• Title/Summary/Keyword: Eigenvalue

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ENHANCING EIGENVALUE APPROXIMATION WITH BANK-WEISER ERROR ESTIMATORS

  • Kim, Kwang-Yeon
    • Korean Journal of Mathematics
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    • v.28 no.3
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    • pp.587-601
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    • 2020
  • In this paper we propose a way of enhancing eigenvalue approximations with the Bank-Weiser error estimators for the P1 and P2 conforming finite element methods of the Laplace eigenvalue problem. It is shown that we can achieve two extra orders of convergence than those of the original eigenvalue approximations when the corresponding eigenfunctions are smooth and the underlying triangulations are strongly regular. Some numerical results are presented to demonstrate the accuracy of the enhanced eigenvalue approximations.

S-Eigenvalue Concept for Linear Continuous-Time Systems with Probabilistic Uncertainties

  • Seo, Young-Bong;Park, Jae-Weon
    • 제어로봇시스템학회:학술대회논문집
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    • 2002.10a
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    • pp.44.5-44
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    • 2002
  • We propose a concept of the S-eigenvalue(stochastic-eigenvalue) along with corresponding eigenvector, and then we define the PDF corresponding to the S-eigenvalue on a complex plane. Based on the S-eigenvalue concept, we will establish the S-stability concept for linear continuous-time systems with probabilistic uncertainties in the system matrix. These results explicitly characterize how the S-eigenvalue in the complex plane may impose S-stability on S-eigenstructure assignment. Finally, we present numerical examples to illustrate the proposed concept.

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Determination of Minimum Eigenvalue in a Continuous-time Weighted Least Squares Estimator (연속시간 하중최소자승 식별기의 최소고우치 결정)

  • Kim, Sung-Duck
    • The Transactions of the Korean Institute of Electrical Engineers
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    • v.41 no.9
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    • pp.1021-1030
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    • 1992
  • When using a least squares estimator with exponential forgetting factor to identify continuous-time deterministic system, the problem of determining minimum eigenvalue is described in this paper. It is well known fact that the convergence rate of parameter estimates relies on various factors consisting of the estimator and especially, theirproperties can be directly affected by all eigenvalues in the parameter error differential equation. Fortunately, there exists only one adjusting eigenvalue in the given estimator and then, the parameter convergence rates depend on this minimum eigenvalue. In this note, a new result to determine the minimum eigenvalue is proposed. Under the assumption that the input has as many spectral lines as the number of parameter estimates, it can be proven that the minimum eigenvalue converges to a constant value, which is a function of the forgetting factor and the parameter estimates number.

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Comparison of the Natural Period Obtained by Eigenvalue Analysis and Ambient Vibration Measurement in Bearing-Wall Type Apartment (고유치해석과 진동계측을 통한 벽식 공동주택의 고유주기 비교)

  • Yoon, Sung-Won;Jeong, Sug-Chang;Lim, In-Sik
    • Journal of Korean Association for Spatial Structures
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    • v.6 no.3 s.21
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    • pp.43-50
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    • 2006
  • This paper is concerned with the natural periods of ambient vibration and eigenvalue analysis. Ambient vibration tests were conducted to four bearing-wall reinforced concrete buildings ranging from twelve to nineteen stories. The performance of modeling in eigenvalue analysis was investigated using consideration of rigidity out of the plane in the slab and the non-structural bearing wall. Measured natural period was also compared with the value by the KBC2005. Natural period of the short direction in eigenvalue analysis is well fitted with the measured one. In the other hand, Natural period of the long direction in eigenvalue analysis is slightly more overestimated than the measured one. Natural period of the long direction in eigenvalue analysis was found to be enhanced by considering the effect of the stiffness out of the plane of the slab and non-structural wall in the structural modeling.

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On lower bounds of eigenvalues for self adjoint operators

  • Lee, Gyou-Bong
    • Journal of the Korean Mathematical Society
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    • v.31 no.3
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    • pp.477-492
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    • 1994
  • For the eigenvalue problem of $Au = \lambda u$ where A is considered as a semi-bounded self-adjoint operator on a Hilbert space, we are used to apply two complentary methods finding upper bounds and lower bounds to the eigenvalues. The most popular method for finding upper bounds may be the Rayleigh-Ritz method which was developed in the 19th century while a method for computing lower bounds may be the method of intermediate eigenvalue problems which has been developed since 1950's. In the method of intermediate eigenvalue problems (IEP), we consider the original operator eigenvalue problem as a perturbation of a simpler, resolvable, self-adjoint eigenvalue problem, called a base problem, that gives rough lower bounds.

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EIGENVALUE MONOTONICITY OF (p, q)-LAPLACIAN ALONG THE RICCI-BOURGUIGNON FLOW

  • Azami, Shahroud
    • Communications of the Korean Mathematical Society
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    • v.34 no.1
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    • pp.287-301
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    • 2019
  • In this paper we study monotonicity the first eigenvalue for a class of (p, q)-Laplace operator acting on the space of functions on a closed Riemannian manifold. We find the first variation formula for the first eigenvalue of a class of (p, q)-Laplacians on a closed Riemannian manifold evolving by the Ricci-Bourguignon flow and show that the first eigenvalue on a closed Riemannian manifold along the Ricci-Bourguignon flow is increasing provided some conditions. At the end of paper, we find some applications in 2-dimensional and 3-dimensional manifolds.

