• The Bulletin of The Korean Astronomical Society
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• v.41 no.1
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• pp.38.2-38.2
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• 2016

The mapping of dark matter clustering from real to redshift spaces introduces the anisotropic property to the measured density power spectrum in redshift space, known as the Redshift Space Distortion (hereafter RSD) effect. The mapping formula is intrinsically non-linear, which is complicated by the higher order polynomials due to the indefinite cross correlations between the density and velocity fields, and the Finger-of-God (hereafter FoG) effect due to the randomness of the peculiar velocity field. Furthermore, the rigorous test of this mapping formula is contaminated by the unknown non-linearity of the density and velocity fields, including their auto- and cross-correlations, for calculating which our theoretical calculation breaks down beyond some scales. Whilst the full higher order polynomials remains unknown, the other systematics can be controlled consistently within the same order truncation in the expansion of the mapping formula, as shown in this paper. The systematic due to the unknown non-linear density and velocity fields is removed by separately measuring all terms in the expansion using simulations. The uncertainty caused by the velocity randomness is controlled by splitting the FoG term into two pieces, 1) the non-local FoG term being independent of the separation vector between two different points, and 2) the local FoG term appearing as an indefinite polynomials which is expanded in the same order as all other perturbative polynomials. Using 100 realizations of simulations, we find that the best fitted non-local FoG function is Gaussian, with only one scale-independent free parameter, and that our new mapping formulation accurately reproduces the observed power spectrum in redshift space at the smallest scales by far, up to k ~ 0.3 h/Mpc, considering the resolution of future experiments.

• Communications of the Korean Mathematical Society
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• v.15 no.1
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• pp.29-36
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• 2000

The T-ideal of F(X) generated by $x^{n}$ for all x $\in$ X, is generated also by the symmetric polynomials. For each symmetric poly-nomial, there corresponds one row of the incidence matrix. Finding the nilpotency of nil-algebra of nil-index n is equivalent to determining the smallest integer N such that the (n, N)-incidence matrix has rank equal to N!. In this work, we show that the (n, (equation omitted)$^{(1,....,n)}$-incidence matrix is center-symmetric.

• Bulletin of the Korean Mathematical Society
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• v.55 no.2
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• pp.573-586
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• 2018

Let R be a noncommutative prime ring of characteristic different from 2, Q be its maximal right ring of quotients and C be its extended centroid. Suppose that $f(x_1,{\ldots},x_n)$ be a noncentral multilinear polynomial over $C,b{\in}Q,F$ a b-generalized derivation of R and d is a nonzero derivation of R such that d([F(f(r)), f(r)]) = 0 for all $r=(r_1,{\ldots},r_n){\in}R^n$. Then one of the following holds: (1) there exists ${\lambda}{\in}C$ such that $F(x)={\lambda}x$ for all $x{\in}R$; (2) there exist ${\lambda}{\in}C$ and $p{\in}Q$ such that $F(x)={\lambda}x+px+xp$ for all $x{\in}R$ with $f(x_1,{\ldots},x_n)^2$ is central valued in R.

• Transactions of the Korean Society of Mechanical Engineers
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• v.18 no.8
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• pp.1967-1973
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• 1994

The free vibration of a thin plate with three different boundary conditions is discussed in this paper. A semi-analytical approach to the plate problems has been exploited using computer algebra system(CAS). The approximate solutions are assumed as algebraic polynomials that satisfy the appropriate boundary conditions. In order to solve problems, Galerkin method is used, which is known as an ineffective tool for practical engineering problems, being involved with a large number of multiple integration and differentiation. All the admissible functions used in this paper are generated automatically by CAS otherwise a tedious algebraic manipulations should be done by hand. One, six and fifteen-term solutions in terms of frequency parameters are presented and compared with exact solutions. Even using one-term solution, the comparison with existing data shows good agreement and accuracy of the present method.

