3D Fractals, Axiom of Algebra (Δn)n = +1

After having laid down the Axiom of Algebra, bringing the creation of the square root of -1 by Euler to the entire circle and thus authorizing a simple notation of the nth roots of unity, the author uses it to organize homogeneous divisions of the limited development of the exponential function, that is opening the way to the use of a whole bunch of new primary functions in Differential Calculus. He then shows how new supercomplex products in dimension 3 make it possible to calculate fractals whose connexity depends on the product considered. We recall the geometry of convex polygons and regular polygons.

Multi-range Fractals in Materials

A new model of multi-range fractals is proposed to explain the experimental results observed on the fractal dimensions of the fracture surfaces in materials.The relationship of multi-range fractals with multi-scaling fractals has been also discussed.

Multiple-cell elements and regular multifractals

Based on fractal super fibers and binary fractal fibers, the following objectives are approached in this paper： First, the concept of multiple-cell elements is induced and abstracted. Second, through multiple-cell elements, the constructability of regular multifractals with strict self-similarities is confirmed, and the universality of the con- struction mode for regular multifractals is proved. Third, through the construction mode and multiple-cell elements, regular multifractals are demonstrated to be equivalent to generalized regular single fractals with multilayer fine structures. On the basis of such equivalence, the dimension formula of the regular single fractal is extended to that of the regular multifractal, and the geometry of regular single fractals is extended to that of regular multifractals. Fourth, through regular multifractals, a few golden fractals are constructed.

CALCULUS ON FRACTALS BASED UPON LOCAL FIELDS─In memory of Founding Editor Professor M.T.Cheng with great respect and deep sorrow

The main contents in this note are: 1. introduction; 2. locally compact groups and local fields; 3. calculus on fractals based upon local fields; 4. fractional calculus and fractals; 5. fractal function spaces and PDE on fractals.

Multirange Fractals in Materials: Applications to Fracture and Mechanical Alloying under Ball Milling

A new model of multirange fractals is proposed to explain the experimental results observed on the fractal dimensions of the fractured surfaces in materials. A new explanation to the Williford's multifractal curve on the relationship of fractal dimension with fracture properties in materials has been given. It shows the importance of fractorizing out the effect of fractal structure from other physical causes and separating the appropriate range of scale from multirange fractals. Mechanical alloying process under ball milling as a non-equilibrium dynamical system has been also analyzed.

Chaotic Fractals at the Root of Relativistic Quantum Physics and Cosmology

At its most basic level physics starts with space-time topology and geometry. On the other hand topology’s and geometry’s simplest and most basic elements are random Cantor sets. It follows then that nonlinear dynamics i.e. deterministic chaos and fractal geometry is the best mathematical theory to apply to the problems of high energy particle physics and cosmology. In the present work we give a short survey of some recent achievements of applying nonlinear dynamics to notoriously difficult subjects such as quantum entanglement as well as the origin and true nature of dark energy, negative absolute temperature and the fractal meaning of the constancy of the speed of light.

STUDY ON MULTI-RANGE FRACTALS IN FRACTURED SURFACE OF ROCK

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Application of Fractals on Microstructure of Neodymium-Doped Yttrium Aluminum (Nd∶YAG) Transparent Ceramics

Nd∶YAG precursor powders were synthesized by homogeneous precipitation and Nd∶YAG transparent ceramics were prepared by vacuum sintering at 1700 ℃ for 5 h. The ceramic materials were characterized by light transmittance, field emission gun-environment scanning microscope. Fractal geometry was used to study the quantitative relationships between light transmittance and fractal dimensions of Nd∶YAG transparent ceramics. It was found that the transmittance of Nd∶YAG with 1 mm in thickness was about 45% and 58% in visible and near-infrared region respectively. The microstructures of Nd∶YAG transparent ceramics were obvious fractal characteristic and fractal dimensions depart a little from two-dimension. The light transmittance decreased with increasing of fractal dimension and nonlinear fit curve was y=1350-1185x+269x2 between fractal dimension and light transmittance of Nd∶YAG transparent ceramics.

FRACTALS OF HYBRID ORBITALS AND THEIR APPLICATIONS IN THE ENZYME MODELS

An enzyme is a kind of protein with catalytic activity and long chain,and its structure and shape are determined by the hybridized state of atomic orbital.The fractal dimension(D_f)is closely related to the hybridization,e.g.D_f=2ln2/ln[2(1+α/(1-α))]for the spa type, where a denotes the fraction of the s orbital in the hybridized molecular orbital.This relationship and the five fractal theorems introduced by the present paper play an important role in the investigations of the model of imitative enzyme.

Urban Growth Prediction Modelling Using Fractals and Theory of Chaos

Urban growth prediction has acquired an important consideration in urban sustainability. An effective approach of urban prediction can be a valuable tool in urban decision making and planning. A large urban development has been occurred during last decade in the touristic village of Pogonia Etoloakarnanias, Greece, where an urban growth of 57.5% has been recorded from 2003 to 2011. The prediction of new urban settlements was achieved using fractals and theory of chaos. More specifically, it was found that the urban growth is taken place within a Sierpinski carpet. Several shapes of Sierpinski carpets were tested in order to find the most appropriate, which produced an accuracy percentage of 70.6% for training set and 81.8% for validation set. This prediction method can be effectively applied in urban growth modelling, once cities are fractals and urban complexity can be successfully described through a Sierpinski tessellation.

Combining Fractals and Box-Counting Dimension

In this paper, the box-counting dimension is used to derive an explicit formula for the dimension of a fractal constructed using several contractions or by combining fractals. This dimension agrees with the Hausdorff dimension in the particular case when the scales factors considered are all the same. A more general sufficient condition for the box-counting dimension and the Hausdorff dimension to be the same is given. It is also shown that the dimension of the fractal obtained by combining two fractals is the weighted average of the dimensions of the two fractals.

