• Industrial Engineering and Management Systems
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• v.13 no.1
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• pp.52-66
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• 2014

Promotion of a closed-loop supply chain requires disassembly systems that recycle end-of-life (EOL) assembled products. To operate the recycling disassembly system, parts selection is environmentally and economically carried out with non-destructive or destructive disassembly, and the recycling rate of the whole EOL product is determined. As the number of disassembled parts increases, the recycling rate basically increases. However, the labor cost also increases and brings lower profit, which is the difference between the recovered material prices and the disassembly costs. On the other hand, since the precedence relationships among disassembly tasks of the product also change with the parts selections, it is also required to optimize allocation of the tasks in designing a disassembly line. In addition, because information is required for such a design, the recycling rate, profit of each part and disassembly task times take precedence among the disassembly tasks. However, it is difficult to obtain that information in advance before collecting the actual EOL product. This study proposes and analyzes an optimal disassembly system design using integer programming with the environmental and economic parts selection (Igarashi et al., 2013), which harmonizes the recycling rate and profit using recyclability evaluation method (REM) developed by Hitachi, Ltd. The first stage involves optimization of environmental and economic parts selection with integer programming with ${\varepsilon}$ constraint, and the second stage involves optimization of the line balancing with integer programming in terms of minimizing the number of stations. The first and second stages are generally and mathematically formulized, and the relationships between them are analyzed in the cases of cell phones, computers and cleaners.

• Journal of Integrative Natural Science
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• v.12 no.2
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• pp.44-46
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• 2019

In this paper, we study a ratio variation of Nicomachus' theorem by computing ${\sum\limits_{k=1}^{n}}[rk]$ and ${\sum\limits_{k=1}^{n}}[rk]^3$, where r is a rational number.

• Journal of applied mathematics & informatics
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• v.24 no.1_2
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• pp.313-323
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• 2007

The equitable total chromatic number ${\chi}_{et}(G)$ of a graph G is the smallest integer ${\kappa}$ for which G has a total ${\kappa}$-coloring such that the number of vertices and edges in any two color classes differ by at most one. In this paper, we determine the equitable total chromatic number of one class of the graphs.

• ETRI Journal
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• v.27 no.5
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• pp.617-627
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• 2005

This paper proposes an efficient scalar multiplication algorithm for hyperelliptic curves, which is based on the idea that efficient endomorphisms can be used to speed up scalar multiplication. We first present a new Frobenius expansion method for special hyperelliptic curves that have Gallant-Lambert-Vanstone (GLV) endomorphisms. To compute kD for an integer k and a divisor D, we expand the integer k by the Frobenius endomorphism and the GLV endomorphism. We also present improved scalar multiplication algorithms that use the new expansion method. By our new expansion method, the number of divisor doublings in a scalar multiplication is reduced to a quarter, while the number of divisor additions is almost the same. Our experiments show that the overall throughputs of scalar multiplications are increased by 15.6 to 28.3 % over the previous algorithms when the algorithms are implemented over finite fields of odd characteristics.

• Journal of Magnetics
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• v.22 no.1
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• pp.29-33
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• 2017

We report a micromagnetic numerical study on different quasi-crystal formations of magnetic vortices in a rich variety of dynamic transient states in soft magnetic nano-disks. Only the application of spin-polarized dc currents to a single magnetic vortex leads to the formation of topological-soliton quasi-crystals composed of different configurations of skyrmions with positive and negative half-integer numbers (magnetic vortices and antivortices). Such topological object formations in soft magnets, not only in the absence of Dzyaloshinskii-Moriya interaction but also without magnetocrystalline anisotropy, are discussed in terms of two different topological charges, the winding number and the skyrmion number. This work offers an insight into the dynamic topological-spin-texture quasi-crystal formations in soft magnets.

