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Littérature scientifique sur le sujet « Mandelbrot sets »

Auteur : Grafiati

Publié le 4 juin 2021

Mis à jour le 30 juillet 2024

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Articles de revues sur le sujet "Mandelbrot sets"

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LIU, XIANG-DONG, ZHI-JIE LI, XUE-YE ANG et JIN-HAI ZHANG. « MANDELBROT AND JULIA SETS OF ONE-PARAMETER RATIONAL FUNCTION FAMILIES ASSOCIATED WITH NEWTON'S METHOD ». Fractals 18, n^o 02 (juin 2010) : 255–63. http://dx.doi.org/10.1142/s0218348x10004841.

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In this paper, general Mandelbrot and Julia sets of one-parameter rational function families associated with Newton's method were discussed. The bounds of these general Mandelbrot sets and two formulas for calculating the number of different periods periodic points of these rational functions were given. The relations between general Mandelbrot sets and common Mandelbrot sets of zn + c (n ∈ Z, n ≥ 2), along with the relations between general Mandelbrot sets and their corresponding Julia sets were investigated. Consequently, the results were found in the study: there are similarities between the Mandelbrot and Julia sets of one-parameter rational function families associated with Newton's method and the Mandelbrot and Julia sets of zn + c (n ∈ Z, n ≥ 2).

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Mu, Beining. « Fuzzy Julia Sets and Fuzzy Superior Julia Sets ». Highlights in Science, Engineering and Technology 72 (15 décembre 2023) : 375–80. http://dx.doi.org/10.54097/5c5hp748.

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This article examine the past study of fuzzy Mandelbrot set and fuzzy superior Mandelbrot set, then give the definition of fuzzy Julia sets and fuzzy superior Julia sets with the idea of utilizing membership functions to represent the escape velocity of Julia sets or superior Julia sets of each complex number in the definition of fuzzy Mandelbrot set and fuzzy superior Mandelbrot set inherited while the membership functions are selected to distinguish complex numbers with different escape velocity and same orbit. Then some examples of fuzzy Julia sets and fuzzy superior Julia sets are presented. With some observation of the examples, some analytical and topological properties of fuzzy Julia sets are demonstrated and sketches of the proofs are also presented. In the part of fuzzy superior Julia sets, some trivial conclusions as well as an example is presented where more complicated properties which involves the study into very chaotic behaviors are left open.

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Jha, Ketan, et Mamta Rani. « Control of Dynamic Noise in Transcendental Julia and Mandelbrot Sets by Superior Iteration Method ». International Journal of Natural Computing Research 7, n^o 2 (avril 2018) : 48–59. http://dx.doi.org/10.4018/ijncr.2018040104.

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Researchers and scientists are attracted towards Julia and Mandelbrot sets constantly. They analyzed these sets intensively. Researchers have studied the perturbation in Julia and Mandelbrot sets which is due to different types of noises, but transcendental Julia and Mandelbrot sets remained ignored. The purpose of this article is to study the perturbation in transcendental Julia and Mandelbrot sets. Also, we made an attempt to control the perturbation in transcendental sets by using superior iteration method.

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Danca, Marius-F. « Mandelbrot Set as a Particular Julia Set of Fractional Order, Equipotential Lines and External Rays of Mandelbrot and Julia Sets of Fractional Order ». Fractal and Fractional 8, n^o 1 (19 janvier 2024) : 69. http://dx.doi.org/10.3390/fractalfract8010069.

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This paper deepens some results on a Mandelbrot set and Julia sets of Caputo’s fractional order. It is shown analytically and computationally that the classical Mandelbrot set of integer order is a particular case of Julia sets of Caputo-like fractional order. Additionally, the differences between the fractional-order Mandelbrot set and Julia sets from their integer-order variants are revealed. Equipotential lines and external rays of a Mandelbrot set and Julia sets of fractional order are determined.

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Tassaddiq, Asifa, Muhammad Tanveer, Muhammad Azhar, Waqas Nazeer et Sania Qureshi. « A Four Step Feedback Iteration and Its Applications in Fractals ». Fractal and Fractional 6, n^o 11 (9 novembre 2022) : 662. http://dx.doi.org/10.3390/fractalfract6110662.

