scholarly journals Using fractal geometry for solving divide-and-conquer recurrences

1995 ◽  
Vol 37 (2) ◽  
pp. 145-171 ◽  
Author(s):  
Simant Dube
Keyword(s):  
Fractal Geometry ◽  
Recursive Function ◽  
Time Complexity ◽  
Recursive Algorithm ◽  
Dynamic Structure ◽  
Divide And Conquer ◽  
Function System ◽  
Fractal Image ◽  
Dimensional Measure ◽  
Self Similar

AbstractA relationship between the fractal geometry and the analysis of recursive (divide-and-conquer) algorithms is investigated. It is shown that the dynamic structure of a recursive algorithm which might call other algorithms in a mutually recursive fashion can be geometrically captured as a fractal (self-similar) image. This fractal image is defined as the attractor of a mutually recursive function system. It then turns out that the Hausdorff-Besicovich dimension D of such an image is precisely the exponent in the time complexity of the algorithm being modelled. That is, if the Hausdorff D-dimensional measure of the image is finite then it serves as the constant of proportionality and the time complexity is of the form Θ(nD), else it implies that the time complexity is of the form Θ(nD logpn), where p is an easily determined constant.

Fractal Geometry as a Bridge between Realms

2013 ◽  
pp. 25-43 ◽  
Author(s):  
Terry Marks-Tarlow
Keyword(s):  
Fractal Geometry ◽  
Open Systems ◽  
Autonomous Systems ◽  
Mathematical Notation ◽  
Deep Space ◽  
The Unconscious ◽  
Self And Other ◽  
Branching Structures ◽  
Self Similar ◽  
Multidimensional Objects

This chapter describes fractal geometry as a bridge between the imaginary and the real, mind and matter, conscious and the unconscious. Fractals are multidimensional objects with self-similar detail across size and/or time scales. Jung conceived of number as the most primitive archetype of order, serving to link observers with the observed. Whereas Jung focused upon natural numbers as the foundation for order that is already conscious in the observer, I offer up the fractal geometry as the underpinnings for a dynamic unconscious destined never to become fully conscious. Throughout nature, fractals model the complex, recursively branching structures of self-organizing systems. When they serve at the edges of open systems, fractal boundaries articulate a paradoxical zone that simultaneously separates as it connects. When modeled by Spencer-Brown’s mathematical notation, full interpenetration between inside and outside edges translates to a distinction that leads to no distinction. By occupying the infinitely deep “space between” dimensions and levels of existence, fractal boundaries contribute to the notion of intersubjectivity, where self and other become most entwined. They also exemplify reentry dynamics of Varela’s autonomous systems, plus Hofstadter’s ever-elusive “tangled hierarchy” between brain and mind.

Orbit of an Image Under Iterated System II

2011 ◽  
Vol 2 (4) ◽  
pp. 57-74
Author(s):  
S. L. Singh ◽  
S. N. Mishra ◽  
Sarika Jain
Keyword(s):  
Fractal Geometry ◽  
Iterative Procedure ◽  
Iterated Function System ◽  
Mathematical Structure ◽  
Program Code ◽  
Computational Mathematics ◽  
Fractal Image ◽  
Iterated Function ◽  
Iterative Procedures

An orbital picture is a mathematical structure depicting the path of an object under Iterated Function System. Orbital and V-variable orbital pictures initially developed by Barnsley (2006) have utmost importance in computer graphics, image compression, biological modeling and other areas of fractal geometry. These pictures have been generated for linear and contractive transformations using function and superior iterative procedures. In this paper, the authors introduce the role of superior iterative procedure to find the orbital picture under an IFS consisting of non-contractive or non-expansive transformations. A mild comparison of the computed figures indicates the usefulness of study in computational mathematics and fractal image processing. A modified algorithm along with program code is given to compute a 2-variable superior orbital picture.

scholarly journals Dimension approximation of attractors of graph directed IFSs by self-similar sets

2018 ◽  
Vol 167 (01) ◽  
pp. 193-207 ◽  
Author(s):  
ÁBEL FARKAS
Keyword(s):  
Iterated Function System ◽  
Black Box ◽  
Function System ◽  
Separation Condition ◽  
Iterated Function ◽  
Strong Separation ◽  
Self Similar ◽  
Strong Separation Condition ◽  
Distance Sets

AbstractWe show that for the attractor (K1, . . ., Kq) of a graph directed iterated function system, for each 1 ⩽ j ⩽ q and ϵ > 0 there exists a self-similar set K ⊆ Kj that satisfies the strong separation condition and dimHKj − ϵ < dimHK. We show that we can further assume convenient conditions on the orthogonal parts and similarity ratios of the defining similarities of K. Using this property as a ‘black box’ we obtain results on a range of topics including on dimensions of projections, intersections, distance sets and sums and products of sets.