• Bulletin of the Korean Mathematical Society
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• v.60 no.4
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• pp.863-872
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• 2023

Let n ⩾ 2 be an integer, we denote the smallest integer b such that gcd {(ⁿ_k) : b < k < n - b} > 1 as b(n). For any prime p, we denote the highest exponent α such that p^α | n as v_p(n). In this paper, we partially answer a question asked by Hong in 2016. For a composite number n and a prime number p with p | n, let n = a_mp^m + r, 0 ⩽ r < p^m, 0 < a_m < p. Then we have $$v_p\(\text{gcd}\{\(n\\k\)\;:\;b(n)1\}\)=\{\array{1,&&a_m=1\text{ and }r=b(n),\\0,&&\text{otherwise.}}$$

• Bulletin of the Korean Mathematical Society
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• v.61 no.4
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• pp.907-915
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• 2024

Let f be an integral quadratic form in k variables, F the Gram matrix corresponding to a ℤ-basis of ℤ^k. For r ∈ F^-1ℤ^k, a rational number n with f(r) ≡ n mod ℤ and a positive integer c, set N_f(n, r; c) := #{x ∈ ℤ^k/cℤ^k : f(x + r) ≡ n mod c}. Siegel showed that for each prime p, there is a number w depending on r and n such that N_f(n, r; p^ν+1) = p^k-1N_f(n, r; p^ν) holds for every integer ν > w and gave a rough estimation on the upper bound for such w. In this short note, we give a more explicit estimation on this bound than Siegel's.

• Honam Mathematical Journal
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• v.37 no.4
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• pp.529-547
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• 2015

In this paper, we proved some properties of the greatest expanded numbers, and give the method to determine the greatest expanded numbers and find the integer x for which $S_{q,p}(x)-x$ is the largest. Additionally, we provide an algorithm to find the greatest expanded number.

• Communications of the Korean Mathematical Society
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• v.31 no.3
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• pp.447-450
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• 2016

Let $F_n$ and $L_n$ be the nth Fibonacci number and Lucas number, respectively. The order of appearance of m in the Fibonacci sequence, denoted by z(m), is the smallest positive integer k such that m divides $F_k$. Marques obtained the formula of $z(L^k_n)$ in some cases. In this article, we obtain the formula of $z(L^k_n)$ for all $n,k{\geq}1$.

• Journal of the Korea Institute of Information and Communication Engineering
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• v.12 no.7
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• pp.1218-1226
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• 2008

With the development of semiconductor integrated technology and with the increasing use of multimedia functions in computer, more functions have been implemented as hardware. Nowadays, most microprocessors beyond 32 bits generally implement an integer multiplier as hardware. However, as for a divider, only specific microprocessor implements traditional SRT algorithm as hardware due to complexity of implementation and slow speed. This paper suggested an algorithm that uses a multiplier, 'w bit $\times$ w bit = 2w bit', to process $\frac{N}{D}$ integer division. That is, the reciprocal number D is first calculated, and then multiply dividend N to process integer division. In this paper, when the divisor D is '$D=0.d{\times}2^L$, 0.5 < 0.d < 1.0', approximate value of ' $\frac{1}{D}$', '$1.g{\times}2^{-L}$', which satisfies ' $0.d{\times}1.g=1+e$, $e<2^{-w}$', is defined as over reciprocal number and then an algorithm for over reciprocal number is suggested. This algorithm multiplies over reciprocal number '$01.g{\times}2^{-L}$' by dividend N to process $\frac{N}{D}$ integer division. The algorithm suggested in this paper doesn't require additional revision, because it can calculate correct reciprocal number. In addition, this algorithm uses only multiplier, so additional hardware for division is not required to implement microprocessor. Also, it shows faster speed than the conventional SRT algorithm and performs operation by word unit, accordingly it is more suitable to make compiler than the existing division algorithm. In conclusion, results from this study could be used widely for implementation SOC(System on Chip) and etc. which has been restricted to microprocessor and size of the hardware.