• Title, Summary, Keyword: Frobenius problem

### THE FROBENIUS PROBLEM FOR NUMERICAL SEMIGROUPS GENERATED BY THE THABIT NUMBERS OF THE FIRST, SECOND KIND BASE b AND THE CUNNINGHAM NUMBERS

• Song, Kyunghwan
• Bulletin of the Korean Mathematical Society
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• v.57 no.3
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• pp.623-647
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• 2020
• The greatest integer that does not belong to a numerical semigroup S is called the Frobenius number of S. The Frobenius problem, which is also called the coin problem or the money changing problem, is a mathematical problem of finding the Frobenius number. In this paper, we introduce the Frobenius problem for two kinds of numerical semigroups generated by the Thabit numbers of the first kind, and the second kind base b, and by the Cunningham numbers. We provide detailed proofs for the Thabit numbers of the second kind base b and omit the proofs for the Thabit numbers of the first kind base b and Cunningham numbers.

### Determination of Parameter Value in Constraint of Sparse Spectrum Fitting DOA Estimation Algorithm (희소성 스펙트럼 피팅 도래각 추정 알고리즘의 제한조건에 포함된 상수 결정법)

• Cho, Yunseung;Paik, Ji-Woong;Lee, Joon-Ho
• The Journal of Korean Institute of Communications and Information Sciences
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• v.41 no.8
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• pp.917-920
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• 2016
• SpSF algorithm is direction-of-arrival estimation algorithm based on sparse representation of incident signlas. Cost function to be optimized for DOA estimation is multi-dimensional nonlinear function, which is hard to handle for optimization. After some manipulation, the problem can be cast into convex optimiztion problem. Convex optimization problem tuns out to be constrained optimization problem, where the parameter in the constraint has to be determined. The solution of the convex optimization problem is dependent on the specific parameter value in the constraint. In this paper, we propose a rule-of-thumb for determining the parameter value in the constraint. Based on the fact that the noise in the array elements is complex Gaussian distributed with zero mean, the average of the Frobenius norm of the matrix in the constraint can be rigorously derived. The parameter in the constrint is set to be two times the average of the Frobenius norm of the matrix in the constraint. It is shown that the SpSF algorithm actually works with the parameter value set by the method proposed in this paper.

### The Sliding Control using Nonlinear Sliding Surfaces (비선형 슬라이딩 평면을 이용한 슬라이딩 제어)

• Han, Jong-Kil
• The Journal of the Korea institute of electronic communication sciences
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• v.7 no.5
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• pp.1133-1138
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• 2012
• In the paper, design of nonlinear sliding surfaces which are based on optimal control is studied, The state trajectory by the input of optimal control was obtained by Frobenius theorem and matrix decomposition method, was set the nonlinear sliding surfaces of the system. The states is maintained to sliding surfaces from initial states. As the result, robustness of the system can be guaranteed throughout an entire response of the system starting form the initial time instance, the uncertainty and external disturbance that can occur during the reaching time is removed, the problem of large control input was solved, and setting the sliding surfaces optimal path was able to reduce the tracking time. The validity of the proposed control scheme is shown in computer simulation for inverted pendulum.

### COMPLEX SUBMANIFOLDS IN REAL HYPERSURFACES

• Han, Chong-Kyu;Tomassini, Giuseppe
• Journal of the Korean Mathematical Society
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• v.47 no.5
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• pp.1001-1015
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• 2010
• Let M be a $C^{\infty}$ real hypersurface in $\mathbb{C}^{n+1}$, $n\;{\geq}\;1$, locally given as the zero locus of a $C^{\infty}$ real valued function r that is defined on a neighborhood of the reference point $P\;{\in}\;M$. For each k = 1,..., n we present a necessary and sufficient condition for there to exist a complex manifold of dimension k through P that is contained in M, assuming the Levi form has rank n - k at P. The problem is to find an integral manifold of the real 1-form $i{\partial}r$ on M whose tangent bundle is invariant under the complex structure tensor J. We present generalized versions of the Frobenius theorem and make use of them to prove the existence of complex submanifolds.

### Free Vibrations and Buckling of Rectangular Plates with Linearly Varying In-Plane Loading

• Chang, Kyong-Ho;Shim, Hyun-Ju;Kang, Jae-Hoon
• Journal of Korean Association for Spatial Structures
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• v.4 no.4
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• pp.99-111
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• 2004
• An exact solution procedure is formulated for the free vibration and buckling analysis of rectangular plates having two opposite edges simply supported when these edges are subjected to linearly varying normal stresses. The other two edges may be clamped, simply supported or free, or they may be elastically supported. The transverse displacement (w) is assumed as sinusoidal in the direction of loading (x), and a power series is assumed in the lateral (y) direction (i.e., the method of Frobenius). Applying the boundary conditions yields the eigenvalue problem of finding the roots of a fourth order characteristic determinant. Care must be exercised to obtain adequate convergence for accurate vibration frequencies and buckling loads, as is demonstrated by two convergence tables. Some interesting and useful results for vibration frequencies and buckling loads, and their mode shapes, are presented for a variety of edge conditions and in-plane loadings, especially pure in-plane moments.

