Relevant bibliographies by topics / Fractal Squares

Contents

  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers
  6. Reports

Academic literature on the topic 'Fractal Squares'

Author: Grafiati
Published: 4 June 2021
Last updated: 12 February 2022

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Journal articles on the topic "Fractal Squares"

1

ZHANG, YAN-FANG. "A LOWER BOUND OF TOPOLOGICAL HAUSDORFF DIMENSION OF FRACTAL SQUARES." Fractals 28, no. 06 (September 2020): 2050115. http://dx.doi.org/10.1142/s0218348x20501157.

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Given an integer [Formula: see text] and a digit set [Formula: see text], there is a self-similar set [Formula: see text] satisfying the set equation [Formula: see text]. This set [Formula: see text] is called a fractal square. By studying the line segments contained in [Formula: see text], we give a lower estimate of the topological Hausdorff dimension of fractal squares. Moreover, we compute the topological Hausdorff dimension of fractal squares whose nontrivial connected components are parallel line segments, and introduce the Latin fractal squares to investigate the question when the topological Hausdorff dimension of a fractal square coincides with its Hausdorff dimension.
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LAU, KA–SING, JUN JASON LUO, and HUI RAO. "Topological structure of fractal squares." Mathematical Proceedings of the Cambridge Philosophical Society 155, no. 1 (March 1, 2013): 73–86. http://dx.doi.org/10.1017/s0305004112000692.

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AbstractGiven an integer n ≥ 2 and a digit set ⊊ {0,1,. . .,n − 1}2, there is a self-similar set F ⊂ ℝ2 satisfying the set equation: F=(F+)/n. We call such F a fractal square. By studying a periodic extension H= F + ℤ2, we classify F into three types according to their topological properties. We also provide some simple criteria for such classification.
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Liang, Zhen, and Huo-Jun Ruan. "Gap sequences of fractal squares." Journal of Mathematical Analysis and Applications 472, no. 2 (April 2019): 1475–86. http://dx.doi.org/10.1016/j.jmaa.2018.12.003.

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RAO, FENG, XIAOHUA WANG, and SHENGYOU WEN. "ON THE TOPOLOGICAL CLASSIFICATION OF FRACTAL SQUARES." Fractals 25, no. 03 (May 11, 2017): 1750028. http://dx.doi.org/10.1142/s0218348x17500281.

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A fractal square is a nonempty compact set in the plane satisfying [Formula: see text], where [Formula: see text] is an integer and [Formula: see text] is nonempty. We give the topological classification of fractal squares with [Formula: see text] and [Formula: see text].
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LUO, JUN JASON, and JING-CHENG LIU. "ON THE CLASSIFICATION OF FRACTAL SQUARES." Fractals 24, no. 01 (March 2016): 1650008. http://dx.doi.org/10.1142/s0218348x16500080.

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In the previous paper [K. S. Lau, J. J. Luo and H. Rao, Topological structure of fractal squares, Math. Proc. Camb. Phil. Soc. 155 (2013) 73–86], Lau, Luo and Rao completely classified the topological structure of so called fractal square [Formula: see text] defined by [Formula: see text], where [Formula: see text]. In this paper, we further provide simple criteria for the [Formula: see text] to be totally disconnected, then we discuss the Lipschitz classification of [Formula: see text] in the case [Formula: see text], which is an attempt to consider non-totally disconnected sets.
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Xiao, Jian-Ci. "Fractal squares with finitely many connected components *." Nonlinearity 34, no. 4 (February 18, 2021): 1817–36. http://dx.doi.org/10.1088/1361-6544/abd611.

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Feng, Rao, Wang Xiaohua, and Zhu Yunjie. "Lipschitz equivalence of a pair of fractal squares." SCIENTIA SINICA Mathematica 48, no. 3 (May 24, 2017): 363. http://dx.doi.org/10.1360/n012016-00180.

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Ruan, Huo-Jun, and Yang Wang. "Topological invariants and Lipschitz equivalence of fractal squares." Journal of Mathematical Analysis and Applications 451, no. 1 (July 2017): 327–44. http://dx.doi.org/10.1016/j.jmaa.2017.02.012.

