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- Biblioteca Digitais de Teses e Dissertações da USP
- Universidade Estadual Paulista
- Associação Brasileira de Engenharia e Ciências Mecânicas - ABCM
- Elsevier B.V.
- UNESP - Universidade Estadual Paulista, Pró-Reitoria de Pesquisa; Programa de Pós-Graduação em Educação Matemática
- Universidade Cornell
- Universidade Autônoma de Barcelona
- Rochester Instituto de Tecnologia
- World Scientific Publishing Company
- Mais Publicadores...

## ‣ Propriedades aritméticas e topológicas de uma classe de fractais de rauzy; Arithmetic and topological properties of a subclass of the so-called Rauzy's fractals

Fonte: Biblioteca Digitais de Teses e Dissertações da USP
Publicador: Biblioteca Digitais de Teses e Dissertações da USP

Tipo: Tese de Doutorado
Formato: application/pdf

Publicado em 09/03/2010
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#Arithmetic and topological properties#Fractal de Rauzy#Propriedades aritméticas e topológicas#Rauzy fractals

Estudamos as propriedades aritméticas, geométricas e topológicas de uma classe dos chamados Fractais de Rauzy. Estudamos partucularmente o azulejamento periódico do plano complexo C induzido por eles, assim como a dimensão de Hausdorff de suas fronteiras. Tal trabalho exige um estudo detalhado da fronteira destes conjuntos, que está associada às propriedades aritméticas da 'alpha' -representação dos números complexos com respeito a um certo número algébrico 'alfa'; We study the arithmetic, geometric and topological properties of a class of the so-called Rauzy's fractals. In particular we study the periodic tiling of the complex plane C induced by them and the Hausdorff dimension of its boundary. Such work is connected to a detailed study of the boundary of such sets and the arithmetic properties of the 'alpha' representation of complex numbers with respect to a certain algebraic number 'alpha'

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## ‣ Tilings induced by a class of cubic Rauzy fractals

Fonte: Universidade Estadual Paulista
Publicador: Universidade Estadual Paulista

Tipo: Artigo de Revista Científica
Formato: 6-31

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#Lattice tiling#Rauzy fractal#Self-replicating tiling#Substitution#Homeomorphic#Rauzy fractals#Topological properties#Substitution reactions#Topology#Fractals

We study aperiodic and periodic tilings induced by the Rauzy fractal and its subtiles associated with beta-substitutions related to the polynomial x3-ax2-bx-1 for a≥b≥1. In particular, we compute the corresponding boundary graphs, describing the adjacencies in the tilings. These graphs are a valuable tool for more advanced studies of the topological properties of the Rauzy fractals. As an example, we show that the Rauzy fractals are not homeomorphic to a closed disc as soon as a≤2b-4. The methods presented in this paper may be used to obtain similar results for other classes of substitutions.© 2012 Elsevier B.V. All rights reserved.

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## ‣ Geometry, dynamics and fractals

Fonte: Associação Brasileira de Engenharia e Ciências Mecânicas - ABCM
Publicador: Associação Brasileira de Engenharia e Ciências Mecânicas - ABCM

Tipo: Artigo de Revista Científica
Formato: text/html

Publicado em 01/03/2008
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Consider a collection of elastic wires folded according to a given pattern induced by a sequence of fractal plane curves. The folded wires can act as elastic springs. Therefore it is easy to build up a corresponding sequence of simple oscillators composed by the elastic springs clamped at one end and carrying a mass at the opposite end. The oscillation periods of the ordered sequence of these oscillators are related following a power law and therefore display a fractal structure. The periods of each oscillator clearly depend on the mechanical properties of the wire, on the mass at the end and on the boundary conditions. Therefore there are infinitely many possibilities to design a dynamical fractal sequence in opposition to the well defined fractal dimension of the underneath geometric sequence. Nevertheless the geometric fractal dimension of the primordial geometric curve is always related somehow to the dynamical fractal dimension characterizing the oscillation period sequence. It is important to emphasize that the dynamical fractal dimension of a given sequence built up after the geometry of a primordial one is not unique. This peculiarity introduces the possibility to have a broader information spectrum about the geometry which is otherwise impossible to achieve. This effect is clearly demonstrated for random fractals. The present paper deals with a particular family of curves...

