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Fractals from simple polynomial composite functions

1993
*
Computers & graphics
*

This paper describes a method of generating

doi:10.1016/0097-8493(93)90122-p
fatcat:px6s6rc5o5ehbga6ce5whx5xqi
*fractals*by composing two*simple**polynomial**functions*. ... Many common*fractals*, such as the Mandelbrot set, the tricorn, and the forced logistic map, as well as new*fractals*can be generated with this technique. ... Conclusions Many interesting*fractals*can be generated*from**composition*of*simple**polynomial**functions*. In many cases, the symmetry of the resulting*fractal*can be easily proved. ...##
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Polynomiography and applications in art, education, and science

2003
*
Educators program from the 30th annual conference on Computer graphics and interactive techniques - GRAPH '03
*

Polynomiography is the art and science of visualizing approximation of the zeros of complex

doi:10.1145/965107.965108
fatcat:76rmsgmtpjg2pf34rqcqisihjq
*polynomials*. Informally speaking, polynomiography allows one to create colorful images of*polynomials*. ... concepts, theorems, and algorithms, e.g., the algebra and geometry of complex numbers, the notions of convergence and continuity, geometric constructs such as Voronoi regions, and modern notions such as*fractals*... It makes use of one or an infinite number of iteration*functions*for*polynomial*root-finding. A polynomiograph may or may not result in a*fractal*image. ...##
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Polynomiography and applications in art, education, and science

2003
*
Educators program from the 30th annual conference on Computer graphics and interactive techniques - GRAPH '03
*

Polynomiography is the art and science of visualizing approximation of the zeros of complex

doi:10.1145/965106.965108
dblp:conf/siggraph/Kalantari03
fatcat:r4h26mh2abgczeyntwpwzkyqre
*polynomials*. Informally speaking, polynomiography allows one to create colorful images of*polynomials*. ... concepts, theorems, and algorithms, e.g., the algebra and geometry of complex numbers, the notions of convergence and continuity, geometric constructs such as Voronoi regions, and modern notions such as*fractals*... It makes use of one or an infinite number of iteration*functions*for*polynomial*root-finding. A polynomiograph may or may not result in a*fractal*image. ...##
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Polynomiography and applications in art, education, and science

2004
*
Computers & graphics
*

Polynomiography is the art and science of visualizing approximation of the zeros of complex

doi:10.1016/j.cag.2004.03.009
fatcat:4lkkm72qdvaa5hply76sze7szi
*polynomials*. Informally speaking, polynomiography allows one to create colorful images of*polynomials*. ... concepts, theorems, and algorithms, e.g., the algebra and geometry of complex numbers, the notions of convergence and continuity, geometric constructs such as Voronoi regions, and modern notions such as*fractals*... It makes use of one or an infinite number of iteration*functions*for*polynomial*root-finding. A polynomiograph may or may not result in a*fractal*image. ...##
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New iteration fractal branch and bound method for stacking sequence optimizations of multiple laminates

2007
*
Composite structures
*

Laminated

doi:10.1016/j.compstruct.2006.09.001
fatcat:mu4jxorvfjap3kv5qswpxrmq3m
*composites*are usually fabricated*from*unidirectional plies of given thickness with limited fiber orientations to a small set of angles (e.g. 0°, +45°, 45°, and 90°). ... An improved version of*fractal*branch and bound method has been proposed to optimize multiple stacking sequences. ... Evaluations of the maximum g value*from*Eq. (10) require only maximization of the quadratic*polynomials*in the reduced triangular region, obtained*from*the*fractal*arrangement analysis of feasible laminates ...##
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An Invitation to Polynomiography via Exponential Series
[article]

2017
*
arXiv
*
pre-print

Here we exhibit some

arXiv:1707.09417v1
fatcat:mlprofckpvcfpgigo333fdk3sm
*fractal*and non-*fractal*polynomiographs of the partial sums while also presenting a brief introduction of the underlying concepts. ... It also leads to new discoveries on*polynomials*and inspires new applications. ... techniques, and operations such as multiplication of*polynomials*, scaling their zeros,*compositions*and more. ...##
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Spectral zeta functions of fractals and the complex dynamics of polynomials

2007
*
Transactions of the American Mathematical Society
*

We obtain formulas for the spectral zeta

doi:10.1090/s0002-9947-07-04150-5
fatcat:2nd33i6dfbcejh27lj4yq6vboq
*function*of the Laplacian on symmetric finitely ramified*fractals*, such as the Sierpinski gasket, and a*fractal*Laplacian on the interval. ... These formulas contain a new type of zeta*function*associated with a*polynomial*(rational*functions*also can appear in this context). ... License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use SPECTRAL ZETA*FUNCTIONS*OF*FRACTALS*AND DYNAMICS ...##
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Fractals for Kernelization Lower Bounds
[article]

