• Journal for History of Mathematics
• /
• v.13 no.2
• /
• pp.49-64
• /
• 2000

Every real number can be expressed as a simple continued fraction. In particular, a number is rational if and only if its simple continued fraction has a finite number of terms. Owing to this property, continued fractions have been a powerful tool which determines a real number to be rational or not. Continued fractions provide not only a series of best estimate for a real number, but also a useful method for finding near commensurabilities between events with different periods. In this paper, we investigate the history and some properties of continued fractions, and then consider their applications in several examples. Also we explain why the Fibonacci numbers and the Golden section appear in nature in terms of continued fractions, with some examples such as the arrangements of petals round a flower, leaves round branches and seeds on seed head.

• Journal of architectural history
• /
• v.6 no.3 s.13
• /
• pp.145-154
• /
• 1997

Le Corbusier had developed the Modulor replacing the Regulating Lines in the 1930's and 1940's. The lines can not produce lyrical ideas or creational concept but make plastic dimension pure only to keep in balance. First, the Grid of Proportions was developed and then the Modulor 1. The Modulor originated in one square, double square, the two golden means added or subtracted, the place of the right angle, human dimension, the Fibonacci series and divided Red series and Blue series. He developed the Modulor 2 for a reconciliation between the foot-and-inch system and the metre. As Corbusier devoted himself to a social and industrial problem, he was sure that a transcendental universality be in cosmos including a human being. His dream was to creat a orderly and harmonious utopia. What intrinsic harmony between Cosmos, nature and a human being be realized by mathematic order was the reason he hinted himself to the proportion.

• Journal of the Korean Operations Research and Management Science Society
• /
• v.23 no.3
• /
• pp.17-26
• /
• 1998

Shortest path problem in road network with turn penalties and prohibitions frequently arises from various transportation optimization models. In this paper, we propose a new algorithm for the shortest Path problem with turn prohibitions and delays. The proposed algorithm maintains distance labels of arcs, which is similar to labels of nodes of Dijkstra's algorithm. Fibonacci heap implementation of the proposed algorithm solves the problem in O(mn ＋ mlogm). We provide a new insight in transforming network with turn penalties and prohibitions into another network in which turn penalties and prohibitions are implicitly considered. The proposed algorithm is implemented using new data structure and compared with Ziliaskopoulos' algorithm. Computational results show that the proposed algorithm is very efficient.

• Honam Mathematical Journal
• /
• v.40 no.3
• /
• pp.549-559
• /
• 2018

In this paper, we consider the congruences involving harmonic numbers and the terms of the sequences {$U_{kn}$} and {$V_{kn}$}. For example, for an odd prime number p, $${\sum\limits_{i=1}^{p-1}}H_i{\frac{U_{k(i+m)}}{V^i_k}}{\equiv}{\frac{(-1)^kU_{k(m+1)}}{_pV^{p-1}_k}}(V^p_k-V_{kp})(mod\;p)$$, where $m{\in}{\mathbb{Z}}$ and $k{\in}{\mathbb{Z}}$ with $p{\nmid}V_k$.

• Journal of applied mathematics & informatics
• /
• v.28 no.5_6
• /
• pp.1439-1443
• /
• 2010

For a positive integer k $\geq$ 2, the k-bonacci sequence {$g^{(k)}_n$} is defined as: $g^{(k)}_1=\cdots=g^{(k)}_{k-2}=0$, $g^{(k)}_{k-1}=g^{(k)}_k=1$ and for n > k $\geq$ 2, $g^{(k)}_n=g^{(k)}_{n-1}+g^{(k)}_{n-2}+{\cdots}+g^{(k)}_{n-k}$. And the k-Lucas sequence {$l^{(k)}_n$} is defined as $l^{(k)}_n=g^{(k)}_{n-1}+g^{(k)}_{n+k-1}$ for $n{\geq}1$. In this paper, we give a representation of nth k-Lucas $l^{(k)}_n$ by using determinant.