Structural Dynamics Optimization by Second Order Sensitivity with respect to Finite Element Parameter (유한요소 구조 인자의 2차 민감도에 의한 동적 구조 최적화)

  • Kim, Yong-Yun
    • Transactions of the Korean Society of Machine Tool Engineers
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    • v.15 no.3
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    • pp.8-16
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    • 2006
  • This paper discusses design sensitivity analysis and its application to a structural dynamics modification. Eigenvalue derivatives are determined with respect to the element parameters, which include intrinsic property parameters such as Young's modulus, density of the material, diameter of a beam element, thickness of a plate element, and shape parameters. Derivatives of stiffness and mass matrices are directly calculated by derivatives of element matrices. The first and the second order derivatives of the eigenvalues are then mathematically derived from a dynamic equation of motion of FEM model. The calculation of the second order eigenvalue derivative requires the sensitivity of its corresponding eigenvector, which are developed by Nelson's direct approach. The modified eigenvalue of the structure is then evaluated by the Taylor series expansion with the first and the second derivatives of eigenvalue. Numerical examples for simple beam and plate are presented. First, eigenvalues of the structural system are numerically calculated. Second, the sensitivities of eigenvalues are then evaluated with respect to the element intrinsic parameters. The most effective parameter is determined by comparing sensitivities. Finally, we predict the modified eigenvalue by Taylor series expansion with the derivatives of eigenvalue for single parameter or multi parameters. The examples illustrate the effectiveness of the eigenvalue sensitivity analysis for the optimization of the structures.

An Application of the Multigrid Method to Eigenvalue problems (복합마디방법의 고유치문제에 응용)

  • Lee, Gyou-Bong;Kim, Sung-Soo;Sung, Soo-Hak
    • The Journal of Natural Sciences
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    • v.8 no.2
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    • pp.9-11
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    • 1996
  • We apply a full mutigrid scheme to computing eigenvalues of the Laplace eigenvalue problem with Dirichlet boundary condition. We use finite difference method to get an algebraic equation and apply inverse power method to estimating the smallest eigenvalue. Our result shows that combined method of inverse power method and full multigrid scheme is very effective in calculating eigenvalue of the eigenvalue problem.

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A Cooperative Spectrum Sensing Method based on Eigenvalue and Superposition for Cognitive Radio Networks (인지무선네트워크를 위한 고유값 및 중첩기반의 협력 스펙트럼 센싱 기법)

  • Miah, Md. Sipon;Koo, Insoo
    • The Journal of the Institute of Internet, Broadcasting and Communication
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    • v.13 no.4
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    • pp.39-46
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    • 2013
  • Cooperative spectrum sensing can improve sensing reliability, compared with single node spectrum sensing. In addition, Eigenvalue-based spectrum sensing has also drawn a great attention due to its performance improvement over the energy detection method in which the more smoothing factor, the better performance is achieved. However, the more smoothing factor in Eignevalue-based spectrum sensing requires the more sensing time. Furthermore, more reporting time in cooperative sensing will be required as the number of nodes increases. Subsequently, we in this paper propose an Eigenvalue and superposition-based spectrum sensing where the reporting time is utilized so as to increase the number of smoothing factors for autocorrelation calculation. Simulation result demonstrates that the proposed scheme has better detection probability in both local as well as global detection while requiring less sensing time as compared with conventional Eigenvalue-based detection scheme.

Students' conceptual development of eigenvalue and eigenvector based on the situation model (상황모델에 기반한 학생들의 고유치와 고유벡터 개념발달)

  • Shin, Kyung-Hee
    • The Mathematical Education
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    • v.51 no.1
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    • pp.77-88
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    • 2012
  • This qualitative research provides a situation model, which is designed for promoting learning of eigenvalue and eigenvector. This study also demonstrates the usefulness of the model through a small groups discussion. Particularly, participants of the discussion were asked to decide the numbers of milk cows in order to make constant amounts of cheese production. Through such discussions, subjects understood the notion of eigenvalue and eigenvector. This study has following implications. First of all, the present research finds significance of situation model. A situation model is useful to promote learning of mathematical notions. Subjects learn the notion of eigenvalue and eigenvector through the situation model without difficulty. In addition, this research demonstrates potentials of small groups discussion. Learners participate in discussion more actively under small group debates. Such active interaction is necessary for situation model. Moreover, this study emphasizes the role of teachers by showing that patience and encouragement of teachers promote students' feeling of achievement. The role of teachers are also important in conveying a meaning of eigenvalue and eigenvector. Therefore, this study concludes that experience of learning the notion of eigenvalue and eigenvector thorough situation model is important for teachers in future.