• Journal of the Korean Mathematical Society
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• v.41 no.4
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• pp.717-735
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• 2004

A problem raised by Selfridge and solved by Pomerance asks to find the pairs (a, b) of natural numbers for which $2^a\;-\;2^b$ divides $n^a\;-\;n^b$ for all integers n. Vajaitu and one of the authors have obtained a generalization which concerns elements ${\alpha}_1,\;{\cdots},\;{{\alpha}_{\kappa}}\;and\;{\beta}$ in the ring of integers A of a number field for which ${\Sigma{\kappa}{i=1}}{\alpha}_i{\beta}^{{\alpha}i}\;divides\;{\Sigma{\kappa}{i=1}}{\alpha}_i{z^{{\alpha}i}}\;for\;any\;z\;{\in}\;A$. Here we obtain a further generalization, proving the corresponding finiteness results in a multidimensional setting.

• Journal of the Korean Mathematical Society
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• v.59 no.6
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• pp.1067-1082
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• 2022

This paper presents a new optimal three-step eighth-order family of iterative methods for finding multiple roots of nonlinear equations. Different from the all existing optimal methods of the eighth-order, the new iterative scheme is constructed using one function and three derivative evaluations per iteration, preserving the efficiency and optimality in the sense of Kung-Traub's conjecture. Theoretical results are verified through several standard numerical test examples. The basins of attraction for several polynomials are also given to illustrate the dynamical behaviour and the obtained results show better stability compared to the recently developed optimal methods.

• Bulletin of the Korean Mathematical Society
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• v.55 no.2
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• pp.625-631
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• 2018

Let R be a commutative ring with identity and let R[x] be the collection of polynomials with coefficients in R. There are a lot of multiplications in R[x] such that together with the usual addition, R[x] becomes a ring that contains R as a subring. These multiplications are from a class of functions ${\lambda}$ from ${\mathbb{N}}_0$ to ${\mathbb{N}}$. The trivial case when ${\lambda}(i)=1$ for all i gives the usual polynomial ring. Among nontrivial cases, there is an important one, namely, the case when ${\lambda}(i)=i!$ for all i. For this case, it gives the well-known Hurwitz polynomial ring $R_H[x]$. In this paper, we completely determine the Krull dimension of $R_H[x]$ when R is a $Pr{\ddot{u}}fer$ domain. Let R be a $Pr{\ddot{u}}fer$ domain. We show that dim $R_H[x]={\dim}\;R+1$ if R has characteristic zero and dim $R_H[x]={\dim}\;R$ otherwise.

• The Journal of Society for e-Business Studies
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• v.11 no.2
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• pp.1-11
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• 2006

This paper proposes an efficient inversion algorithm for Galois field GF(2n) by using a modified multi-bit shifting method based on the Montgomery algorithm. It is well known that the efficiency of arithmetic algorithms depends on the basis and many foregoing papers use either polynomial or optimal normal basis. An inversion algorithm, which modifies a multi-bit shifting based on the Montgomery algorithm, is studied. Trinomials and AOPs (all-one polynomials) are tested to calculate the inverse. It is shown that the suggested inversion algorithm reduces the computation time up to 26 % of the forgoing multi-bit shifting algorithm. The modified algorithm can be applied in various applications and is easy to implement.

• Journal of the Korea Institute of Information Security & Cryptology
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• v.9 no.4
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• pp.3-10
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• 1999

More concerns are concentrated in finite fields arithmetic as finite fields being applied for Elliptic curve cryptosystem coding theory and etc. Finite fields arithmetic is affected in represen -tation of those. Optimal normal basis is effective in hardware implementation and polynomial field which is effective in the basis conversion with optimal normal basis and show that the arithmetic of finite field with the basis is effective in software implementation.

• Journal of the Korea Institute of Information Security & Cryptology
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• v.18 no.3
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• pp.45-50
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• 2008

The choice of an irreducible polynomial and the representation of elements have influence on the efficiency of operators for finite fields. This paper suggests two serial multiplier for the extention field GF$(p^n)$ where p is odd prime. A serial multiplier using an irreducible binomial consists of (2n+5) resisters, 2 MUXs, 2 multipliers of GF(p), and 1 adder of GF(p). It obtains the mulitplication result after $n^2+n$ clock cycles. A serial multiplier using an AOP consists of (2n+5) resisters, 1 MUX, 1 multiplier of CF(p), and 1 adder of GF(p). It obtains the mulitplication result after $n^2$+3n+2 clock cycles.