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## ‣ A class of cubic Rauzy fractals

Fonte: Elsevier B.V.
Publicador: Elsevier B.V.

Tipo: Artigo de Revista Científica
Formato: 114-130

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Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq); Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP); Processo FAPESP: 2013/24541-0; Processo FAPESP: 2008/02841-4; Processo FAPESP: 2010108654-1; In this paper, we study arithmetical and topological properties for a class of Rauzy fractals R-a given by the polynomial x(3) - ax(2) + x - 1 where a >= 2 is an integer. In particular, we prove the number of neighbors of R-a in the periodic tiling is equal to 8. We also give explicitly an automaton that generates the boundary of R-a. As a consequence, we prove that R-2 is homeomorphic to a topological disk.

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## ‣ A Cross-age study of students' understanding of fractals

Fonte: UNESP - Universidade Estadual Paulista, Pró-Reitoria de Pesquisa; Programa de Pós-Graduação em Educação Matemática
Publicador: UNESP - Universidade Estadual Paulista, Pró-Reitoria de Pesquisa; Programa de Pós-Graduação em Educação Matemática

Tipo: Artigo de Revista Científica
Formato: text/html

Publicado em 01/12/2013
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The purpose of this study is to examine how students understand fractals depending on age. Students' understandings were examined in four dimensions: defining fractals, determining fractals, finding fractal patterns rules and mathematical operations with fractals. The study was conducted with 187 students (grades 8, 9, 10) by using a two-tier test consisting of nine questions prepared based on the literature and Turkish mathematics and geometry curriculums. The findings showed that in all grades, students may have misunderstandings and lack of knowledge about fractals. Moreover, students can identify and determine the fractals, but when the grade level increased, this success decreases. Although students were able to intuitively determine a shape as fractal or not, they had some problems in finding pattern rules and formulizing them.

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## ‣ Disconnected Julia sets and gaps in the spectrum of Laplacians on symmetric finitely ramified fractals

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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It is known that Laplacian operators on many fractals have gaps in their
spectra. This fact precludes the possibility that a Weyl-type ratio can have a
limit and is also a key ingredient in proving that the Fourier series on such
fractals can have better convergence results than in the classical setting. In
this paper we prove that the existence of gaps is equivalent to the total
disconnectedness of the Julia set of the spectral decimation function for the
class of fully symmetric p.c.f. fractals, and for self-similar fully symmetric
finitely ramified fractals with regular harmonic structure. We also formulate
conjectures related to geometry of finitely ramified fractals with spectral
gaps, to complex spectral dimensions, and to convergence of Fourier series on
such fractals.; Comment: 16 pages

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## ‣ The structure factor of fat deterministic Vicsek fractals: a small-angle scattering study

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 16/07/2015
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We study here the small-angle scattering structure factor for deterministic
fat fractals in the reciprocal space. It is shown that fat fractals are exact
self-similar in the range of iterations having the same values of the scaling
factor, and therefore in each of these ranges all the properties of regular
fractals can be inferred to fat fractals. In order to illustrate the above
findings we introduce deterministic fattened versions of Vicsek deterministic
fractals. We calculate the mono- and polydisperse structure factor and study
its scattering properties.

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## ‣ Statistical Mechanics and Quantum Fields on Fractals

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 25/10/2012
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Fractals define a new and interesting realm for a discussion of basic
phenomena in quantum field theory and statistical mechanics. This interest
results from specific properties of fractals, e.g., their dilatation symmetry
and the corresponding absence of Fourier mode decomposition. Moreover, the
existence of a set of distinct dimensions characterizing the physical
properties (spatial or spectral) of fractals make them a useful testing ground
for dimensionality dependent physical problems. This paper addresses specific
problems including the behavior of the heat kernel and spectral zeta functions
on fractals and their importance in the expression of spectral properties in
quantum physics. Finally, we apply these results to specific problems such as
thermodynamics of quantum radiation by a fractal blackbody.; Comment: 21 pages, 2 figures, 1 table. Proceedings of the conference :
Applications of Fractals and Dynamical Systems in Science and Economics
Edited by: David Carfi, Michel L. Lapidus, Erin P. J. Pearse, and Machiel van
Frankenhuijsen. Contemporary Mathematics (CONM) book series

