korean Mathematical Society
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• v.32 no.1
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• pp.19-23
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• 1995

We will estimate the lower bound of the first nonzero Neumann and Dirichlet eigenvalue of Laplacian equation on compact Riemannian manifold M with boundary. In case that the boundary of M has positive second fundamental form elements, Ly-Yau[3] gave the lower bound of the first nonzero neumann eigenvalue $\eta_1$. In case that the second fundamental form elements of $\partial$M is bounded below by negative constant, Roger Chen[4] investigated the lower bound of $\eta_1$. In [1], [2], we obtained the lower bound of the first nonzero Neumann eigenvalue is estimated under the condtion that the second fundamental form elements of boundary is bounded below by zero. Moreover, I realize that "the interior rolling $\varepsilon$ - ball condition" is not necessary when the first Dirichlet eigenvalue was estimated in [1].ed in [1].

• Bulletin of the Korean Mathematical Society
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• v.30 no.2
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• pp.229-238
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• 1993

Let M be an n-dimensional compact Riemannian manifold with boundary .part.M. We consider the Neumann eigenvalue problem on M of the equation (Fig.) where .upsilon. is the unit outward normal vector to the boundary .part.M. due to the importance of Poincare inequality for analysis on manifolds, one wishes to obtain the lower bound of the first non-zero eigenvalue .eta.$_{1}$ of (1.1). For the purpose of applications, it is important to relax the dependency of the lower bound on the geometric quantities. For general compact manifolds with convex boundary, Li-Yau [3] obtained the lower bound of .eta.$_{1}$. Recently, Roger Chen [1] investigated the lower bound of the first Neumann eigenvalue .eta.$_{1}$ on compact manifold M with nonconvex boundary. We investigate the lower bound .eta.$_{1}$ with the same conditions as those of Roger chen. But, using the different auxiliary function, we have the following theorem.

• Journal of the Chungcheong Mathematical Society
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• v.23 no.4
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• pp.629-633
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• 2010

In this paper we provide a sharp lower bound of the first Neumann eigenvalue of a compact hypersurface $\Sigma$ inside a convex set C in a Riemannian manifold under the assumption that ${\partial}{\Sigma}$ meets ${\partial}C$ orthogonally.

• Bulletin of the Korean Mathematical Society
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• v.58 no.6
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• pp.1315-1325
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• 2021

In this paper, we firstly prove some general inequalities for the Neumann eigenvalues for domains contained in a Euclidean n-space ℝⁿ. Using the general inequalities, we can derive some new Neumann eigenvalues estimates which include an upper bound for the (k + 1)^th eigenvalue and a new estimate for the gap of the consecutive eigenvalues. Moreover, we give sharp lower bound for the first eigenvalue of two kinds of eigenvalue problems of the biharmonic operator with Navier boundary condition on compact Riemannian manifolds with boundary and positive Ricci curvature.

• Communications of the Korean Mathematical Society
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• v.13 no.3
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• pp.579-587
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• 1998

In this paper we will estimate the lower bound of k-th Dirichlet eigenvalue $ \lambda_{k}$ / of Laplace equation on bounded domain in sphere.

• Honam Mathematical Journal
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• v.45 no.1
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• pp.82-91
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• 2023

In this paper, we study the nonlinear eigenvalue problem for some of the (p, q)-Laplacian on compact Finsler manifolds with zero boundary condition, and estimate the lower bound of the first eigenvalues for (p, q)-Laplace operators on Finsler manifolds.

• Structural Engineering and Mechanics
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• v.88 no.5
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• pp.493-500
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• 2023

The multiparameter eigenvalue method can be used to solve the damped finite element model updating problems. This method transforms the original problems into multiparameter eigenvalue problems. Comparing with the numerical methods based on various optimization methods, a big advantage of this method is that it can provide all possible choices of physical parameters. However, when solving the transformed singular multiparameter eigenvalue problem, the proposed method based on the generalised inverse of a singular matrix has some computational challenges and may fail. In this paper, more details on the transformation from the dynamic model updating problem to the multiparameter eigenvalue problem are presented and the structure of the transformed problem is also exposed. Based on this structure, the rigorous mathematical deduction gives the upper bound of the number of possible choices of the physical parameters, which confirms the singularity of the transformed multiparameter eigenvalue problem. More importantly, we present a row and column compression method to overcome the defect of the proposed numerical method based on the generalised inverse of a singular matrix. Also, two numerical experiments are presented to validate the feasibility and effectiveness of our method.

• 제어로봇시스템학회:학술대회논문집
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• 1993.10b
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• pp.425-427
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• 1993

Some upper bounds for the solution of the continuous algebraic Riccati equation are presented. These consist of bounds for summations of eigenvalues, products of eigenvalues, individual eigenvalues and the minimum eigenvalue of the solution matrix. Among these bounds, the first three are the first results for the upper bound of each case, while bounds for the minimum eigenvalue supplement the existing ones and require no side conditions for their validities.

• Journal of applied mathematics & informatics
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• v.33 no.5_6
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• pp.627-633
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• 2015

In this paper, we generalize the concept of the energy of Seidel matrix S(G) which denoted by S^α(G) and obtain some results related to this matrix. Also, we obtain an upper and lower bound for S^α(G) related to all of graphs with |detS(G)| ≥ (n - 1); n ≥ 3.

• Journal of the Korean Mathematical Society
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• v.48 no.2
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• pp.253-266
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• 2011

In this paper we give an upper bound of the first eigenvalue of the Laplace operator on a complete stable minimal hypersurface M in the hyperbolic space which has finite $L^2$-norm of the second fundamental form on M. We provide some sufficient conditions for minimal hypersurface of the hyperbolic space to be stable. We also describe stability of catenoids and helicoids in the hyperbolic space. In particular, it is shown that there exists a family of stable higher-dimensional catenoids in the hyperbolic space.