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2: Ray tracing. 3: The Sierpinski Pyramid.

2: Ray tracing. 3: The Sierpinski Pyramid.

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Visually complex objects with infinitesimally fine features, naturally call for mathematical representations. The geometrical property of self-similarity - the whole similar to its parts - when iterated to infinity generates such features. Finite sets of affine contractions called Iterated Function Systems (IFS), with their compact attractors IFS...

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... The space of possible parametric families of tipset connected n-ary complex trees is incredibly vast and rich. Nonetheless we believe that the theoretical basis set in this work will contribute to the study of fractal dendrites [33], topological spaces admitting a unique fractal structure [34] [35], and the geometry and dimension of fractals [36] [37] [38]. We also believe that the notion of complex tree can be extended to hypercomplex spaces in three and higher dimensions [39] [40]. ...
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The theory of complex trees is introduced as a new approach to study a broad class of self-similar sets. Systems of equations encoded by complex trees tip-to-tip equivalence relations are used to obtain one-parameter families of connected self-similar sets Fₐ(z). In order to study topological changes of Fₐ(z) in regions R⊂ℂ where these families are defined, we introduce a new kind of set M⊆R which extends the usual notion of connectivity locus for a parameter space. Moreover we consider another set M₀⊆M related to a special type of connectivity for which we provide a theorem. Among other things, the present theory provides a unified framework to families of self-similar sets traditionally studied as separate with elements Fₐ(z) disconnected for parameters z∈R∖M.
... The space of possible parametric families of tipset connected n-ary complex trees is incredibly vast and rich. Nonetheless we believe that the theoretical basis set in this work will contribute to the study of fractal dendrites [42] [54], topological spaces admitting a unique fractal structure [7] [21], and the geometry and dimension of fractals [4] [26] [52]. We also believe that the notion of complex tree can be extended to hypercomplex spaces in three and higher dimensions [24] [27]. ...
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Advanced Mathematics Master Thesis, Universitat de Barcelona http://hdl.handle.net/2445/129385 The theory of complex trees is introduced as a new approach to study a broad class of self-similar sets which includes Cantor sets, Koch curves, Lévy curves, Sierpiński gaskets, Rauzy fractals, and fractal dendrites. We note a fundamental dichotomy for n-ary complex trees that allows us to study topological changes in regions $\mathcal{R}$ where one-parameter families of connected self-similar sets are defined. Moreover, we show how to obtain these families from systems of equations encoded by tip-to-tip equivalence relations. As far as we know, these families and the sets $M , M_{0}$, and $\mathcal{K}$ that we introduce to study $\mathcal{R}$ are new. We provide a theorem, and a necessary condition, for certifying if a given tipset (self-similar set associated to a complex tree) is a fractal dendrite. We highlight a special class of totally connected tipsets that we call root-connected. And we provide a pair of theorems related to them. For a given one-parameter family we also define the set of root-connected trees $M_{0}$ which presents an asymptotic similarity between its boundary and their associated tipsets. By adapting the notion of post-critically finite self-similar set (p.c.f. for short), the open set condition, and the Hausdorff dimension, we arrive to an upper bound for the existence of p.c.f. trees in a given one-parameter family. We also provide a theorem that allows us to discard non-p.c.f. trees just by looking at some local properties. In relation to this theorem, we set a conjecture of an interesting observation that has been consistent in numerous computational experiments. The space of one-parameter families of tipset-connected complex trees has just begun to be explored. For the family $TS(z) := T \{z, 1/2, 1/4z\}$ we prove that there is a pair of regions contained in the set $\mathcal{K}$ with a piece-wise smooth boundary. We show that this piece-wise smooth boundary is a rather exceptional case by considering a closely related family, $T S(z) := T {z, -1/2, 1/4z}$. Finally we indicate how the general framework works for one-parameter families with non-fixed mirror-symmetric trees.