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Fractals play a vital role in modeling the natural environment. The present aim is to investigate the escape criterion to generate specific fractals such as Julia sets, Mandelbrot sets and Multi-corns via F-iteration using complex functions h(z)=zn+c, h(z)=sin(zn)+c and h(z)=ezn+c, n≥2,c∈C. We observed some beautiful Julia sets, Mandelbrot sets and Multi-corns for n = 2, 3 and 4. We generalize the algorithms of the Julia set and Mandelbrot set to visualize some Julia sets, Mandelbrot sets and Multi-corns. Moreover, we calculate image generation time in seconds at different values of input parameters.

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Yan, De Jun, Xiao Dan Wei, Hong Peng Zhang, Nan Jiang et Xiang Dong Liu. « Fractal Structures of General Mandelbrot Sets and Julia Sets Generated from Complex Non-Analytic Iteration F_m(z)=z¯^m+c ». Applied Mechanics and Materials 347-350 (août 2013) : 3019–23. http://dx.doi.org/10.4028/www.scientific.net/amm.347-350.3019.

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In this paper we use the same idea as the complex analytic dynamics to study general Mandelbrot sets and Julia sets generated from the complex non-analytic iteration . The definition of the general critical point is given, which is of vital importance to the complex non-analytic dynamics. The general Mandelbrot set is proved to be bounded, axial symmetry by real axis, and have (m+1)-fold rotational symmetry. The stability condition of periodic orbits and the boundary curve of stability region of one-cycle are given. And the general Mandelbrot sets are constructed by the escape-time method and the periodic scanning algorithm, which present a better understanding of the structure of the Mandelbrot sets. The filled-in Julia sets Km,c have m-fold structures. Similar to the complex analytic dynamics, the general Mandelbrot sets are kinds of mathematical dictionary or atlas that map out the behavior of the filled-in Julia sets for different values of c.

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KOZMA, ROBERT T., et ROBERT L. DEVANEY. « Julia sets converging to filled quadratic Julia sets ». Ergodic Theory and Dynamical Systems 34, n^o 1 (21 août 2012) : 171–84. http://dx.doi.org/10.1017/etds.2012.115.

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AbstractIn this paper we consider singular perturbations of the quadratic polynomial $F(z) = z^2 + c$ where $c$ is the center of a hyperbolic component of the Mandelbrot set, i.e., rational maps of the form $z^2 + c + \lambda /z^2$. We show that, as $\lambda \rightarrow 0$, the Julia sets of these maps converge in the Hausdorff topology to the filled Julia set of the quadratic map $z^2 + c$. When $c$ lies in a hyperbolic component of the Mandelbrot set but not at its center, the situation is much simpler and the Julia sets do not converge to the filled Julia set of $z^2 + c$.

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Al-Salami, Hassanein Q. « Some Properties of the Mandelbrot Sets M(Q_α) ». JOURNAL OF UNIVERSITY OF BABYLON for Pure and Applied Sciences 31, n^o 2 (29 juin 2023) : 263–69. http://dx.doi.org/10.29196/jubpas.v31i2.4683.

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The aim of this work is to study the concept of the parameter plane, so we are interested in introducing the Mandebrot set,as well as the Mandelbrot set is a fractals ,whereas, if we zoom in on any small piece in the border regions of the shape, we notice that it is similar to the original shape, that is the Mandelbrot set, with special topological specifications, and we introduce of some properties of the Mandelbrot sets for the map in the form , where is a complex constant, and we prove that the conjecture is the Duadi-Hubbard theorem . By studying of the parameter plane that the Mandelbrot set is a component form in one of its details in the dynamical plane it is the Julia set , in other words every small detail of the Mandelbrot set represents the Julia set, so there is more than one characteristic of the Julia set, we will limit ourselves only to presenting the connection of the Julia set for .

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Sekovanov, Valeriy S., Larisa B. Rybina et Kseniya Yu Strunkina. « The study of the frames of Mandelbrot sets of polynomials of the second degree as a means of developing the originality of students' thinking ». Vestnik Kostroma State University. Series : Pedagogy. Psychology. Sociokinetics, n^o 4 (2019) : 193–99. http://dx.doi.org/10.34216/2073-1426-2019-25-4-193-199.