Quantum Fractals and the Casimir-Dark Energy Duality—The Road to a Clean Quantum Energy Nano Reactor

Based on Witten’s T-duality and mirror symmetry we show, following earlier work, the fundamental complimentarity of the Casimir energy and dark energy. Such a conclusion opens new vistas in cold fusion technology in the wider sense of the word which we tackle via fractal nano technologies leading to some design proposals for a nano Casimir-dark energy reactor.

Two-Dimensional Animal-Like Fractals in Thin Films

we present a few unique animal-like fractal patterns in ionized-clnster-beam deposited fullerene-tetracyanoquinodimethane thin films.The fractal patterns consisting of animal-like aggregates such as"fishes"and"quasi-seahorses"have been characterized by transnission electron microscopy.The results indicate that the sall aggregates ofthe aninmal-like body are composed of many single crystals whose crystalline directions are generally different.The formation of tle fractal patterns can be attributed to the cluster-diffusion-lirnited aggregation.

Fractals in Quantum Information Process

In the recent work of Kiss et al.[Phys.Rev.Lett.107(2011)100501],the evolvement of two-qubit quantum states in a measurement-based purification process is studied.As they pointed out,the purification results manifest sensitivity to the applied initial states.The convergence regions to different stable circles are depicted on a complex plane.Because of the result patterns'likeness to typical fractals,we make further study on the interesting patterns'connection to fractals.Finally,through a numerical method we conclude that the boundaries of different islands of the patterns are fractals,which possess a non-integral fractal dimension.Also,we show that the fractal dimension would vary with the change of the portion of the noise added to the initial states.

Fractals of Protein Backbone Structures

Ths paper，based on the principles of geometric self-similarity of fractal theory and some research results of rotein chemistry，improved the method of comput－ing protein fractal dimensions，and computed fractal dimensions of some protein back bone，secondary and assumed folding structures．The relationship between protein back-bone strucrural fractal dimensions and its spatial structures was investigated．The results indicated that protein backbone fractal dimensions not only have a close relation with protein secondary structure，but also with its folding．In addition，the folding of protein Polypeptide chains in 3-D space may be similar to the other macromolecular chain be haviour described by the self-avoiding walks（SAW）model．

Fractals Generated by Statistical Contraction Operators

In the theory of random fractal, there are two important classes of random sets, one is the class of fractals generated by the paths of stochastic processes and another one is the class of factals generated by statistical contraction operators. Now we will introduce some things about the probability basis and fractal properties of fractals in the last class. The probability basis contains (1) the convergence and measurability of a random recursive setK(ω) as a random element, (2) martingals property. The fractal properties include (3) the character of various similarity, (4) the separability property, (5) the support and zero-one law of distributionP k =P·K ?1, (6) the Hausdorff dimension and Hausdorff exact measure function.

Mathematical and Physical Fractals

A review of the concepts developed about mathematical and physical fractals is presented followed by experimental results of the latter, considered to be a fourth state of matter which pervades the universe from galaxies to submicroscopic systems. A model of multiple fractal aggregation via a computer code is shown to closely simulate physical fractals experiments carried out in simulated and in real low gravity.

Self-Affine Fractals and the Fractal Dimension of Fractured Rock Profiles

The measured profiles of laboratory fractured rocks should be self-affine fractal.The scaling properties of these profiles are described by two parameters-the fractal dimension D and the crossover length tc The D values of eight profiles are calculated by the ruler method and by the standard deviation method respectively.It is shown that if tc is far greater than the sampling step tc two methods yield the same results,although if it is far smaller than r,the D by the standard method will be about 1.20,while D by the ruler method will very close to 1.0,because two fractal dimensions,local and global,exist on two sides of tc In order to obtain the local fractal dimension which may be close to that of the standard deviation method,the ruler method must be modified.We propose a way to estimate the tc and to modify the ruler method.Finally,a profile having given D is generated in terms of the principle of non-integer order differential,through which the above two methods are verified and lead to the same

DYNAMIC TRAVERSAL CRITERION FOR GEOMETRIC FRACTALS

This paper presents a new generating criterion for self-similar geometric fractalsDynamic Traversal Criterion (DTC) and the principle to practice it. According to the principle,symbol shifting technique is put forward which can control the traversal symbols dynamically in recursive procession. The Dynamic Traversal Criterion inherits the mechanism for generating self-similar fractals from traditional way and creates more fractal images from one initiator and generator than Static traversal strategy.

Geodesic metrics on fractals and applications to heat kernel estimates

It is well known that for a Brownian motion, if we change the medium to be inhomogeneous by a measure μ, then the new motion(the time-changed process) will diffuse according to a different metric D(·, ·).In 2009, Kigami initiated a general scheme to construct such metrics through some self-similar weight functions g on the symbolic space. In order to provide concrete models to Kigami’s theoretical construction, in this paper,we give a thorough study of his metric on two classes of fractals of primary importance: the nested fractals and the generalized Sierpinski carpets;we further assume that the weight functions g := ga are generated by“symmetric” weights a. Let M be the domain of a such that Dgadefines a metric, and let S be the boundary of M. One of our main results is that the metrics from ga satisfy the metric chain condition if and only if a ∈ S.To determine M and S, we provide a recursive weight transfer construction on the nested fractals, and a basic symmetric argument on the Sierpinski carpet. As an application, we use the metric chain condition to obtain the lower estimate of the sub-Gaussian heat kernel. This together with the upper estimate obtained by Kigami allows us to have a concrete class of metrics for the time change, and the two-sided sub-Gaussian heat kernel estimate on the fundamental fractals.