• Bulletin of the Korean Mathematical Society
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• v.55 no.3
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• pp.763-776
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• 2018

In this paper we consider the element-wise (Hadamard) product (or sum) of elliptic divisibility sequences and study the periodic structure of these sequences. We obtain that the element-wise product (or sum) of elliptic divisibility sequences are periodic modulo a prime p like linear recurrence sequences. Then we study periodicity properties of product sequences. We generalize our results to the case of modulo $p^l$ for some prime p > 3 and positive integer l. Finally we consider the p-adic behavior of product sequences and give a generalization of [9, Theorem 4].

• The Mathematical Education
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• v.58 no.2
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• pp.319-335
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• 2019

In this paper, we examine the effectiveness of calculation according to automation, which is one of Computational Thinking, by coding the conceptual process into Python language, focusing on the concept of divisibility in elementary school textbooks. The educational implications of these considerations are as follows. First, it is possible to make a field of learning that can revise the new mathematical concept through the opportunity to reinterpret the Conceptual Thinking learned in school mathematics from the perspective of Computational Thinking. Second, from the analysis of college students, it can be seen that many students do not have mathematical concepts in terms of efficiency of computation related to the divisibility. This phenomenon is a characteristic of the mathematics curriculum that emphasizes concepts. Therefore, it is necessary to study new mathematical concepts when considering the aspect of utilization. Third, all algorithms related to the concept of divisibility covered in elementary mathematics textbooks can be found to contain the notion of iteration in terms of automation, but little recursive activity can be found. Considering that recursive thinking is frequently used with repetitive thinking in terms of automation (in Computational Thinking), it is necessary to consider low level recursive activities at elementary school. Finally, it is necessary to think about mathematical Conceptual Thinking from the point of view of Computational Thinking, and conversely, to extract mathematical concepts from computer science's Computational Thinking.

• Communications of the Korean Mathematical Society
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• v.39 no.2
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• pp.437-460
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• 2024

In this paper we present the weighted shift operators having the property of moment infinite divisibility. We first review the monotone theory and conditional positive definiteness. Next, we study the infinite divisibility of sequences. A sequence of real numbers γ is said to be infinitely divisible if for any p > 0, the sequence γ^p = {γ^p_n}^∞_n=0 is positive definite. For sequences α = {α_n}^∞_n=0 of positive real numbers, we consider the weighted shift operators W_α. It is also known that W_α is moment infinitely divisible if and only if the sequences {γ_n}^∞_n=0 and {γ_n+1}^∞_n=0 of W_α are infinitely divisible. Here γ is the moment sequence associated with α. We use conditional positive definiteness to establish a new criterion for moment infinite divisibility of W_α, which only requires infinite divisibility of the sequence {γ_n}^∞_n=0. Finally, we consider some examples and properties of weighted shift operators having the property of (k, 0)-CPD; that is, the moment matrix M_γ(n, k) is CPD for any n ≥ 0.

• The Journal of the Institute of Internet, Broadcasting and Communication
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• v.14 no.5
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• pp.87-93
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• 2014

For n/m=qm+r, there is no simple divisibility rule for simple m=7 such that is the n multiply by m? This problem can be more complex for two or more digits of m. The Dunkels method has been known for generalized divisibility test method, but this method can not compute very large digits number that can not processed by computer. This paper suggests simple and exact divisibility method for m completely irrelevant n and m of digits. The proposed method sets $r_1=n_1n_2{\cdots}n_l(mod m)$ for $n=n_1n_2n_3{\cdots}n_k$, $m=m_1m_2{\cdots}m_l$. Then this method computes $r_i=r_{i-1}{\times}10+n_i(mod m)$, $i=2,3,{\cdots}k-l+1$ and reduces the digits of n one-by-one. The proposed method can be get the quotient and remainder with easy, fast and correct for various n,m experimental data.

• Journal of KIISE:Computer Systems and Theory
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• v.30 no.7_8
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• pp.408-416
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• 2003

An electronic cash system has to provide the security, to prevent the double spending and to support the divisibility of electronic cash for the easy of use. Divisible electronic cash system allows an electronic cash to be divided into subdivisions. Each subdivision is worth any desired value, but all values must add up to the original cash value. Divisible scheme brings some advantages. It reduces to make the change and also there is no necessity that a customer must withdraw a cash of the desired value whenever transactions occur. In this paper, we present an electronic cash protocol which provides the divisibility based on the double hash chain technique. Electronic cash is constructed in the form of coins. Coins, generated by the double hush chain, have different denominations. The divisibility based on the double hash chain technique. Electronic cash is constructed in the form of coins. Coins, generated by the double hash chain, have different denominations. The divisibility of an electronic cash is satisfied by the payment certificate, which is a pair of bank´s proxy signature received from the bank. When a customer pays the coin of subdivision, the fairness of that coin is certified by a customer´s signing instead of a bank. Although the proposed method does not guarantee user´s anonymity, it generates coins which cannot be forged, and the customer can use an electronic cash conveniently and efficiently with its divisibility.

• Journal of the Korean Mathematical Society
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• v.39 no.5
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• pp.765-773
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• 2002

Let $textsc{k}$＝$F_{q}$(T) be a rational function field. Let $\ell$ be a prime number with ($\ell$, q-1) ＝ 1. Let K/$textsc{k}$ be an elmentary abelian $\ell$-extension which is contained in some cyclotomic function field. In this paper, we study the $\ell$-divisibility of ideal class number $h_{K}$ of K by using cyclotomic units.s.s.

• Bulletin of the Korean Mathematical Society
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• v.50 no.4
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• pp.1357-1365
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• 2013

We find a lower bound on the number of real/inert imagi-nary/ramified imaginary quadratic extensions of the function field $\mathbb{F}_q(t)$ whose ideal class groups have an element of a fixed order, where $q$ is a power of 2.

• Proceedings of the Korea Institutes of Information Security and Cryptology Conference
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• 1994.11a
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• pp.51-66
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• 1994

A divisible off-line electronic cash system based on cut-and-choose has first been proposed by [OO91] and recently more efficient single term divisible cash system was presented in [EO94] which is based on Brand's scheme [Bra93]. In this paper, we present a different type of single term divisible electronic cash system which is more efficient than previously proposed systems such as [OO91], [YLR93], and [EO94] in the standpoint of the amount of communication, the number of modular multiplications required in the payment transactions, and the storage requirement in the withdrawal protocol. Our scheme is a modified version of [LL93], where the major improvement has been made in its withdrawal transaction to introduce untraceability and multi-spendability. We have borrowed the idea of the withdrawal protocol of our scheme from [EO94] with minor modifications. Transferability in our scheme allows only a finite number of transfer. Our scheme satisfies an the desirable properties of an electronic cash system such as untraceability, divisibility and transferability. In addition, we present a n-spendable cash. The basic idea of extension to multi-spendability has been borrowed from [Bra93] with minor modifications.

• Korean Journal of Mathematics
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• v.2
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• pp.35-38
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• 1994

• The Mathematical Education
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• v.22 no.2
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• pp.35-36
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• 1984