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The article presents a methodology for studying the frames of Mandelbrot sets of polynomials of the second degree of a complex variable, based on the integration of analytical methods, mathematical programming and the use of computer graphics. A connection is established between the frames of the first and second orders of Mandelbrot sets of functions and with the curves – cardioid, lemniscate and circle. Algorithms for constructing the frames of the Mandelbrot sets of the functions under consideration in the MathCad mathematical package are presented. The task is to describe 3-order frames (where) of the Mandelbrot sets of functions and, which correspond to the existence of attracting fixed points of period 3. It is shown that the establishment of associative relations between classes of various mathematical objects (polynomials of a complex variable, curves, Mandelbrot sets) contributes to the development of original thinking and creative potential of students.

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Wang, Feng Ying, Li Ming Du et Zi Yang Han. « The Construction for Generalized Mandelbrot Sets of the Frieze Group ». Advanced Materials Research 756-759 (septembre 2013) : 2562–66. http://dx.doi.org/10.4028/www.scientific.net/amr.756-759.2562.

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By an analysis of symmetric features of equivalent mappings of the frieze group, a definition of their generalized Mandelbrot sets is given and a novel method for constructing generalized Mandelbrot sets of equivalent mappings of frieze group is presented via utilizing the Ljapunov exponent as the judgment standard. Based on generating parameter space of dynamical system, lots of patterns of generalized Mandelbrot sets are produced.

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Thèses sur le sujet "Mandelbrot sets"

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Tingen, Larry L. « The Julia and Mandelbrot sets for the Hurwitz zeta function ». View electronic thesis (PDF), 2009. http://dl.uncw.edu/etd/2009-3/tingenl/larrytingen.pdf.

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Jones, Rafe. « Galois martingales and the hyperbolic subset of the p-adic Mandelbrot set / ». View online version ; access limited to Brown University users, 2005. http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&res_dat=xri:pqdiss&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&rft_dat=xri:pqdiss:3174623.

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Tolmie, Julie. « Visualisation, navigation and mathematical perception : a visual notation for rational numbers mod 1 ». View thesis entry in Australian Digital Theses Program, 2000. http://thesis.anu.edu.au/public/adt-ANU20020313.101505/index.html.

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Poirier, Schmitz Alfredo. « Invariant measures on polynomial quadratic Julia sets with no interior ». Pontificia Universidad Católica del Perú, 2014. http://repositorio.pucp.edu.pe/index/handle/123456789/96022.

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We characterize invariant measures for quadratic polynomial Julia sets with no interior. We prove that besides the harmonic measure —the only one that is even and invariant—, all others are generated by a suitable odd measure.
En este artículo caracterizamos medidas invariantes sobre conjuntos de Julia sin interior asociados con polinomios cuadráticos. Probamos que más allá de la medida armónica —la única par e invariante—, el resto son generadas por su parte impar.

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Kuo, Li-Feng, et 郭立峰. « Mandelbrot Sets, Julia Sets and Their Algorithms ». Thesis, 2019. http://ndltd.ncl.edu.tw/handle/6n28d7.

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碩士
國立中央大學
數學系
107
In this thesis, we survey the big theme of fractals － Mandelbrot sets. We start to study Julia sets before study Mandelbrot sets, and the goal is generating figures of fractals and applying to arts. Hence, we introduce the definition and properties of Julia sets firstly, and use this theory to arrange some useful algorithms for generating the figures of Julia sets. After we survey Julia sets, we can study Mandelbrot sets, since the definition of Mandelbrot sets is all of the points such that the Julia set is onnected. However, we obtain the obstacle when generating andelbrot sets, that is, how to check the Julia set is connected or not? The answer of this question is － the fundamental theorem of Mandelbrot sets, we can generate the figures of Mandelbrot sets by this theorem. Finally, we give some examples of Mandelbrot sets and Julia sets, and introduce 3-dimensional Mandelbrot sets and Julia sets.

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Fitzgibbon, Elizabeth Laura. « Rational maps : the structure of Julia sets from accessible Mandelbrot sets ». Thesis, 2014. https://hdl.handle.net/2144/15111.