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Rezende, Veridiana, Mariana Moran, Thais Michele Mártires, and Fabricia Carvalho Paixão. "O Fractal Árvore Pitagórica e Diferentes Representações: uma Investigação com Alunos do Ensino Médio." Jornal Internacional de Estudos em Educação Matemática 11, no. 2 (September 11, 2018): 160. http://dx.doi.org/10.17921/2176-5634.2018v11n2p160-171.

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A Geometria dos Fractais em sala de aula propicia diversas possibilidades de estudos de conceitos matemáticos, incentiva o uso de recursos tecnológicos, proporciona surpresas pela beleza e complexidade dos fractais. Apresentamos neste texto uma investigação acerca das possibilidades do uso de diferentes registros de representação semiótica aliado à Geometria dos Fractais, no que concerne ao ensino de matemática. E, para exemplificar estas possibilidades em sala de aula, relatamos uma situação de ensino, que foi implementada com quinze (15) alunos do 3º ano do Ensino Médio de uma escola pública do interior do Paraná, relacionada à construção do fractal Árvore Pitagórica. Como procedimentos metodológicos elaboramos cinco tarefas associadas ao fractal Árvore Pitagórica que tiveram por finalidade explorar diversos conceitos matemáticos emergidos no processo de construção deste fractal, nas quais foram contempladas as representações figural, numérica, algébrica e linguagem natural. As análises dos registros mostram que a implementação das tarefas possibilitou aos alunos: o estudo de diversos elementos matemáticos tais como: áreas e perímetros de quadrados e triângulos, teorema de Pitágoras, ângulos, congruência de triângulos, frações, potências, números decimais entre outros; as construções figurais da Árvore Pitagórica por meio de diferentes representações e a visualização das principais características de um fractal, bem como a compreensão de seu processo de construção.Palavras-chave: Ensino de Matemática. Representação Semiótica. Geometria dos Fractais.AbstractThe geometry of the fractures in the classroom provides several possibilities for studying mathematical concepts, encourages the use of technological resources, provides surprises for the beauty and complexity of the fractals. We present in this text an investigation about the possibilities of the use of different registers of semiotic representation allied to the Geometry of the Fractais, in what concerns the teaching of mathematics. And, to exemplify these possibilities in the classroom, we report a teaching situation, which was implemented with fifteen (15) students of the 3rd year of High School in a public school in the interior of Paraná, related to the construction of the Pythagorean Tree fractal. As methodological procedures we elaborated five tasks associated to the fractal Pythagorean Tree that had as purpose to explore several mathematical concepts emerged in the process of construction of this fractal, in which figural, numerical, algebraic and natural language representations were contemplated. The analysis of the registers shows that the implementation of the tasks enabled the students to study several mathematical elements such as: areas and perimeters of squares and triangles, Pythagorean theorem, angles, congruence of triangles, fractions, powers, decimals, among others; the figurative constructions of the Pythagorean Tree by means of different representations and the visualization of the main characteristics of a fractal, as well as the understanding of its process of construction.Keywords: Mathematics Teaching. Semiotic Representation. Geometry of Fractions.
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PALAGALLO, JUDITH, and MARIA SALCEDO. "SYMMETRIES OF FRACTAL TILINGS." Fractals 16, no. 01 (March 2008): 69–78. http://dx.doi.org/10.1142/s0218348x08003806.

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Self-similar tilings of the plane can be generated by an iterative replacement property using squares and triangles. We show tilings with tiles that are topological disks and others whose tiles are connected but have an interior that is disconnected. In each example we note the symmetries of the individual tiles and describe how each tiling exhibits a property common to crystallographic tilings.
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Dissertations / Theses on the topic "Fractal Squares"

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Roinestad, Kristine A. "Geometry of Fractal Squares." Diss., Virginia Tech, 2010. http://hdl.handle.net/10919/26883.

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This paper will examine analogues of Cantor sets, called fractal squares, and some of the geometric ways in which fractal squares raise issues not raised by Cantor sets. Also discussed will be a technique using directed graphs to prove bilipschitz equivalence of two fractal squares.
Ph. D.
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Omilion, Alexis Kathleen. "The Effect of Multiple Scales on Fractal-Grid-Generated Turbulence." Cleveland State University / OhioLINK, 2018. http://rave.ohiolink.edu/etdc/view?acc_num=csu1528635554940034.

