Fractals

The main goal of this paper is to provide a generalized definition of fractal dimension for any space equipped with a fractal structure. This novel theory generalizes the classical box-counting dimension theory on the more general context of GF-spaces.
In this way, if we select the so-called natural fractal structure on any Euclidean space, then the box-counting dimension becomes just a particular case. This idea allows to consider a wide range of fractal structures to calculate the effective fractal dimension for any subset of this space. Unlike it happens with the classical theory of fractal dimension, the new definitions we provide may be calculated in contexts where the box-counting one can have no sense or cannot be calculated. Nevertheless, the new models can be computed for any space admitting a fractal structure, just
as easy as the box-counting dimension in empirical applications.

Receiving of remote sensed data's signals in urban space information reception centers is usually difficult, because of complex electromagnetic situation in cities and insufficient EMC. Traditional methods for digital reconstruction of images use smoothing and autoregressive forecasting. In this case anomalous spikes on the image are lost. Alternative method based on Kolmogorov-Wiener's filters and fractal properties of satellite images is

Our previous works suggest that fractal texture feature is useful to detect pediatric brain tumor in multimodal MRI. In this study, we systematically investigate efficacy of using several different image features such as intensity, fractal texture, and level-set shape in segmentation of posterior-fossa (PF) tumor for pediatric patients. We explore effectiveness of using four different feature selection and three different segmentation techniques, respectively, to discriminate tumor regions from normal tissue in multimodal brain MRI. We further study the selective fusion of these features for improved PF tumor segmentation. Our result suggests that Kullback-Leibler divergence measure for feature ranking and selection and the expectation maximization algorithm for feature fusion and tumor segmentation offer the best results for the patient data in this study. We show that for T1 and fluid attenuation inversion recovery (FLAIR) MRI modalities, the best PF tumor segmentation is obtained using the texture feature such as multifractional Brownian motion (mBm) while that for T2 MRI is obtained by fusing level-set shape with intensity features. In multimodality fused MRI (T1, T2, and FLAIR), mBm feature offers the best PF tumor segmentation performance. We use different similarity metrics to evaluate quality and robustness of these selected features for PF tumor segmentation in MRI for ten pediatric patients.

Detrended fluctuation analysis (DFA) is used to investigate correlations between the monthly average of the maximum daily temperatures for different locations in the continental United States and the different climates these locations have. When we plot the scaling exponents obtained from the DFA versus the standard deviation of the temperature fluctuations, we observe crowding of data points belonging to the same climates. Thus, we conclude that by observing the long-time trends in the fluctuations of temperature it would be possible to distinguish between different climates.

BACKGROUND Fractal analysis has been shown to be capable of characterizing irregular time series generated in non-linear systems. Fluctuations in skin flow signals have a fractal nature, but to date there has been no analysis of subjects with hypertension. The aim of this study is to assess the fractal dimensions of skin microcirculation flows in subjects with a familial predisposition or newly diagnosed hypertension. METHODS A four-minutes rest flow (RF), minimal flow (BZ) during three-minutes ischemia and eight-minutes heat flow (HF) were recorded (using laser Doppler flowmetry) in patients with untreated hypertension, and in normotensives with no [NT(-)] or with a familial predisposition to hypertension [NT(+)]. Average one-minute surface areas under the curve of flow records and box dimensions (D) were calculated. Anova Kruskall-Wallis, c2 statistic and multivariate reverse regression analysis were used for calculation. RESULTS We studied 70 people (average age 36.1 ± 10.3 years...

Fractal measurements using the Cantor's dust method in a linear one-dimensional analysis mode were made on the fracture patterns revealed on two-dimensional, planar surfaces in four granites. This method allows one to conclude that: (1) The fracture systems seen on two-dimensional surfaces in granites are consistent with the part of fractal theory that predicts a repetition of patterns on different scales of observation, self similarity. Fractal analysis gives essentially the same values of D on the scale of kilometres, metres and centimetres (five orders of magnitude) using mapped, surface fracture patterns in a Sierra Nevada granite batholith (Mt. Abbot quadrangle, Calif.). (2) Fractures show the same fractal values at different depths in a given batholith. Mapped fractures (main stage ore veins) at three mining levels (over a 700 m depth interval) of the Boulder batholith, Butte, Mont. show the same fractal values although the fracture disposition appears to be different at different levels. (3) Different sets of fracture planes in a granite batholith, Central France, and in experimental deformation can have different fractal values. In these examples shear and tension modes have the same fractal values while compressional fractures follow a different fractal mode of failure. The composite fracture patterns are also fractal but with a different, median, fractal value compared to the individual values for the fracture plane sets. These observations indicate that the fractal method can possibly be used to distinguish fractures of different origins in a complex system. It is concluded that granites fracture in a fractal manner which can be followed at many scales. It appears that fracture planes of different origins can be characterized using linear fractal analysis.