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For the family of complex rational maps F_λ(z)=z^n+λ/z^d, where λ is a complex parameter and n, d ≥ 2 are integers, many small copies of the well-known Mandelbrot set are visible in the parameter plane. An infinite number of these are located around the boundary of the connectedness locus and are accessible by parameter rays from the Cantor set locus. Maps taken from main cardioid of these accessbile Mandelbrot sets have attracting periodic cycles. A method for constructing models of the Julia sets corresponding to such maps is described. These models are then used to explore the existence of dynamical conjugacies between maps taken from distinct accessible Mandelbrot sets in the upper halfplane.

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Hannah, Walter. « Internal rays of the Mandelbrot set ». Thesis, 2006. http://www.ithaca.edu/hs/depts/math/docs/theses/whannahthesis.pdf.

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Lauber, Arnd. « On the Stability of Julia Sets of Functions having Baker Domains ». Doctoral thesis, 2004. http://hdl.handle.net/11858/00-1735-0000-0006-B3DE-F.

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Livres sur le sujet "Mandelbrot sets"

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Mandelbrot, Benoit B. Fractals and chaos : The Mandelbrot set and beyond. New York, NY : Springer, 2004.

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Tomboulian, Sherryl. Indirect addressing and load balancing for faster solution to Mandelbrot Set on SIMD architectures. Hampton, Va : ICASE, 1989.

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Banaś, Marian. Analiza teoretyczna i badania właściwości zawiesin nieziarnistych w zastosowaniu do projektowsnia i eksploatacji wielostrumieniowych urządzeń sedymentacyjnych : Theoretical analysis and investigations of the properties of the non-grainy suspensions in terms to design and use of the lamella settling devices. Kraków : Wydawnictwa AGH, 2012.

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Devaney, Robert, dir. Complex Dynamical Systems : The Mathematics Behind the Mandelbrot and Julia Sets. Providence, Rhode Island : American Mathematical Society, 1995. http://dx.doi.org/10.1090/psapm/049.

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1948-, Devaney Robert L., et Branner Bodil, dir. Complex dynamical systems : The mathematics behind the Mandelbrot and Julia sets. Providence, R.I : American Mathematical Society, 1994.

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1945-, Stewart Ian, et Clarke Arthur Charles 1917-, dir. The colours of infinity : The beauty and power of fractals. [S.l.] : Clear Books, 2004.

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Lesmoir-Gordon, Nigel. The colours of infinity : The beauty and power of fractals. London : Springer Verlag, 2010.

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1945-, Kauffman Louis H., et Sandin Daniel J, dir. Hypercomplex iterations : Distance estimation and higher dimensional fractals. River Edge, NJ : World Scientific, 2002.

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Milnor, John W. Dynamical systems (1984-2012). Sous la direction de Bonifant Araceli 1963-. Providence, Rhode Island : American Mathematical Society, 2014.

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Universal Mandelbrot Set : Beginning of the Story. World Scientific Publishing Co Pte Ltd, 2006.

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Chapitres de livres sur le sujet "Mandelbrot sets"

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Agarwal, Ravi P., Kanishka Perera et Sandra Pinelas. « Julia and Mandelbrot Sets ». Dans An Introduction to Complex Analysis, 316–20. Boston, MA : Springer US, 2011. http://dx.doi.org/10.1007/978-1-4614-0195-7_49.

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Korsch, H. J., et H. J. Jodl. « Mandelbrot and Julia Sets ». Dans Chaos, 227–48. Berlin, Heidelberg : Springer Berlin Heidelberg, 1999. http://dx.doi.org/10.1007/978-3-662-03866-6_11.

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Korsch, H. J., et H. J. Jodl. « Mandelbrot and Julia Sets ». Dans Chaos, 227–48. Berlin, Heidelberg : Springer Berlin Heidelberg, 1994. http://dx.doi.org/10.1007/978-3-662-02991-6_11.

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Douady, Adrien. « Julia Sets and the Mandelbrot Set ». Dans The Beauty of Fractals, 161–74. Berlin, Heidelberg : Springer Berlin Heidelberg, 1986. http://dx.doi.org/10.1007/978-3-642-61717-1_13.

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Reeve, Dominic E. « Mandelbrot, Julia Sets and Nonlinear Mappings ». Dans Fractals and Chaos, 35–42. New York, NY : Springer New York, 1991. http://dx.doi.org/10.1007/978-1-4612-3034-2_3.

