• Title/Summary/Keyword: fractal mathematics

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Paradigm and Pan-paradigm in Mathematics and Architecture (수학과 건축의 패러다임과 범 패러다임)

  • Kye, Young Hee
    • Communications of Mathematical Education
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    • v.27 no.2
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    • pp.165-177
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    • 2013
  • Mathematics teaching is often more effective when teachers connect the contents of mathematics with history, culture, and social events. In the history of mathematics, the 'paradigm' theory from Thomas Kuhn's scientific revolution is very effective to explain the revolutionary process of development in mathematics, and his theory has been widely quoted in the history of science and economics. However, it has not been appropriate to use his theory in the other fields. This is due to the fact that the scope of Kuhn's paradigm theory is limited to mathematics and science. In this study, this researcher introduced pan-paradigm as a general concept that encompasses all, since through any relation in the field of mathematics and architecture, Thomas Kuhn's theory of paradigm does not explain the phenomena. That is, at the root of various cultures there exist always a 'collective unconsciousness' and 'demands of the times,' and these two factors by synergism form values and controlling principles common to various parts of the culture, and this synergism leads the cultural activities, the process of which is a phenomenon called pan-paradigm.

MEASURE DERIVATIVE AND ITS APPLICATIONS TO $\sigma$-MULTIFRACTALS

  • Kim, Tae-Sik;Ahn, Tae-Hoon;Kim, Gwang-Il
    • Journal of the Korean Mathematical Society
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    • v.36 no.1
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    • pp.229-241
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    • 1999
  • The fractal space is often associated with natural phenomena with many length scales and the functions defined on this space are usually not differentiable. First we define a $\sigma$-multifractal from $\sigma$-iterated function systems with probability. We introduce the measure derivative through the invariant measure of the $\sigma$-multifractal. We show that the non-differentiable function on the $\sigma$-multifractal can be differentiable with respect to this measure derivative. We apply this result to some examples of ordinary differential equations and diffusion processes on $\sigma$-multifractal spaces.

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FOURIER'S TRANSFORM OF FRACTIONAL ORDER VIA MITTAG-LEFFLER FUNCTION AND MODIFIED RIEMANN-LIOUVILLE DERIVATIVE

  • Jumarie, Guy
    • Journal of applied mathematics & informatics
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    • v.26 no.5_6
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    • pp.1101-1121
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    • 2008
  • One proposes an approach to fractional Fourier's transform, or Fourier's transform of fractional order, which applies to functions which are fractional differentiable but are not necessarily differentiable, in such a manner that they cannot be analyzed by using the so-called Caputo-Djrbashian fractional derivative. Firstly, as a preliminary, one defines fractional sine and cosine functions, therefore one obtains Fourier's series of fractional order. Then one defines the fractional Fourier's transform. The main properties of this fractal transformation are exhibited, the Parseval equation is obtained as well as the fractional Fourier inversion theorem. The prospect of application for this new tool is the spectral density analysis of signals, in signal processing, and the analysis of some partial differential equations of fractional order.

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Hausdorff dimension of some sub-similar sets

  • Kim, Tae-Sik
    • Bulletin of the Korean Mathematical Society
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    • v.35 no.3
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    • pp.397-408
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    • 1998
  • We often use the Hausdorff dimension as a tool of measuring how complicate the fractal is. But it is usually very difficult to calculate that value. So there have been many tries to find the dimension of the given set and most of these are related to the density theorem of invariant measure. The aims of this paper are to introduce the k-irreducible subsimilar sets as a generalization of the set defined by V.Drobot and J.Turner in ([1]) and calculate their Hausdorff dimensions by using algebraic methods.

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DIMENSIONS OF A DERANGED CANTOR SET WITH SPECIFIC CONTRACTION RATIOS

  • Baek, In-Soo
    • Bulletin of the Korean Mathematical Society
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    • v.41 no.2
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    • pp.269-274
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    • 2004
  • We investigate a deranged Cantor set (a generalized Cantor set) using the similar method to find the dimensions of cookie-cutter repeller. That is, we will use a Gibbs measure which is a weak limit of a subsequence of discrete Borel measures to find the dimensions. The deranged Cantor set that will be considered is a generalized form of a perturbed Cantor set (a variation of the symmetric Cantor set) and a cookie-cutter repeller.