In pathological brain, a variety of morphological forms exist that reflect differences in functional requirements. To better understand microglia function in neurological disease, it is important to identify and quantify microglia morphology associated with specific neuropathologies. Traditional feature parameters such as area or cell diameter are not sufficient. In this study microglia were quantified by the box-counting fractal dimension (DB). One hundred and four cells from post-mortem tissue were analyzed comprising cells of controls, Alzheimer's disease, schizophrenia and affective disorder. The DB was significantly different from the control (1.36) compared to schizophrenia (1.41), Alzheimer's disease (1.41) and affective disorder (1.43) with p < 0.01. Thus fractal analysis provides a useful quantitative and objective measure of microglial form associated with normal function and diverse neuropathology. The distribution of fractal dimensions associated with microgli...

Reconstructions of foraging behavior and diet are central to our understanding of fossil hominin ecology and evolution. Current hypotheses for the evolution of the genus Homo invoke a change in foraging behavior to include higher quality foods. Recent microwear texture analyses of fossil hominin teeth have suggested that the evolution of Homo erectus may have been marked by a transition to a more variable diet. In this study, we used microwear texture analysis to examine the occlusal surface of 2 molars from Dmanisi, a 1.8 million year old fossil hominin site in the Republic of Georgia. The Dmanisi molars were characterized by a moderate degree of surface complexity (Asfc), low textural fill volume (Tfv), and a relatively low scale of maximum complexity (Smc), similar to specimens of early African H. erectus. While caution must be used in drawing conclusions from this small sample (n = 2), these results are consistent with continuity in diet as H. erectus expanded into Eurasia.

Given the inverse of the Golden Mean $ \tau^{ -1} = \phi  = { 1\over 2} (\sqrt 5  - 1)$, 
it is known that the continuous fraction expansion of $ \phi^{ -1} = 1 + \phi = \tau$ is  $ ( 1, 1, 1, \cdots )$.  Integer solutions
for the pairs of numbers $ ( d_i, n_i ), i = 1, 2, 3, \cdots $ are found obeying the equation
$ ( 1 + \phi)^n = d + \phi^n$.  The latter equation was inspired from El Naschie's formulation
of fractal Cantorian space time $ {\cal E}_\infty$,  and such that it furnishes the continuous fraction expansion of $  ( 1 + \phi )^n ~= ~  (d, d, d, d, \cdots  )$, generalizing the original expression for the Golden mean.    Hardy showed that is possible to demonstrate nonlocality without using Bell inequalities for two particles prepared in $nonmaximally$ entangled states.  The maximal probability of obtaining his nonlocality proof was found to be precisely $\phi^5$.  Zheng showed that three-particle nonmaximally entangled states revealed quantum nonlocality without using inequalities, and the maximal probability of obtaining the nonlocality proof was found to be $ 0.25 \sim \phi^3 = 0.236$.  Given that the two-parameter $ p, q$ quantum-calculus deformations of the integers $ [ n ]_{ p, q}  = F_n $  $coincide$ precisely with the Fibonacci numbers, as a result of Binet's formula when
$ p = ( 1 + \phi) = \tau,  q  =  - \phi =  - \tau^{ -1} $,  we explore further the implications of these results in the quantum entanglement of two-particle spin-$s$ states.

A presentation of the Czech translation of a seminal book on architectural theory by Nikos A. Salingaros and others, with a conversation between the author and the Czech editor, and two chapters (in Czech). Published in Kontexty [Contexts, Brno] IX, 2017, No. 1, pp. 49–66.

We present a fractal coding method to recognize online handwritten Tamil characters and propose a novel technique to increase the efficiency in terms of time while coding and decoding. This technique exploits the redundancy in data, thereby achieving better compression and usage of lesser memory. It also reduces the encoding time and causes little distortion during reconstruction. Experiments have been conducted to use these fractal codes to classify the online handwritten Tamil characters from the IWFHR 2006 competition dataset. In one approach, we use coding and decoding process. A recognition accuracy of 90% has been achieved by using DTW for distortion evaluation during classification and encoding processes as compared to 78% using nearest neighbor classifier. In other experiments, we use the fractal code, fractal dimensions and features derived from fractal codes as features in separate classifiers. Whereas the fractal code was successful as a feature, the other two features ar...