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## ‣ Micro and Macro Fractals generated by multi-valued dynamical systems

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 28/04/2013
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#Mathematics - General Topology#Mathematics - Dynamical Systems#Mathematics - Metric Geometry#54H20, 37M05, 54C60

Given a multi-valued function $\Phi$ on a topological space $X$ we study the
properties of its fixed fractal, which is defined as the closure of the orbit
$\Phi^\omega(Fix(\Phi))=\bigcup_{n\in\omega}\Phi^n(Fix(\Phi))$ of the set
$Fix(\Phi)=\{x\in X:x\in\Phi(x)\}$ of fixed points of $\Phi$. A special
attention is paid to the duality between micro-fractals and macro-fractals,
which are fixed fractals for a contracting compact-valued function $\Phi$ on a
complete metric space $X$ and its inverse multi-function $\Phi^{-1}$. With help
of algorithms (described in this paper) we generate various images of
macro-fractals which are dual to some well-known micro-fractals like the
fractal cross, the Sierpinski triangle, Sierpinski carpet, the Koch curve, or
the fractal snowflakes. The obtained images show that macro-fractals have a
large-scale fractal structure, which becomes clearly visible after a suitable
zooming.; Comment: 16 pages + 32 pages of Appendix with Gallery of Macro-Fractals

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## ‣ Using blinking fractals for mathematical modeling of processes of growth in biological systems

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 14/03/2012
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Many biological processes and objects can be described by fractals. The paper
uses a new type of objects - blinking fractals - that are not covered by
traditional theories considering dynamics of self-similarity processes. It is
shown that both traditional and blinking fractals can be successfully studied
by a recent approach allowing one to work numerically with infinite and
infinitesimal numbers. It is shown that blinking fractals can be applied for
modeling complex processes of growth of biological systems including their
season changes. The new approach allows one to give various quantitative
characteristics of the obtained blinking fractals models of biological systems.; Comment: 19 pages, 12 figures

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## ‣ Smooth bumps, a Borel theorem and partitions of smooth functions on p.c.f. fractals

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 03/09/2009
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We provide two methods for constructing smooth bump functions and for
smoothly cutting off smooth functions on fractals, one using a probabilistic
approach and sub-Gaussian estimates for the heat operator, and the other using
the analytic theory for p.c.f. fractals and a fixed point argument. The heat
semigroup (probabilistic) method is applicable to a more general class of
metric measure spaces with Laplacian, including certain infinitely ramified
fractals, however the cut off technique involves some loss in smoothness. From
the analytic approach we establish a Borel theorem for p.c.f. fractals, showing
that to any prescribed jet at a junction point there is a smooth function with
that jet. As a consequence we prove that on p.c.f. fractals smooth functions
may be cut off with no loss of smoothness, and thus can be smoothly decomposed
subordinate to an open cover. The latter result provides a replacement for
classical partition of unity arguments in the p.c.f. fractal setting.; Comment: 26 pages. May differ slightly from published (refereed) version

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## ‣ Scaling Range and Cutoffs in Empirical Fractals

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 13/01/1998
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Fractal structures appear in a vast range of physical systems. A literature
survey including all experimental papers on fractals which appeared in the six
Physical Review journals (A-E and Letters) during the 1990's shows that
experimental reports of fractal behavior are typically based on a scaling range
$\Delta$ which spans only 0.5 - 2 decades. This range is limited by upper and
lower cutoffs either because further data is not accessible or due to crossover
bends. Focusing on spatial fractals, a classification is proposed into (a)
aggregation; (b) porous media; (c) surfaces and fronts; (d) fracture and (e)
critical phenomena. Most of these systems, [except for class (e)] involve
processes far from thermal equilibrium. The fact that for self similar fractals
[in contrast to the self affine fractals of class (c)] there are hardly any
exceptions to the finding of $\Delta \le 2$ decades, raises the possibility
that the cutoffs are due to intrinsic properties of the measured systems rather
than the specific experimental conditions and apparatus. To examine the origin
of the limited range we focus on a class of aggregation systems. In these
systems a molecular beam is deposited on a surface, giving rise to nucleation
and growth of diffusion-limited-aggregation-like clusters. Scaling arguments
are used to show that the required duration of the deposition experiment
increases exponentially with $\Delta$. Furthermore...