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Prehl, geb Balg Janett. "Diffusion on Fractals." Master's thesis, Universitätsbibliothek Chemnitz, 2007. http://nbn-resolving.de/urn:nbn:de:swb:ch1-200701033.

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We study anomalous diffusion on fractals with a static external field applied. We utilise the master equation to calculate particle distributions and from that important quantities as for example the mean square displacement . Applying different bias amplitudes on several regular Sierpinski carpets we obtain maximal drift velocities for weak field strengths. According to ~t^(2/d_w), we determine random walk dimensions of d_w<2 for applied external fields. These d_w corresponds to superdiffusion, although diffusion is hindered by the structure of the carpet, containing dangling ends. This seems to result from two competing effects arising within an external field. Though the particles prefer to move along the biased direction, some particles get trapped by dangling ends. To escape from there they have to move against the field direction. Due to the by the bias accelerated particles and the trapped ones the probability distribution gets wider and thus d_w<2
In dieser Arbeit untersuchen wir anomale Diffusion auf Fraktalen unter Einwirkung eines statisches äußeres Feldes. Wir benutzen die Mastergleichung, um die Wahrscheinlichkeitsverteilung der Teilchen zu berechnen, um daraus wichtige Größen wie das mittlere Abstandsquadrat zu bestimmen. Wir wenden unterschiedliche Feldstärken bei verschiedenen regelmäßigen Sierpinski-Teppichen an und erhalten maximale Driftgeschwindigkeiten für schwache Feldstärken. Über ~t^{2/d_w} bestimmen wir die Random-Walk-Dimension d_w als d_w<2. Dieser Wert für d_w entspricht der Superdiffusion, obwohl der Diffusionsprozess durch Strukturen des Teppichs, wie Sackgassen, behindert wird. Es schient, dass dies das Ergebnis zweier konkurrierender Effekte ist, die durch das Anlegen eines äußeren Feldes entstehen. Einerseits bewegen sich die Teilchen bevorzugt entlang der Feldrichtung. Andererseits gelangen einige Teilchen in Sackgassen. Um die Sackgassen, die in Feldrichtung liegen, zu verlassen, müssen sich die Teilchen entgegen der Feldrichtung bewegen. Somit sind die Teilchen eine gewisse Zeit in der Sackgasse gefangen. Infolge der durch das äußere Feld beschleunigten und der gefangenen Teilchen, verbreitert sich die Wahrscheinlichkeitsverteilung der Teilchen und somit ist d_w<2
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Wu, I.-lin, and 吳易霖. "A Study on Particle Swarm Optimization based Resistant Fractal Image Compression using Least Trimmed Squares." Thesis, 2009. http://ndltd.ncl.edu.tw/handle/91232641875182701078.

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碩士
義守大學
資訊工程學系碩士班
97
Fractal image compression (FIC) is a lossy coding scheme. It possesses the advantages of high quality of retrieved image, zoom invariant, and high compression ratio. It is used for many applications recently in the filed of image reconstruction, watermark, medical image, feature recognition, and so on. However, if a corrupted image is encoded by FIC, the quality of retrieved image will be poor. Self-similarity and partitioned iterated function system are the underlying idea of FIC. Practically, we find the similarity between range blocks and domain blocks in the encoding process of FIC. The scheme of least squares is used to calculate contrast and brightness between a range block and a domain block in the conventional FIC. The least squares estimator is the best linear unbiased estimator under assumptions that the random variable is zero-mean, constant variance, and uncorrected random variable. As is well known in regression theory that linear regressor is sensitive to outliers. That’s reason why the quality of retrieved image will be poor. Robust regression is usually used against noises in regression theory. Least trimmed squares (LTS) method with resistance from the robust regression is proposed in this thesis. It is embedded into the encoding procedure of the FIC. Recursive weighted least squares with simple architecture and fast convergence is used in this thesis. To effectively improve the encoding speed, particle swarm optimization (PSO) is utilized to reduce the search space.
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Prehl, geb Balg Janett. "Diffusion on Fractals." Master's thesis, 2006. https://monarch.qucosa.de/id/qucosa%3A18745.