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Peitgen, Heinz-Otto, Hartmut Jürgens et Dietmar Saupe. « The Mandelbrot Set : Ordering the Julia Sets ». Dans Fractals for the Classroom, 415–73. New York, NY : Springer New York, 1992. http://dx.doi.org/10.1007/978-1-4612-4406-6_8.

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Peitgen, Heinz-Otto, Hartmut Jürgens et Dietmar Saupe. « The Mandelbrot Set : Ordering the Julia Sets ». Dans Chaos and Fractals, 841–901. New York, NY : Springer New York, 1992. http://dx.doi.org/10.1007/978-1-4757-4740-9_15.

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Peitgen, Heinz-Otto, Hartmut Jürgens et Dietmar Saupe. « The Mandelbrot Set : Ordering the Julia Sets ». Dans Chaos and Fractals, 783–837. New York, NY : Springer New York, 2004. http://dx.doi.org/10.1007/0-387-21823-8_15.

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McClure, Mark. « Complex Dynamics:Julia Sets and the Mandelbrot Set ». Dans Mathematica in Action, 277–300. New York, NY : Springer New York, 2010. http://dx.doi.org/10.1007/978-0-387-75477-2_12.

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Ochkov, Valery, Alan Stevens et Anton Tikhonov. « Iterations and Fractal Sets of Mandelbrot and Julia ». Dans STEM Problems with Mathcad and Python, 263–91. Boca Raton : Chapman and Hall/CRC, 2022. http://dx.doi.org/10.1201/9781003228356-14.

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Actes de conférences sur le sujet "Mandelbrot sets"

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Kumar, Suthikshn. « Public Key Cryptographic System Using Mandelbrot Sets ». Dans MILCOM 2006. IEEE, 2006. http://dx.doi.org/10.1109/milcom.2006.302396.

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Dejun, Yan, Yang Rijing, Xin Huijie et Zheng Jiangchao. « Generalized Mandelbrot Sets and Julia Sets for Non-analytic Complex Maps ». Dans 2010 International Workshop on Chaos-Fractals Theories and Applications (IWCFTA). IEEE, 2010. http://dx.doi.org/10.1109/iwcfta.2010.42.

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Yan, Dejun, Junxing Zhang, Nan Jiang et Lidong Wang. « General Mandelbrot Sets and Julia Sets Generated from Non-analytic Complex Iteration ⨍m(z)=z^n+c ». Dans 2009 International Workshop on Chaos-Fractals Theories and Applications (IWCFTA 2009). IEEE, 2009. http://dx.doi.org/10.1109/iwcfta.2009.89.

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Seytov, Sh J., N. B. Narziyev, A. I. Eshniyozov et S. N. Nishonov. « The algorithms for developing computer programs for the sets of Julia and Mandelbrot ». Dans PROBLEMS IN THE TEXTILE AND LIGHT INDUSTRY IN THE CONTEXT OF INTEGRATION OF SCIENCE AND INDUSTRY AND WAYS TO SOLVE THEM : (PTLICISIWS-2022). AIP Publishing, 2023. http://dx.doi.org/10.1063/5.0145456.

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Yan, Dejun, Xiaodan Wei, Hongpeng Zhang, Nan Jiang et Xiangdong Liu. « Fractal Structures of General Mandelbrot Sets and Julia Sets Generated From Complex Non-Analytic Iteration Fm(Z)=Zm+c ». Dans 2nd International Symposium on Computer, Communication, Control and Automation. Paris, France : Atlantis Press, 2013. http://dx.doi.org/10.2991/isccca.2013.42.

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Ganikhodzhayev, Rasul, et Shavkat Seytov. « An analytical description of mandelbrot and Julia sets for some multi-dimensional cubic mappings ». Dans INTERNATIONAL UZBEKISTAN-MALAYSIA CONFERENCE ON “COMPUTATIONAL MODELS AND TECHNOLOGIES (CMT2020)” : CMT2020. AIP Publishing, 2021. http://dx.doi.org/10.1063/5.0058341.

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Dawkins, Jeremy J., David M. Bevly et Robert L. Jackson. « Multiscale Terrain Characterization Using Fourier and Wavelet Transforms for Unmanned Ground Vehicles ». Dans ASME 2009 Dynamic Systems and Control Conference. ASMEDC, 2009. http://dx.doi.org/10.1115/dscc2009-2718.