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## ‣ Controlling the efficiency of trapping in treelike fractals

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 02/07/2013
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Efficiently controlling the trapping process, especially the trapping
efficiency, is central in the study of trap problem in complex systems, since
it is a fundamental mechanism for diverse other dynamic processes. Thus, it is
of theoretical and practical significance to study the control technique for
trapping problem. In this paper, we study the trapping problem in a family of
proposed directed fractals with a deep trap at a central node. The directed
fractals are a generalization of previous undirected fractals by introducing
the directed edge weights dominated by a parameter. We characterize all the
eigenvalues and their degeneracies for an associated matrix governing the
trapping process. The eigenvalues are provided through an exact recursive
relation deduced from the self-similar structure of the fractals. We also
obtain the expressions for the smallest eigenvalue and the mean first-passage
time (MFPT) as a measure of trapping efficiency, which is the expected time for
the walker to first visit the trap. The MFPT is evaluated according to the
proved fact that it is approximately equal to reciprocal of the smallest
eigenvalue. We show that the MFPT is controlled by the weight parameter, by
modifying which, the MFPT can scale superlinealy...

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## ‣ Linear and branched polymers on fractals

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 15/08/2005
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This is a pedagogical review of the subject of linear polymers on
deterministic finitely ramified fractals. For these, one can determine the
critical properties exactly by real-space renormalization group technique. We
show how this is used to determine the critical exponents of self-avoiding
walks on different fractals. The behavior of critical exponents for the
$n$-simplex lattice in the limit of large $n$ is determined. We study
self-avoiding walks when the fractal dimension of the underlying lattice is
just below 2. We then consider the case of linear polymers with attractive
interactions, which on some fractals leads to a collapse transition. The
fractals also provide a setting where the adsorption of a linear chain near on
attractive substrate surface and the zipping-unzipping transition of two
mutually interacting chains can be studied analytically. We also discuss
briefly the critical properties of branched polymers on fractals.; Comment: 46 pages, 23 figures

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## ‣ Determining global mean-first-passage time of random walks on Vicsek fractals using eigenvalues of Laplacian matrices

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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The family of Vicsek fractals is one of the most important and
frequently-studied regular fractal classes, and it is of considerable interest
to understand the dynamical processes on this treelike fractal family. In this
paper, we investigate discrete random walks on the Vicsek fractals, with the
aim to obtain the exact solutions to the global mean first-passage time
(GMFPT), defined as the average of first-passage time (FPT) between two nodes
over the whole family of fractals. Based on the known connections between FPTs,
effective resistance, and the eigenvalues of graph Laplacian, we determine
implicitly the GMFPT of the Vicsek fractals, which is corroborated by numerical
results. The obtained closed-form solution shows that the GMFPT approximately
grows as a power-law function with system size (number of all nodes), with the
exponent lies between 1 and 2. We then provide both the upper bound and lower
bound for GMFPT of general trees, and show that leading behavior of the upper
bound is the square of system size and the dominating scaling of the lower
bound varies linearly with system size. We also show that the upper bound can
be achieved in linear chains and the lower bound can be reached in star graphs.
This study provides a comprehensive understanding of random walks on the Vicsek
fractals and general treelike networks.; Comment: Definitive version accepted for publication in Physical Review E

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## ‣ V-variable fractals and superfractals

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 16/12/2003
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Deterministic and random fractals, within the framework of Iterated Function
Systems, have been used to model and study a wide range of phenomena across
many areas of science and technology. However, for many applications
deterministic fractals are locally too similar near distinct points while
standard random fractals have too little local correlation. Random fractals are
also slow and difficult to compute. These two major problems restricting
further applications are solved here by the introduction of V-variable fractals
and superfractals.; Comment: 17 pages, 10 figures

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## ‣ Determining mean first-passage time on a class of treelike regular fractals