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A dopagem química é uma rota promissora para projetar e controlar as propriedades eletrônicas de nanofibras de grafeno em zigue-zague (ZGNR). Utilizando os primeiros princípios dos cálculos da teoria funcional da densidade (DFT), no B3LYP / 6-31G, implementado no software Gaussian 09, várias propriedades, como a estrutura geométrica, o espectro de infravermelho DOS, HOMO, LUMO e o gap de energia do ZGNR foram investigadas com vários locais e concentrações de fósforo (P). Observou-se que o ZGNR pode ser convertido da dimensão linear para a fractal usando impurezas de fósforo (P). Também se destaca a árvore binária fractal das estruturas ZGNR e P-ZGNR. Os resultados demonstraram que a diferença de energia possui valores diferentes, localizados nessa faixa de 0,51eV a 1,158 eV para estruturas ZGNR e P-ZGNR intocadas. Essa faixa de gap de energia é variável de acordo com o uso de GNRs em qualquer aparelho. Então, o P-ZGNR tem comportamento semicondutor. Além disso, não há números de onda imaginários no espectro vibracional avaliado, confirmando que o modelo corresponde à energia mínima. Então, esses resultados fazem com que o P-ZGNR possa ser utilizado em várias aplicações, devido a essa estrutura ter se tornado mais estável e com menor reatividade. Palavras-chave: Nanofibras de grafeno, diferença de energia, Densidade de estado, espectros de infravermelho. ABSTRACT Chemical doping is a promising route to engineering and controlling the electronic properties of the zigzag graphene nanoribbon (ZGNR). By using the first-principles of the density functional theory (DFT) calculations at the B3LYP/ 6-31G, which implemented in the Gaussian 09 software, various properties, such as the geometrical structure, DOS, HOMO, LUMO infrared spectra, and energy gap of the ZGNR, were investigated with various sites and concentrations of the phosphorus (P). It was observed that the ZGNR could be converted from linear to fractal dimension by using phosphorus (P) impurities. Also, the fractal binary tree of the ZGNR and P-ZGNR structures is a highlight. The results demonstrated that the energy gap has different values, which located at this range from 0.51eV to 1.158 eV for pristine ZGNR and P-ZGNR structures. This range of energy gap is variable according to the use of GNRs in any apparatus. Then, the P-ZGNR has semiconductor behavior. Moreover, there are no imaginary wavenumbers on the evaluated vibrational spectrum confirms that the model corresponds to minimum energy. Then, these results make P-ZGNR can be utilized in various applications due to this structure became more stable and lower reactivity.
Article
Full-text available
Chemical doping is a promising route to engineering and controlling the electronic properties of the zigzag graphene nanoribbon (ZGNR). By using the first-principles of the density functional theory (DFT) calculations at the B3LYP/ 6-31G, which implemented in the Gaussian 09 software, various properties, such as the geometrical structure, DOS, HOMO, LUMO infrared spectra, and energy gap of the ZGNR, were investigated with various sites and concentrations of the phosphorus (P). It was observed that the ZGNR could be converted from linear to fractal dimension by using phosphorus (P) impurities. Also, the fractal binary tree of the ZGNR and P-ZGNR structures is a highlight. The results demonstrated that the energy gap has different values, which located at this range from 0.51eV to 1.158 eV for pristine ZGNR and P-ZGNR structures. This range of energy gap is variable according to the use of GNRs in any apparatus. Then, the P-ZGNR has semiconductor behavior. Moreover, there are no imaginary wavenumbers on the evaluated vibrational spectrum confirms that the model corresponds to minimum energy. Then, these results make P-ZGNR can be utilized in various applications due to this structure became more stable and lower reactivity.
Article
We present the first 3D algorithm capable of answering the question: what would a Mandelbrot-like set in the shape of a bunny look like? More concretely, can we find an iterated quaternion rational map whose potential field contains an isocontour with a desired shape? We show that it is possible to answer this question by casting it as a shape optimization that discovers novel, highly complex shapes. The problem can be written as an energy minimization, the optimization can be made practical by using an efficient method for gradient evaluation, and convergence can be accelerated by using a variety of multi-resolution strategies. The resulting shapes are not invariant under common operations such as translation, and instead undergo intricate, non-linear transformations.
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Bounding sets to IFS fractals are useful largely due to their property of iterative containment, in both theoretical and computational settings. The tightest convex bounding set, the convex hull has revealed itself to be particularly relevant in the literature. The problem of its exact determination may have been overshadowed by various approximation methods, so our aim is to emphasize its relevance and beauty. The finiteness of extrema is examined a priori from the IFS parameters - a property of the convex hull often taken for granted. Former methods are surveyed and improved upon, and a new "outside-in" approach is introduced and crystallized for practical applicability. Periodicity in the address of extremal points will emerge to be the central idea.