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This paper investigates the use of the Fourier transform and Wavelet transform as methods to supplement the more common root mean squared elevation and power spectral density methods of terrain characterization. Two dimensional terrain profiles were generated using the Weierstrass-Mandelbrot fractal equation. The Fourier and Wavelet transforms were used to decompose these terrains into a parameter set. A two degree of freedom quarter car model was used to evaluate the vehicle response before and after the terrain characterization. It was determined that the Fourier transform can be used to reduce the profile into the key frequency components. The Wavelet transform can effectively detect discontinuities of the profile and changes in the roughness of the profile. These two techniques can be added to current methods to yield a more robust terrain characterization.

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Shahinpoor, Mohsen. « An Introduction to Smart Fractal Structures and Mechanisms ». Dans ASME 1993 Design Technical Conferences. American Society of Mechanical Engineers, 1993. http://dx.doi.org/10.1115/detc1993-0160.

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Abstract Fractal structures are unique in the sense that they are highly expandible or collapsible and yet they are capable of preserving their basic structural geometry in a dynamic fashion. This dynamic geometric invariance opens up a new territory in fractal solids, i.e., fractal structures, mechanisms and robot manipulators. Some of these structure are in the form of highly deployable mechanisms and possibly redundant, multi-axis, multi-arm, multi-finger robot manipulators whose kinematic structure is fractal. Thus, simple fractal structures, such as triadic cantor set, and fractal functions, such as the Weirstraus-Mandelbrot functions, govern the structural branching of such robots and essentially define their kinematic structure. These deployable fractal structures, mechanisms and robot manipulators are shown to be capable of generating unique, and yet unparalleled properties such as computer-controlled microsensing even down to molecular level (micromachining) and computer-controlled dynamics such as the creation of hypervelocity fractons with speeds in the range of hundreds of kilometers per second. A number of structures and mechanisms and their unique properties are presented in this paper and a simple kinematic model is presented and briefly discussed.

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Michopoulos, John G., et Athanasios Iliopoulos. « High Dimensional Full Inverse Characterization of Fractal Volumes ». Dans ASME 2012 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/detc2012-71050.

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The present paper describes a methodology for the inverse identification of the complete set of parameters associated with the Weirstrass-Mandelbrot (W-M) function that can describe any fractal scalar field distribution of measured data defined within a volume. Our effort is motivated by the need to be able to describe a scalar field quantity distribution in a volume in order to be able to represent analytically various non-homogeneous material properties distributions for engineering and science applications. Our method involves utilizing a refactoring of the W-M function that permits defining the characterization problem as a high dimensional singular value decomposition problem for the determination of the so-called phases of the function. Coupled with this process is a second level exhaustive search that enables the determination of the density of the frequencies involved in defining the trigonometric functions involved in the definition of the W-M function. Numerical applications of the proposed method on both synthetic and actual volume data, validate the efficiency and the accuracy of the proposed approach. This approach constitutes a radical departure from the traditional fractal dimension characterization studies and opens the road for a very large number of applications and generalizes the approach developed by the authors for fractal surfaces to that of fractal volumes.

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Michopoulos, John G., et Athanasios Iliopoulos. « Complete High Dimensional Inverse Characterization of Fractal Surfaces ». Dans ASME 2011 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2011. http://dx.doi.org/10.1115/detc2011-47784.

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The present paper describes a methodology for the inverse identification of the complete set of parameters associated with the Weirstrass-Mandelbrot (W-M) function that can describe any rough surface known by its profilometric or topographic data. Our effort is motivated by the need to determine the mechanical, electrical and thermal properties of contact surfaces between deformable materials that conduct electricity and heat and require an analytical representation of the surfaces involved. Our method involves utilizing a refactoring of the W-M function that permits defining the characterization problem as a high dimensional singular value decomposition problem for the determination of the so-called phases of the function. Coupled with this process is a second level exhaustive search that enables the determination of the density of the frequencies involved in defining the trigonometric functions involved in the definition of the W-M function. Our approach proves that this is the only additional parameter that needs to be determined for full characterization of the W-M function as the rest can be selected arbitrarily. Numerical applications of the proposed method on both synthetic and actual elevation data, validate the efficiency and the accuracy of the proposed approach. This approach constitutes a radical departure from the traditional fractal dimension characterization studies and opens the road for a very large number of applications.

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