A PROBABILISTIC POWER DOMAIN ALGORITHM FOR FRACTAL IMAGE DECODING

2002 ◽  
Vol 02 (02) ◽  
pp. 161-173
Author(s):  
V. DRAKOPOULOS ◽  
A. KAKOS ◽  
N. NIKOLAOU
Keyword(s):  
Complexity Analysis ◽  
Iterated Function System ◽  
Function System ◽  
Fractal Image ◽  
Power Domain ◽  
Iterated Function ◽  
Image Decoding

A new algorithm, called herein the random power domain algorithm, is discussed; it generates the image corresponding to an iterated function system with probabilities, a technique used in fractal image decoding. A simple complexity analysis for the algorithm is also derived.

scholarly journals Lattice self-similar sets on the real line are not Minkowski measurable

2018 ◽  
Vol 40 (1) ◽  
pp. 221-232
Author(s):  
SABRINA KOMBRINK ◽  
STEFFEN WINTER
Keyword(s):  
Real Line ◽  
Iterated Function System ◽  
Function System ◽  
Open Set Condition ◽  
The Real ◽  
Open Set ◽  
Puzzle Piece ◽  
Iterated Function ◽  
Self Similar ◽  
Special Classes

We show that any non-trivial self-similar subset of the real line that is invariant under a lattice iterated function system (IFS) satisfying the open set condition (OSC) is not Minkowski measurable. So far, this has only been known for special classes of such sets. Thus, we provide the last puzzle-piece in proving that under the OSC a non-trivial self-similar subset of the real line is Minkowski measurable if and only if it is invariant under a non-lattice IFS, a 25-year-old conjecture.

An Efficient Multibody Divide and Conquer Algorithm

2005 ◽  
Author(s):  
James H. Critchley
Keyword(s):  
Multibody Dynamics ◽  
Time Complexity ◽  
Future Generation ◽  
Parallel Computers ◽  
Parallel Processors ◽  
Divide And Conquer ◽  
Parallel Computer ◽  
Divide And Conquer Algorithm ◽  
Computer Resources ◽  
Theoretical Minimum

A new and efficient form of Featherstone’s multibody Divide and Conquer Algorithm (DCA) is presented. The DCA was the first algorithm to achieve theoretically optimal logarithmic time complexity with a theoretical minimum of parallel computer resources for general problems of multibody dynamics, however the DCA is extremely inefficient in the presence of small to modest parallel computers. The new efficient DCA approach (DCAe) demonstrates that large DCA subsystems can be constructed using fast sequential techniques and realize substantial speed increases in the presence of as few as two parallel processors. Previously the DCA was a tool intended for a future generation of parallel computers, this enhanced version promises practical and competitive performance with the parallel computers of today.

Application of Fractal Geometry to Fracture Forensics, Research and Production

2006 ◽  
Vol 45 ◽  
pp. 1646-1651 ◽  
Author(s):  
J.J. Mecholsky Jr.
Keyword(s):  
Fractal Analysis ◽  
Fractal Geometry ◽  
Crack Size ◽  
Structural Parameter ◽  
New Materials ◽  
Scale Invariant ◽  
Prepared Material ◽  
Reasonable Approach ◽  
Self Similar ◽  
Production Problems

The fracture surface records past events that occur during the fracture process by leaving characteristic markings. The application of fractal geometry aids in the interpretation and understanding of these events. Quantitative fractographic analysis of brittle fracture surfaces shows that these characteristic markings are self-similar and scale invariant, thus implying that fractal analysis is a reasonable approach to analyzing these surfaces. The fractal dimensional increment, D*, is directly proportional to the fracture energy, γ, during fracture for many brittle materials, i.e., γ = ½ E a0 D* where E is the elastic modulus and a0 is a structural parameter. Also, D* is equal to the crack-size-to-mirror-radius ratio. Using this information can aid in identifying toughening mechanisms in new materials, distinguishing poorly fabricated from well prepared material and identifying stress at fracture for field failures. Examples of the application of fractal analysis in research, fracture forensics and solving production problems are discussed.

scholarly journals MULTIPLE REPRESENTATIONS OF REAL NUMBERS ON SELF-SIMILAR SETS WITH OVERLAPS

Fractals ◽  
2019 ◽  
Vol 27 (04) ◽  
pp. 1950051 ◽  
Author(s):  
KAN JIANG ◽  
XIAOMIN REN ◽  
JIALI ZHU ◽  
LI TIAN
Keyword(s):  
Convex Hull ◽  
Iterated Function System ◽  
Multiple Representations ◽  
Function System ◽  
Real Numbers ◽  
Iterated Function ◽  
Self Similar

Let [Formula: see text] be the attractor of the following iterated function system (IFS) [Formula: see text] where [Formula: see text] and [Formula: see text] is the convex hull of [Formula: see text]. The main results of this paper are as follows: [Formula: see text] if and only if [Formula: see text] where [Formula: see text]. If [Formula: see text], then [Formula: see text]As a consequence, we prove that the following conditions are equivalent:(1) For any [Formula: see text], there are some [Formula: see text] such that [Formula: see text].(2) For any [Formula: see text], there are some [Formula: see text] such that [Formula: see text](3) [Formula: see text].

Similarity Hashing: A Computer Vision Solution to the Inverse Problem of Linear Fractals

Fractals ◽  
1997 ◽  
Vol 05 (supp01) ◽  
pp. 39-50 ◽  
Author(s):  
John C. Hart ◽  
Wayne O. Cochran ◽  
Patrick J. Flynn
Keyword(s):  
Computer Vision ◽  
Inverse Problem ◽  
Heuristic Search ◽  
Fractal Geometry ◽  
Self Similarity ◽  
Geometric Hashing ◽  
Numerical Minimization ◽  
Self Similar ◽  
Vision Algorithm ◽  
Fractal Representation

The difficult task of finding a fractal representation of an input shape is called the inverse, problem of fractal geometry. Previous attempts at solving this problem have applied techniques from numerical minimization, heuristic search and image compression. The most appropriate domain from which to attack this problem is not numerical analysis nor signal processing, but model-based computer vision. Self-similar objects cause an existing computer vision algorithm called geometric hashing to malfunction. Similarity hashing capitalizes on this observation to not only detect a shape's morphological self-similarity but also find the parameters of its self-transformations.

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