• Journal of Korean Society for Quality Management
• /
• v.21 no.1
• /
• pp.181-189
• /
• 1993

Over the past two decades, forecasting has gained widespread acceptance as an integral part of business planning and decision making. Accurate forecasting is a prerequisite to successful planning. Accordingly, recent advances in forecasting techniques are of exceptional value to corporate planners. But most of forecasting mothods are reveal its limit and problem for precision and reliability duing to each relationship for raw data and possibility of explanation for each variable. Therefore, to construct the Integrated Forecasting Model(IFM) for Information Factor Analysis, it shoud be considered that whether law data has time lag and variables are explained. For this. following several method can be used : Least Square Method, Markov Process, Fibonacci series, Auto-Correlation, Cross-Correlation, Serial Correlation and Random Walk Theory. Thus, the unified property of these several functions scales the safety and growth of the system which may be varied time-to-time.

• Kyungpook Mathematical Journal
• /
• v.58 no.4
• /
• pp.711-732
• /
• 2018

In this paper, we give a new approach to solving the 2-variable subnormal completion problem (SCP for short). To this aim, we extend the notion of recursively generated weighted shifts, introduced by R. Curto and L. Fialkow, to 2-variable case. We next provide "concrete" necessary and sufficient conditions for the existence of solutions to the 2-variable SCP with minimal Berger measure. Furthermore, a short alternative proof to the propagation phenomena, for the subnormal weighted shifts in 2-variable, is given.

• Korean Journal of Mathematics
• /
• v.29 no.4
• /
• pp.733-740
• /
• 2021

Let (F_n)_n≥0 be the Fibonacci sequence and p be a prime number. For 1≤k≤m, the Fibonomial coefficient is defined as $$\[\array{m\\k}\]_F=\frac{F_{m-k+1}{\ldots}{F_{m-1}F_m}}{{F_1}{\ldots}{F_k}}$$ and $\[\array{m\\k}\]_F=0$ whan k>m. Let a and n be positive integers. In this paper, we find the conditions of prime number p which divides Fibonomial coefficient $\[\array{P^{a+n}\\{p^a}}\]_F$. Furthermore, we also find the conditions of p when $\[\array{P^{a+n}\\{p^a}}\]_F$ is not divisible by p.

• International journal of advanced smart convergence
• /
• v.13 no.2
• /
• pp.35-43
• /
• 2024

In this paper, we propose two 127-bit LFSR (Linear Feedback Shift Register)-based OTP (One-Time Password) generators. One is a 9-digit decimal OTP generator with thirty taps, while the other is a 12-digit OTP generator with forty taps. The 9-digit OTP generator includes only the positions of Fibonacci numbers to enhance randomness, whereas the 12-digit OTP generator includes the positions of prime numbers and odd numbers. Both proposed OTP generators are implemented on an Arduino module, and randomness evaluations indicate that the generators perform well across six criteria and are straightforward to implement with Arduino.

• The Journal of Korea Institute of Information, Electronics, and Communication Technology
• /
• v.14 no.4
• /
• pp.269-274
• /
• 2021

In secret communication, steganography is the sending and receiving of secret messages without being recognized by a third party. In the spatial domain method bitwise information is inserted into the virtual bit plane of the decomposed pixel values of the image. That is, the bitwise secret message is sequentially inserted into the least significant bit(LSB) of the image, which is a cover medium. In terms of application, the LSB is simple, but has a drawback that can be easily detected by a third party. If the upper bit plane is used to increase security, the image quality may deteriorate. In this paper, I present a method for concealing Hangul secret messages in image steganography based on the lo-th bit plane and the decomposition of modified pixel intensity values. After decomposing the Hangeul message to be hidden into choseong, jungseong and jongseong, then a shuffling process is applied to increase confidentiality and robustness. PSNR was used to confirm the efficiency of the proposed method. It was confirmed that the proposed technique has a smaller effect in terms of image quality than the method applying BCD and Fibonacci when inserting a secret message in the upper bit plane. When compared with the reference value, it was confirmed that the PSNR value of the proposed method was appropriate.