• Bulletin of the Korean Mathematical Society
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• v.55 no.1
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• pp.115-137
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• 2018

Let $R_{u{^2},v^2,w^2,p}$ be a finite non chain ring ${\mathbb{F}}_p[u,v,w]{\langle}u^2,\;v^2,\;w^2,\;uv-vu,\;vw-wv,\;uw-wu{\rangle}$, where p is a prime number. This ring is a part of family of Frobenius rings. In this paper, we explore the structures of cyclic codes over the ring $R_{u{^2},v^2,w^2,p}$ of arbitrary length. We obtain a unique set of generators for these codes and also characterize free cyclic codes. We show that Gray images of cyclic codes are 8-quasicyclic binary linear codes of length 8n over ${\mathbb{F}}_p$. We also determine the rank and the Hamming distance for these codes. At last, we have given some examples.

• ETRI Journal
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• v.29 no.6
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• pp.769-778
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• 2007

This paper proposes a multiplication algorithm for $F_{p^m}$, which can be efficiently applied to many pairs of characteristic p and extension degree m except for the case that 8p divides m(p-1). It uses a special class of type- Gauss period normal bases. This algorithm has several advantages: it is easily parallelized; Frobenius mapping is easily carried out since its basis is a normal basis; its calculation cost is clearly given; and it is sufficiently practical and useful when parameters k and m are small.

• 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.

• Journal of applied mathematics & informatics
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• v.33 no.1_2
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• pp.145-155
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• 2015

In this paper, we study the bounds of the coefficients of the characteristic polynomial of the Frobenius endomorphism of the Jacobian of dimension three over a finite field. We provide explicitly computable bounds for the coefficients of the characteristic polynomial. In addition, we present the counting points algorithm for computing a group of the Jacobian of genus 3 hyperelliptic curves over a finite field with large characteristic. Based on these bounds, we found an efficient search space that was used in the counting points algorithm on genus 3 curves. The algorithm was explained and verified through simple examples.

• Journal of applied mathematics & informatics
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• v.30 no.1_2
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• pp.101-109
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• 2012

In this paper, we present an algorithm for computing the number of points on the Jacobian varieties of one-dimensional family of genus 3 nonhyperelliptic curves over finite fields. We also provide the explicit formula of the characteristic polynomial of the Frobenius endomorphism of the Jacobian of $C:y^3=x^4+{\alpha}$ over a finite field $\mathbb{F}_p$ with $p{\equiv}1$ (mod 3) and $p{\neq}1$ (mod 4). Moreover, we give some implementation results using Gaudry-Schost method. A 162-bit order is computed in 97 s on a Pentium IV 2.13 GHz computer using our algorithm.

• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.25 no.6
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• pp.420-425
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• 2015

In this paper a transfer matrix method is developed to solve for bending vibration of beam with linearly reduced width, and subsequently used to determine the exact natural frequencies for such problems. The differential equation, shear force, and bending moment are derived from Hamilton's principle, and the roots of the differential equation are computed using the power series solution of the Frobenius method. The effect of various taper ratio for bending vibration of beam with linearly reduced width is investigated in detail, and to validate the accuracy of the proposed method the results computed are compared with those given from commercial software(ANSYS).

• 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.

• Kyungpook Mathematical Journal
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• v.54 no.4
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• pp.607-617
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• 2014

Let S = C(r,m), the finite cyclic semigroup with index r and period m. Each subsemigroup of S is cyclic if and only if either r = 1; r = 2; or r = 3 with m odd. For $r{\neq}1$, the maximum value of the minimum number of elements in a (minimal) generating set of a subsemigroup of S is 1 if r = 3 and m is odd; 2 if r = 3 and m is even; (r-1)/2 if r is odd and unequal to 3; and r/2 if r is even. The number of cyclic subsemigroups of S is $r-1+{\tau}(m)$. Formulas are also given for the number of 2-generated subsemigroups of S and the total number of subsemigroups of S. The minimal generating sets of subsemigroups of S are characterized, and the problem of counting them is analyzed.

• Journal of Communications and Networks
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• v.18 no.3
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• pp.320-326
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• 2016

In existing literature on multiple-input multiple-output (MIMO) relaying communication systems, antenna selection is often implemented by maximizing the channel capacity or the output single-to-noise ratio (SNR). In this paper, we propose an energy-efficient low-complexity antenna selection scheme for MIMO relaying communication systems. The proposed algorithm is based on beamforming and maximizing the Frobenius norm to jointly optimize the transmit power, number of active antennas, and antenna subsets at the source, relaying and destination. We maximize the energy efficiency between the link of source to relay and the link of relay to destination to obtain the maximum energy efficiency of the system, subject to the SNR constraint. Compared to existing antenna selection methods forMIMO relaying communication systems, simulation results demonstrate that the proposed method can save more power in term of energy efficiency, while having lower computational complexity.

• Kyungpook Mathematical Journal
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• v.56 no.2
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• pp.329-342
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• 2016

Recently, Korobov polynomials have been received a lot of attention, which are discrete analogs of Bernoulli polynomials. In particular, these polynomials are used to derive some interpolation formulas of many variables and a discrete analog of the Euler summation formula. In this paper, we extend these family of polynomials to consider the Korobov polynomials of the fifth kind and of the sixth kind. We present several explicit formulas and recurrence relations for these polynomials. Also, we establish a connection between our polynomials and several known families of polynomials.