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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Relatively general techniques for computing mean first-passage time (MFPT) of
random walks on networks with a specific property are very useful, since a
universal method for calculating MFPT on general graphs is not available
because of their complexity and diversity. In this paper, we present techniques
for explicitly determining the partial mean first-passage time (PMFPT), i.e.,
the average of MFPTs to a given target averaged over all possible starting
positions, and the entire mean first-passage time (EMFPT), which is the average
of MFPTs over all pairs of nodes on regular treelike fractals. We describe the
processes with a family of regular fractals with treelike structure. The
proposed fractals include the $T$ fractal and the Peano basin fractal as their
special cases. We provide a formula for MFPT between two directly connected
nodes in general trees on the basis of which we derive an exact expression for
PMFPT to the central node in the fractals. Moreover, we give a technique for
calculating EMFPT, which is based on the relationship between characteristic
polynomials of the fractals at different generations and avoids the computation
of eigenvalues of the characteristic polynomials. Making use of the proposed
methods, we obtain analytically the closed-form solutions to PMFPT and EMFPT on
the fractals and show how they scale with the number of nodes. In addition...

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## ‣ Description of diffusive and propagative behavior on fractals

Fonte: Universidade Autônoma de Barcelona
Publicador: Universidade Autônoma de Barcelona

Tipo: Artigo de Revista Científica
Formato: application/pdf

Publicado em //2004
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The known properties of diffusion on fractals are reviewed in order to give a general outlook of these dynamic processes. After that, we propose a description developed in the context of the intrinsic metric of fractals, which leads us to a differential equation able to describe diffusion in real fractals in the asymptotic regime. We show that our approach has a stronger physical justification than previous works on this field. The most important result we present is the introduction of a dependence on time and space for the conductivity in fractals, which is deduced by scaling arguments and supported by computer simulations. Finally, the diffusion equation is used to introduce the possibility of reaction-diffusion processes on fractals and analyze their properties. Specifically, an analytic expression for the speed of the corresponding travelling fronts, which can be of great interest for application purposes, is derived.

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## ‣ A Recursive Fractal Design Generator for Dimensions Zero to Two Implemented within a Two Dimensional Core Graphics Package

Fonte: Rochester Instituto de Tecnologia
Publicador: Rochester Instituto de Tecnologia

Tipo: Tese de Doutorado

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#Computer science#Core Graphics System#Fractals#Graphics on Microcomputers#Hausdorff Besicovitch dimension#Interactive computer graphics#Thesis#T385.L45 1985#Mathematical models#Geometry#Fractals

This thesis incorporates the technique developed by Benoit Mandelbrot to describe recursive fractals into an interactive graphics package based on the Core Graphics System (Core) produced by an ACM SIGGRAPH Committee (1977, 1979). The graphics package encompasses simple standard geometric shapes as well as the recursive fractals. To draw those fractals requires knowing both the basic shape or generator, and the points of recursion. These two pieces are acquired through the using of two windows which allow the generator and the points of recursion to be built. Once built, the fractal recursion level is chosen interactively on the main drawing. The conclusion I reached as a result of this project is that it is possible to integrate fractals in a systematic way into a standard graphics package, much as rectangles and circles are today in most graphics systems.

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## ‣ A fractal valued random iteration algorithm and fractal hierachy

Fonte: World Scientific Publishing Company
Publicador: World Scientific Publishing Company

Tipo: Artigo de Revista Científica

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#Keywords: Algorithms#Approximation theory#Computational methods#Computer graphics#Hierarchical systems#Interpolation#Iterative methods#Markov processes#Mathematical transformations#Monte Carlo methods#Random processes

We describe new families of random fractals, referred to as "V-variable", which are intermediate between the notions of deterministic and of standard random fractals. The parameter V describes the degree of "variability": at each magnification level any V-variable fractals has at most V key "forms" or "shapes". V-variable random fractals have the surprising property that they can be computed using a forward process. More precisely, a version of the usual Random Iteration Algorithm, operating on sets (or measures) rather than points, can be used to sample each family. To present this theory, we review relevant results on fractals (and fractal measures), both deterministic and random. Then our new results are obtained by constructing an iterated function system (a super IPS) from a collection of standard IFSs together with a corresponding set of probabilities. The attractor of the super IFS is called a superfractal; it is a collection of V-variable random fractals (sets or measures) together with an associated probability distribution on this collection. When the underlying space is for example ℝ2, and the transformations are computationally straightforward (such as affine transformations), the superfractal can be sampled by means of the algorithm...

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