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We study anomalous diffusion on fractals with a static external field applied. We utilise the master equation to calculate particle distributions and from that important quantities as for example the mean square displacement . Applying different bias amplitudes on several regular Sierpinski carpets we obtain maximal drift velocities for weak field strengths. According to ~t^(2/d_w), we determine random walk dimensions of d_w<2 for applied external fields. These d_w corresponds to superdiffusion, although diffusion is hindered by the structure of the carpet, containing dangling ends. This seems to result from two competing effects arising within an external field. Though the particles prefer to move along the biased direction, some particles get trapped by dangling ends. To escape from there they have to move against the field direction. Due to the by the bias accelerated particles and the trapped ones the probability distribution gets wider and thus d_w<2.
In dieser Arbeit untersuchen wir anomale Diffusion auf Fraktalen unter Einwirkung eines statisches äußeres Feldes. Wir benutzen die Mastergleichung, um die Wahrscheinlichkeitsverteilung der Teilchen zu berechnen, um daraus wichtige Größen wie das mittlere Abstandsquadrat zu bestimmen. Wir wenden unterschiedliche Feldstärken bei verschiedenen regelmäßigen Sierpinski-Teppichen an und erhalten maximale Driftgeschwindigkeiten für schwache Feldstärken. Über ~t^{2/d_w} bestimmen wir die Random-Walk-Dimension d_w als d_w<2. Dieser Wert für d_w entspricht der Superdiffusion, obwohl der Diffusionsprozess durch Strukturen des Teppichs, wie Sackgassen, behindert wird. Es schient, dass dies das Ergebnis zweier konkurrierender Effekte ist, die durch das Anlegen eines äußeren Feldes entstehen. Einerseits bewegen sich die Teilchen bevorzugt entlang der Feldrichtung. Andererseits gelangen einige Teilchen in Sackgassen. Um die Sackgassen, die in Feldrichtung liegen, zu verlassen, müssen sich die Teilchen entgegen der Feldrichtung bewegen. Somit sind die Teilchen eine gewisse Zeit in der Sackgasse gefangen. Infolge der durch das äußere Feld beschleunigten und der gefangenen Teilchen, verbreitert sich die Wahrscheinlichkeitsverteilung der Teilchen und somit ist d_w<2.
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Mahfouth, Altayeb. "Fractal grid-turbulence and its effects on a performance of a model of a hydrokinetic turbine." Thesis, 2016. http://hdl.handle.net/1828/7728.

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This thesis focuses on generating real world turbulence levels in a water tunnel rotor test using fractal grids and characterizing the effect of the fractal grid generated-turbulence on the performance of hydrokinetic turbines. The research of this thesis is divided into three studies: one field study and two laboratory studies. The field study was conducted at the Canadian Hydro Kinetic Turbine Test Centre (CHTTC) on the Winnipeg River. An Acoustic Doppler Velocimeter (ADV) was used in the field study to collect flow measurements in the river. The laboratory studies were conducted at the University of Victoria (UVic) fluids research lab and the Sustainable Systems Design Lab (SSDL). In addition, the Particle Image Velocimetry (PIV) technique was used in the experiential studies to obtain quantitative information about the vector flow field along the test section, both upstream and downstream of the rotor’s plane. The first study is a field study aiming to provide real flow characteristics and turbulence properties at different depths from the free-surface to boundary layer region of a fast river current by conducting a field study in the Winnipeg River using ADV. A novel technique to deploy and control an ADV from free-surface to boundary layer in a fast-current channel is introduced in this work. Flow characteristics in the river, including mean flow velocities and turbulence intensity profiles are analyzed. The obtained results indicate that the maximum mean velocity occurs below the free-surface, suggesting that the mean velocity is independent of the channel depth. From the free-surface to half depth, it was found that changes in both the mean velocity and turbulence intensity are gradual. From mid-depth to the river bed, the mean velocity drops rapidly while the turbulence intensity increases at a fast rate. The turbulent intensity varied from 9% at the free-surface to around 17.5% near the river bed. The results of this study were used in the second lab study to help designing a fractal grid for a recirculating water flume tank. The goal was to modify the turbulence intensity in the water tunnel such that the generated turbulence was similar to that in the river at a location typical of a hydrokinetic device. The properties of fractal-generated turbulence were experimentally investigated by means of 2D Particle Image Velocimetry (PIV). The streamwise turbulent intensity profiles for different grids along the channel are presented. Additionally, visualization of the average and fluctuating flow fields are also presented. The results are in good agreement with results in literature. The third and final study investigated the power coefficient of a scale hydrokinetic turbine rotor in controlled turbulent flow (7.4 % TI), as well as in the low-turbulence smooth flow (0.5% TI) typical of lab scale testing. PIV was employed for capturing the velocity field. The results show that using realistic TI levels in the water tunnel significantly decrease the turbine’s power coefficient compared to smooth flow, highlighting the importance of considering this effect in future experimental campaigns.
Graduate
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Books on the topic "Fractal Squares"

