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### Fractals from simple polynomial composite functions

Ken Shirriff
1993 Computers & graphics
This paper describes a method of generating 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.  ...

### Polynomiography and applications in art, education, and science

Bahman Kalantari
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 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.  ...

### Polynomiography and applications in art, education, and science

Bahman Kalantari
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 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.  ...

### Polynomiography and applications in art, education, and science

Bahman Kalantari
2004 Computers & graphics
Polynomiography is the art and science of visualizing approximation of the zeros of complex 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.  ...

### New iteration fractal branch and bound method for stacking sequence optimizations of multiple laminates

Akira Todoroki, Masato Sekishiro
2007 Composite structures
Laminated 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  ...

### An Invitation to Polynomiography via Exponential Series [article]

Bahman Kalantari
2017 arXiv   pre-print
Here we exhibit some 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.  ...

### Spectral zeta functions of fractals and the complex dynamics of polynomials

Alexander Teplyaev
2007 Transactions of the American Mathematical Society
We obtain formulas for the spectral zeta 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  ...

### Fractals for Kernelization Lower Bounds [article]

Till Fluschnik, Danny Hermelin, André Nichterlein, Rolf Niedermeier
2017 arXiv   pre-print
We present a new technique exploiting triangle-based 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.  ...

### The calculus of fractal interpolation functions

Michael F Barnsley, Andrew N Harrington
1989 Journal of Approximation Theory
The calculus of deterministic 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.  ...

### Good labeling property of simple nested fractals [article]

Mikołaj Nieradko, Mariusz Olszewski
2021 arXiv   pre-print
We show various criteria to verify if a given nested 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).  ...

### Fractal mathematical over extended finite fields Fp[x]/(f(x))

Cecilia E Sandoval-Ruiz
2021 Proyecciones
Thus, the reduced coefficients of the generating 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)) = &  ...

### Power-law energy level spacing distributions in fractals

Askar A. Iliasov, Mikhail I. Katsnelson, Shengjun Yuan
2019 Physical review B
We show that the energy-level distribution of a 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.  ...

### 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.  ...

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

Rajeshri Rana .
2012 International Journal of Research in Engineering and Technology
plane 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  ...

### Transport and elastic properties of fractal media

Anthony P. Roberts, Mark A. Knackstedt
1996 Physica A: Statistical Mechanics and its Applications
We investigate the influence of 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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