2017
*
arXiv
*
pre-print

We present a new technique exploiting triangle-based

arXiv:1512.00333v3
fatcat:yfxcjd5q7fa2tbev7ud7we6ovy
*fractal*structures for extending the range of applicability of*compositions*. ... The*composition*technique is a popular method for excluding*polynomial*-size problem kernels for NP-hard parameterized problems. ... Direct all edges in the paths towards*from*σ ′ to σ and*from*τ to Acknowledgement We thank Manuel Sorge (TU Berlin) for his help with the proof of Theorem 3. ...##
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The calculus of fractal interpolation functions

1989
*
Journal of Approximation Theory
*

The calculus of deterministic

doi:10.1016/0021-9045(89)90080-4
fatcat:bqnsmzdaejabvmlmyvb64j3cti
*fractal**functions*is introduced. ... The*fractal*dimension for a class of*fractal*interpolation*functions*is explicitly computed. ... If we are looking for a*simple*mathematical model for a given*fractal**function*, we would specify that a sequence of points match and use an approximation of the*fractal*dimension to choose CI. ...##
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Good labeling property of simple nested fractals
[article]

2021
*
arXiv
*
pre-print

We show various criteria to verify if a given nested

arXiv:2110.15921v1
fatcat:5zsnqe62fjezzbhfra7gv3vvii
*fractal*has a good labeling property, inter alia we present a characterization of GLP for*fractals*with an odd number of essential fixed points. ... We prove that if a number of essential fixed points is a power of two, then a*fractal*must have GLP and that there are no values other than primes or powers of two guaranteeing GLP. ... Definition 2.3 (*Simple*nested*fractal*). ...##
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Fractal mathematical over extended finite fields Fp[x]/(f(x))

2021
*
Proyecciones
*

Thus, the reduced coefficients of the generating

doi:10.22199/issn.0717-6279-4322
fatcat:qi76glpqcfgd7mmx3x6wudamra
*polynomial*of G contain embedded the modular reduction and thus simplify operations that contain basic finite fields. ... The algorithm describes the process of construction of the GF multiplier, it can start at any stage of LFSR; it is shift the sequence of operation,*from*this point on, thanks to the concurrent adaptation ... Let the*function*f be an operation defined as the product in finite fields on the*polynomial*p(x), and the*function*g be a concatenation operation of products on the*polynomial*g(x), then: g(f (x)) = & ...##
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Power-law energy level spacing distributions in fractals

2019
*
Physical review B
*

We show that the energy-level distribution of a

doi:10.1103/physrevb.99.075402
fatcat:h4v3liyprvbz3fye3d5fcz2vha
*fractal*follows a power-law behaviour, if its energy spectrum is a limit set of piece-wise linear*functions*. ... In this article we investigate the energy spectrum statistics of*fractals*at the quantum level. ... The calculations were numerically checked for the*simple*model of piecewise linear*function*. The second approach is connected with the geometry of the paths in*fractals*. ...##
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Page 961 of Mathematical Reviews Vol. , Issue 91B
[page]

1991
*
Mathematical Reviews
*

*Fractals*: Surprising, complicated, colorful, rich and beautiful shapes can be generated

*from*exceedingly

*simple*geometric forms or formulas via

*simple*algorithms using available computer graph- ics. ... dyu(z) for qua- dratic

*polynomials*is given. There exists a preprint [“Bases of

*functions*, generated by iter- ations of

*polynomials*” (Russian), Preprint; per revr.] containing more details. ...

##
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RELATIVE SUPERIOR MANDELBROT AND JULIA SETS FOR INTEGER AND NON-INTEGER VALUES

2012
*
International Journal of Research in Engineering and Technology
*

plane

doi:10.15623/ijret.2012.0102011
fatcat:vquhqotdonadhkognufenq5aoy
*fractal*images generated*from*the iteration of these*functions*using Ishikawa iteration for integer and non-integer values. ... A great revolution of ideas separates the The*fractals*generated*from*the self-squared*function*, classical mathematics of the 19th century*from*the modern mathematics ...##
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Transport and elastic properties of fractal media

1996
*
Physica A: Statistical Mechanics and its Applications
*

We investigate the influence of

doi:10.1016/s0378-4371(96)00198-7
fatcat:zmtbcex5krdoxoc7jmnedso6da
*fractal*structure on material properties. We calculate the statistical correlation*functions*of*fractal*media defined by level-cut Gaussian random fields. ... We find that a*fractally*rough interface has a relatively strong influence on the properties of*composites*. In contrast a*fractal*volume (mass) has little effect on material properties. ... A*simple*field-field correlation*function*which gives rise to this behaviour in the level-cut GRF is g(r) = 1 + 2 A r 2 l 2 −A/2 . ( 9 ) This field-field correlation*function*actually leads to a spectral ...
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