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Nagarajan, Vijaya. Embodied Mathematics. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780195170825.003.0007.

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This chapter introduces ethnomathematics and discusses the multiple relationships between the kōlam and mathematics. Some of these mathematical properties align with women’s implicit framing knowledge of the kōlam. These ritual patterns are relevant to four key mathematical aspects: symmetry, fractals, array grammars and picture languages, and infinity. This chapter presents the concept of embodied mathematics and argues that Chandralekha’s choreographies embody the three dimensional kōlam. The dot kōlams and the square kōlams are symmetrical. Using geometric algorithms, mathematicians have found that the kōlam is created by transforming and superimposing basic subunits into fractals. Picture languages use sets of basic units combined with formal rules to make larger and seemingly infinite patterns, which computer scientists use for programming computer languages. The kōlam’s connection to infinity serves as a vehicle for auspiciousness. This chapter also discusses how Chandralekha’s choreographies expand the two-dimensional kōlam into three dimensions.
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Crato, Nuno. Figuring It Out. Springer, 2010.

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FIGURING IT OUT: Entertaining Encounters with Everyday Maths. New York, USA: SPRINGER, 2010.

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Book chapters on the topic "Fractal Squares"

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Dudbridge, F. "Least-Squares Block Coding by Fractal Functions." In Fractal Image Compression, 229–41. New York, NY: Springer New York, 1995. http://dx.doi.org/10.1007/978-1-4612-2472-3_12.

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Mandelbrot, Benoit B. "The inexhaustible function z squared plus c." In Fractals and Chaos, 259–67. New York, NY: Springer New York, 2004. http://dx.doi.org/10.1007/978-1-4757-4017-2_23.

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Mandelbrot, Benoit B. "Cantor and Fatou dusts; self-squared dragons." In Fractals and Chaos, 52–72. New York, NY: Springer New York, 2004. http://dx.doi.org/10.1007/978-1-4757-4017-2_4.

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Mandelbrot, Benoit B. "Bifurcation points and the “n-squared” approximation and conjecture, illustrated by M.L Frame and K Mitchell." In Fractals and Chaos, 96–99. New York, NY: Springer New York, 2004. http://dx.doi.org/10.1007/978-1-4757-4017-2_6.

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Popescu, Dan C., Alex Dimca, and Hong Yan. "Generalized square isometries — an improvement for fractal image coding." In Image Analysis and Processing, 637–42. Berlin, Heidelberg: Springer Berlin Heidelberg, 1995. http://dx.doi.org/10.1007/3-540-60298-4_325.

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Jumarie, Guy. "Quantum Entropies of Non-Probabilistic Square Matrices." In Maximum Entropy, Information Without Probability and Complex Fractals, 83–128. Dordrecht: Springer Netherlands, 2000. http://dx.doi.org/10.1007/978-94-015-9496-7_5.

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Upmanu, Vishal, Ajay Kumar Yadav, Nitin Kathuria, and Pradyot Kala. "RIS-Based Multiband Microstrip Patch Antenna Using Square and Giuseppe Peano Fractals." In Lecture Notes in Electrical Engineering, 121–31. Singapore: Springer Singapore, 2018. http://dx.doi.org/10.1007/978-981-13-0665-5_10.