RELATION BETWEEN FRACTAL MEASURES AND CANTOR MEASURES

  • Baek, In-Soo
    • Communications of the Korean Mathematical Society
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    • v.22 no.2
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    • pp.241-246
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    • 2007
  • We investigate the relation between Hausdorff(packing) measure and lower(packing) Cantor measure on a deranged Cantor set. If the infimum of some distortion of contraction ratios is positive, then Hausdorff(packing) measure and lower(packing) Cantor measure of a deranged Cantor set are equivalent except for some singular behavior for packing measure case. It is a generalization of already known result on the perturbed Cantor set.

ASSOUAD DIMENSION: ANTIFRACTAL METRIZATION, POROUS SETS, AND HOMOGENEOUS MEASURES

  • Luukkainen, Jouni
    • Journal of the Korean Mathematical Society
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    • v.35 no.1
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    • pp.23-76
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    • 1998
  • We prove that a non-empty separable metrizable space X admits a totally bounded metric for which the metric dimension of X in Assouad's sense equals the topological dimension of X, which leads to a characterization for the latter. We also give a characterization based on this Assouad dimension for the demension (embedding dimension) of a compact set in a Euclidean space. We discuss Assouad dimension and these results in connection with porous sets and measures with the doubling property. The elementary properties of Assouad dimension are proved in an appendix.

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Applying Stochastic Fractal Search Algorithm (SFSA) in Ranking the Determinants of Undergraduates Employability: Evidence from Vietnam

  • DINH, Hien Thi Thu;CHU, Ngoc Nguyen Mong;TRAN, Van Hong;NGUYEN, Du Van;NGUYEN, Quyen Le Hoang Thuy To
    • The Journal of Asian Finance, Economics and Business
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    • v.7 no.12
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    • pp.583-591
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    • 2020
  • Employability has recently become the first target of the national higher education. Its model has been updated to catch the new trend of Industry 4.0. This paper aims at analyzing and ranking the determinants of undergraduate employability, focusing on business and economics majors in Ho Chi Minh City, Vietnam. In-depth interviews with content analysis have been primarily conducted to reach an agreement on a key group of factors: human capital, social capital, and identity. The Stochastic Fractal Search Algorithm (SFSA) is then applied to rank the sub-factors. Human capital is composed of three major elements: attitude, skill, and knowledge. Social capital is approached at both structural and cognitive aspects with three typical types: bonding, bridging, and linking. The analysis has confirmed the change of priority in employability determinants. Human capital is still a driver but the priority of attitude has been confirmed in the contemporary context. Then, social capital with the important order of linking, bridging, and bonding is emphasized. Skill, knowledge, and identity share the least weight in the model. It is noted that identity is newly proposed in the model but a certain role has been found. The findings are crucial for education strategies to enhance university graduate employability.

Exploration of the educational possibilities of one-stroke drawing problems of complex figure using programming (프로그래밍을 이용한 복잡한 도형의 한붓그리기 문제의 교육적 가능성 탐색 )

  • Cheong, Yong Wook
    • Communications of Mathematical Education
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    • v.38 no.2
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    • pp.247-261
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    • 2024
  • This study propose the educational potential of an activity that solves the task of one-stroke drawing of complex figures using a drag-and-drop type educational programming language such as Scratch. The problem of determining whether a given shape is capable of one-stroke drawing is a separate problem from actually finding the path of one-stroke drawing and implementing it through programming. In particular, finding a path that allows one-stroke drawing of complex shapes with regularity and implementing it through programming requires problem-solving capabilities based on the convergence of various mathematical knowledge. Accordingly, in this study, problems related to one-stroke drawing concerning polygon-related shapes, tessellation-related shapes, and fractal shapes were presented, and the results of one-stroke drawing programming of the shapes were exemplified. In addition, the mathematical knowledge and computational thinking elements necessary for the solution of the illustrated problem were analyzed. This study is significant as a new example of the mathematics education that combines mathematics and information.

On the symmetric sierpinski gaskets

  • Song, Hyun-Jong;Kang, Byung-Sik
    • Communications of the Korean Mathematical Society
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    • v.12 no.1
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    • pp.157-163
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    • 1997
  • Based on a n-regular polygon $P_n$, we show that $r_n = 1/(2 \sum^{[(n-4)/4]+1}_{j=0}{cos 2j\pi/n)}$ is the ratio of contractions $f_i(1 \leq i \leq n)$ at each vertex of $P_n$ yielding a symmetric gasket $G_n$ associated with the just-touching I.F.S. $g_n = {f_i $\mid$ 1 \leq i \leq n}$. Moreover we see that for any odd n, the ratio $r_n$ is still valid for just-touching I.F.S $H_n = {f_i \circ R $\mid$ 1 \leq i \leq n}$ yielding another symmetric gasket $H_n$ where R is the $\pi/n$-rotation with respect to the center of $P_n$.

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