### Direction-of-Arrival Estimation Using Linear Prediction Method in Conjunction with Signal Enhancement Approach (신호부각법과 결합된 선형예측방법을 이용한 도래각 추정)

• 오효성
• The Journal of Korean Institute of Electromagnetic Engineering and Science
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• v.10 no.6
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• pp.959-967
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• 1999
• In this paper, we propose a Linear Prediction Method(LPM) in conjunction with signal enhancement for solving the direction-of-arrival estimation problem of multiple incoherent plane waves incident on a uniform linear array. The basic idea of signal enhancement is that of finding the covariance matrix of given rank that lies closest to a given estimated matrix in Frobenius norm sense. It is well known that LPM has a high-resolution performance in general applications, while it provides a lower statistical performance in lower SNR environment. To solve this problem, the LPM combined with signal enhancement approach is herein proposed. Simulation results are illustrated to demonstrate the better performance of the proposed method than conventional LPM.

### THE PERIODIC JACOBI MATRIX PROCRUSTES PROBLEM

• Li, Jiao-Fen;Hu, Xi-Yan
• Journal of applied mathematics & informatics
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• v.28 no.3_4
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• pp.569-582
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• 2010
• The following "Periodic Jacobi Procrustes" problem is studied: find the Periodic Jacobi matrix X which minimizes the Frobenius (or Euclidean) norm of AX - B, with A and B as given rectangular matrices. The class of Procrustes problems has many application in the biological, physical and social sciences just as in the investigation of elastic structures. The different problems are obtained varying the structure of the matrices belonging to the feasible set. Higham has solved the orthogonal, the symmetric and the positive definite cases. Andersson and Elfving have studied the symmetric positive semidefinite case and the (symmetric) elementwise nonnegative case. In this contribution, we extend and develop these research, however, in a relatively simple way. Numerical difficulties are discussed and illustrated by examples.

### AN ITERATIVE ALGORITHM FOR SOLVING THE LEAST-SQUARES PROBLEM OF MATRIX EQUATION AXB+CYD=E

• Shen, Kai-Juan;You, Chuan-Hua;Du, Yu-Xia
• Journal of applied mathematics & informatics
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• v.26 no.5_6
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• pp.1233-1245
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• 2008
• In this paper, an iterative method is proposed to solve the least-squares problem of matrix equation AXB+CYD=E over unknown matrix pair [X, Y]. By this iterative method, for any initial matrix pair [$X_1,\;Y_1$], a solution pair or the least-norm least-squares solution pair of which can be obtained within finite iterative steps in the absence of roundoff errors. In addition, we also consider the optimal approximation problem for the given matrix pair [$X_0,\;Y_0$] in Frobenius norm. Given numerical examples show that the algorithm is efficient.

### High-Resolution Algorithm for Direction Finding of Multiple Incoherent Plane Waves (다중 인코히어런트 평면파의 도래각 추정을 위한 고분해능 알고리즘)

• 김영수;이성윤
• The Journal of Korean Institute of Communications and Information Sciences
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• v.24 no.9A
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• pp.1322-1328
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• 1999
• In this paper, we propose a Multiple Signal Classification(MUSIC) in conjunction with signal enhancement (SE-MUSIC) for solving the direction-of-arrival estimation problem of multiple incoherent plane waves incident on a uniform linear array. The proposed SE-MUSIC algorithms involve the following main two-step procedure : ( i )to find the enhanced matrix that possesses the prescribed properties and which lies closest to a given covariance matrix estimate in the Frobenius norm sense and (ii) to apply the MUSIC to the enhanced matrix. Simulation results are illustrated to demonstrate the better resolution and statistical performance of the proposed method than MUSIC at lower SNR.

### On the dynamics of rotating, tapered, visco-elastic beams with a heavy tip mass

• Zeren, Serkan;Gurgoze, Metin
• Structural Engineering and Mechanics
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• v.45 no.1
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• pp.69-93
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• 2013
• The present study deals with the dynamics of the flapwise (out-of-plane) vibrations of a rotating, internally damped (Kelvin-Voigt model) tapered Bernoulli-Euler beam carrying a heavy tip mass. The centroid of the tip mass is offset from the free end of the beam and is located along its extended axis. The equation of motion and the corresponding boundary conditions are derived via the Hamilton's Principle, leading to a differential eigenvalue problem. Afterwards, this eigenvalue problem is solved by using Frobenius Method of solution in power series. The resulting characteristic equation is then solved numerically. The numerical results are tabulated for a variety of nondimensional rotational speed, tip mass, tip mass offset, mass moment of inertia, internal damping parameter, hub radius and taper ratio. These are compared with the results of a conventional finite element modeling as well, and excellent agreement is obtained.