• ETRI Journal
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• v.40 no.1
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• pp.111-121
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• 2018

To find the emerging patterns (EPs) in streaming transaction data, the streaming is first divided into some time windows containing a number of transactions. Itemsets are generated from transactions in each window, and then the emergence of itemsets is evaluated between two windows. In the tilted-time windows model (TTWM), it is assumed that people need support data with finer accuracy from the most recent windows, while accepting coarser accuracy from older windows. Therefore, a limited array's elements are used to maintain all support data in a way that condenses old windows by merging them inside one element. The capacity of elements that accommodates the windows inside is modeled using a particular number sequence. However, in a stream, as new data arrives, the current array updating mechanisms lead to many null elements in the array and cause data incompleteness and inaccuracy problems. Two models derived from TTWM, logarithmic TTWM and Fibonacci windows model, also inherit the same problems. This article proposes a novel push-front Fibonacci windows model as a solution, and experiments are conducted to demonstrate its superiority in finding more EPs compared to other models.

• Honam Mathematical Journal
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• v.41 no.3
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• pp.569-579
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• 2019

Cusp (parabolic) points in the extended modular group ${\bar{\Gamma}}$ are basically the images of infinity under the group elements. This implies that the cusp points of ${\bar{\Gamma}}$ are just rational numbers and the set of cusp points is $Q_{\infty}=Q{\cup}\{{\infty}\}$.The Farey graph F is the graph whose set of vertices is $Q_{\infty}$ and whose edges join each pair of Farey neighbours. Each rational number x has an integer continued fraction expansion (ICF) $x=[b_1,{\cdots},b_n]$. We get a path from ${\infty}$ to x in F as $<{\infty},C_1,{\cdots},C_n>$ for each ICF. In this study, we investigate relationships between Fibonacci numbers, Farey graph, extended modular group and ICF. Also, we give a computer program that computes the geodesics, block forms and matrix represantations.

• Honam Mathematical Journal
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• v.45 no.3
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• pp.418-432
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• 2023

In this study, we consider the Hankel and Gram matrices which are defined by the elements of special number sequences. Firstly, the eigenvalues, determinant, and norms of the Hankel matrix defined by special number sequences are obtained. Afterwards, using the relationship between the Gram and Hankel matrices, the eigenvalues, determinants, and norms of the Gram matrices defined by number sequences are given.

• Journal of the Korean Mathematical Society
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• v.55 no.2
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• pp.425-445
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• 2018

A set {$a_1,\;a_2,{\ldots},\;a_m$} of positive integers is called a Diophantine m-tuple if $a_ia_j+1$ is a perfect square for all $1{\leq}i$ < $j{\leq}m$. In this paper, we show that the form of third element in Diophantine pairs and develop some results which are needed to prove the extendibility of the Diophantine pair {a, b} with some conditions. By using this result, we prove the extendibility of Diophantine pairs {$F_{k-2}F_{k+1},\;F_{k-1}F_{k+2}$} and {$F_{k-2}F_{k-1},\;F_{k+1}F_{k+2}$}, where $F_n$ is the n-th Fibonacci number.

• Journal of the Korean Mathematical Society
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• v.40 no.6
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• pp.921-932
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• 2003

A number of variants of Hofstadter's original sequence have been investigated. Here we investigate a collection of similarly defined such sequences which give rise to intriguingly different results.

• Journal of the Chungcheong Mathematical Society
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• v.32 no.4
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• pp.475-490
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• 2019

Let C^(q) be the q-commuting arithmetic table of (x + y)ⁿ with noncommuting variables. We study sequential properties of diagonal sums of C^(q) with various q > 0. One of our results shows that each diagonal sum plays like Fibonacci number with some minor conditions.

• KSBB Journal
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• v.29 no.5
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• pp.336-342
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• 2014

Asexual benthic polyp reproduction plays a major role in the jellyfish bloom. Recent studies found that temperature is the most important factor to regulate the budding rate of the polyps. We established a simple dynamic model to count the number of polyps depending on the variation of temperature with two data sets from different places. The population of polyps was counted through the budding rate and the number of budding times by Fibonacci sequence. It is assumed that the budding rate depends on the temperature only. The budding rate of the asexual reproduction shows very sensitive to the distribution of the seawater temperature. The model was tested to the temperature data of Ansan located in the west sea of Korea. The results indicate that this model can be useful to predict the blooms of Aurelia aurita polyps, which may have considerable influence on the bloom of medusa. The shape of temperature curve plays a key role in the predicting the bloom of Aurelia aurita polyps.

• Communications of the Korean Mathematical Society
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• v.37 no.4
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• pp.977-988
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• 2022

The Padovan sequence is the third-order linear recurrence (𝓟_n)_n≥0 defined by 𝓟_n = 𝓟_n-2 + 𝓟_n-3 for all n ≥ 3 with initial conditions 𝓟₀ = 0 and 𝓟₁ = 𝓟₂ = 1. In this paper, we investigate a generalization of the Padovan sequence called the k-generalized Padovan sequence which is generated by a linear recurrence sequence of order k ≥ 3. We present recurrence relations, the generalized Binet formula and different arithmetic properties for the above family of sequences.

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

• Korean Journal of Mathematics
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• v.28 no.3
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• pp.539-554
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• 2020

A set {a₁, a₂, …, a_m} of m distinct positive integers is called a D(-1)-m-tuple if the product of any distinct two elements in the set decreased by one is a perfect square. In this paper, we find a solution of Pellian equations which is constructed by D(-1)-triples and using this result, we prove the extendibility of D(-1)-pair with some conditions.