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Jeanneret, B., Ph Flückiger, R. Meyer, J. L. Gavilano, Ch Leemann, and P. Martinoli. "Influence of Phase Fluctuations in Dynamical Magnetoconductance Measurements of Both Square and Fractal Wire Networks." In Science and Technology of Thin Film Superconductors 2, 481. Boston, MA: Springer US, 1990. http://dx.doi.org/10.1007/978-1-4684-1345-8_70.

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Jones, William J. "Skew Squares." In The Pattern Book: Fractals, Art, and Nature, 268. WORLD SCIENTIFIC, 1995. http://dx.doi.org/10.1142/9789812832061_0104.

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Szyszkowicz, Mieczyslaw. "Patterns Composed with Squares." In The Pattern Book: Fractals, Art, and Nature, 362–64. WORLD SCIENTIFIC, 1995. http://dx.doi.org/10.1142/9789812832061_0139.

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Conference papers on the topic "Fractal Squares"

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Ambika, P. S., P. K. Rajendrakumar, and Rijil Ramchand. "An Approach to Rolling Bearing Fault Diagnosis using Fractal Descriptors and Regularized Least Squares." In 2019 Second International Conference on Advanced Computational and Communication Paradigms (ICACCP). IEEE, 2019. http://dx.doi.org/10.1109/icaccp.2019.8882984.

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Dehkhoda, P., and A. Tavakoli. "A crown square microstrip fractal antenna." In IEEE Antennas and Propagation Society Symposium, 2004. IEEE, 2004. http://dx.doi.org/10.1109/aps.2004.1331855.

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Wang Yong and Liu Shaobin. "A New Modified Crown Square Fractal Antenna." In 2008 International Conference on Microwave and Millimeter Wave Technology (ICMMT). IEEE, 2008. http://dx.doi.org/10.1109/icmmt.2008.4540400.

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Bukkawar, Sheetal, and Vasif Ahmed. "Square Shaped Fractal Antenna for Multiband Applications." In 2018 International Conference on Smart City and Emerging Technology (ICSCET). IEEE, 2018. http://dx.doi.org/10.1109/icscet.2018.8537305.

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Elkamchouchi, H. M., and M. N. Abd El-Salam. "Square loop antenna miniaturization using fractal geometry." In Proceedings of the Twentieth National Radio Science Conference (NRSC'2003). IEEE, 2003. http://dx.doi.org/10.1109/nrsc.2003.157313.

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Chhabra, Avinashi, Chahat Jain, and Saroop Singh. "Analysis of square slotted fractal antenna on a square slotted metasurface." In 2017 International Conference on Computing, Communication and Automation (ICCCA). IEEE, 2017. http://dx.doi.org/10.1109/ccaa.2017.8230022.

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Safia, Ousama Abu, and Mourad Nedil. "Ultra-broadband V-band fractal T-square antenna." In 2017 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting. IEEE, 2017. http://dx.doi.org/10.1109/apusncursinrsm.2017.8073348.

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Aslan, Ekin, Semih Korkmaz, Sabri Kaya, and Mustafa Turkmen. "Rotated First Iteration Square Fractal Shaped Perfect Absorbers." In Optical Sensors. Washington, D.C.: OSA, 2015. http://dx.doi.org/10.1364/sensors.2015.sew1b.6.

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Wang, Li-na, Shu Lin, Run-nan Cai, Guan-long Huang, and Wen-bin Zhang. "Multiband printed monopole antenna with square-nested fractal." In 2011 6th International ICST Conference on Communications and Networking in China (CHINACOM). IEEE, 2011. http://dx.doi.org/10.1109/chinacom.2011.6158289.

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Kaur, Rupleen, Sahil Saini, Satbir Singh, and Naveen Kumar. "A multiband fractal square patch antenna for aerospace navigation." In 2015 Annual IEEE India Conference (INDICON). IEEE, 2015. http://dx.doi.org/10.1109/indicon.2015.7443235.

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Reports on the topic "Fractal Squares"

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Gureghian, A. B. FRACVAL: Validation (nonlinear least squares method) of the solution of one-dimensional transport of decaying species in a discrete planar fracture with rock matrix diffusion. Office of Scientific and Technical Information (OSTI), August 1990. http://dx.doi.org/10.2172/6468991.

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