In this paper, fractal stochastic Langevin equations are suggested, providing a mathematical model for random walks on the middle-τ Cantor set. The fractal mean square displacement of different random walks on the middle-τ Cantor set are presented. Fractal under-damped and over-damped Langevin equations, fractal scaled Brownian motion, and ultra-slow fractal scaled Brownian motion are suggested and the corresponding fractal mean square displacements are obtained. The results are plotted to show the details.

Ion exchange, adsorption, and chromatography are examples of separation processes frequently used in today’s biorefineries. The particular tasks for which these technologies can be successfully applied are highly influenced by capital cost and efficiency. There exists a potential for significantly increasing the efficiency of these processes whereas simultaneously decreasing their size and capital cost. This potential for process intensification can be realized with the use of engineered fractal equipment. The cost savings potential and the possibilities for broadening the use of fractal-based separation technologies in future biorefinery concepts is illustrated by examples of full-scale implementation in the sugar and sweetener industries.

Multivalued fractals are considered as fixed-points of certain induced union operators, called the Hutchinson–Barns-ley operators, in hyperspaces of compact subsets of the original spaces endowed with the Hausdorff metric. Various approaches are presented for obtaining the existence results, jointly with the information concerning the topological structure of the set of multivalued fractals. According to the applied fixed-point principles, we distinguish among metric , topological and TarskiÕs multivalued fractals. Finite families of condensing and (locally) compact maps as well as of different sorts of contractions are examined with this respect. In particular, continuation principle for multivalued frac-tals is established for (locally) compact maps. Multivalued fractals are also generated implicitly by means of differential inclusions. A randomization of the deterministic results is indicated. Numerical aspects of computer generated multi-valued fractals are discussed in detail.

This is an excerpt from the Pictorial Compendium of The Secret Doctrine
Volume IV, The Nature of Everything.

Follow me down the rabbit hole into the magical world of the Nature of Everything. It is like nothing you have ever imagined! It is filled with wonder and awe! The Universe is all of time and Space and its contents. It includes planets, moons, minor planets, stars, galaxies, the contents of intergalactic space, and all matter and energy.

В статье рассмотрены основные положения теорий динамики рынков, в частности, гипотезы эффективного рынка, гипотезы когерентного рынка, гипотезы фрактального рынка и гипотезы синергетического рынка. Проанализированы инструменты фрактального рынка для принятия на их основе обоснованных инвестиционных решений. Приведены и обоснованы инструменты технического анализа, основанного на игре за трендом для прогнозирования динамики рынка, согласно гипотезе фрактального рынка, в частности, волновую модель Эллиотта и применения нейро-нечетких моделей.
Ключевые слова: эффективный рынок, когерентный рынок, фрактальный рынок, синергетический рынок, технический анализ

Various methods for estimating the self-similarity parameter and/or the intensityof long-range dependence in a time series are available. Some are morereliable than others. To discover the ones that work best, we apply the differentmethods to simulated sequences of fractional Gaussian noise and fractionalARIMA(0; d; 0). We also provide here a theoretical justification for the methodof residuals of regression.1 IntroductionTime series

De la science normale à la science marginale. Analyse d'une bifurcation de trajectoire scientifique: le cas de la Théorie de la Relativité d'Echelle Résumé. Dans le champ scientifique, les agents peuvent choisir de collaborer à la science 'normale', se placer à l'avant-garde la plus légitime (les 'supercordes', la 'matière noire', etc.), ou encore développer leurs recherches dans un nouveau cadre théorique, avec tous les risques que cela comporte. La marginalité d'une théorie soulève la question de la stratégie de ceux qui y collaborent même au détriment de leur 'intérêt' à court terme, qui oriente plutôt vers la compétition immédiate pour occuper les positions centrales dans les domaines déjà constitués. La théorie de la relativité d'échelle (TRE) présente l'intérêt d'une telle situation car elle ouvre une possibilité qu'il faut créer de toutes pièces. S'y investir engage davantage que le choix d'un projet 'risqué ' (par sa difficulté même) dans le cadre d'un paradigme existant car, d'une part, la TRE innove par rapport aux bases conceptuelles déjà acceptées par tous et, d'autre part, se trouve aussi marginalisée par rapport à l'avant-garde la plus légitime (comme celle des 'supercordes'). Ainsi, le cas de la TRE permet d'étudier une région du champ scientifique peu explorée par une sociologie des sciences qui fixe surtout son regard sur les cas extrêmes: histoire de théories devenues reconnues ou controverses spectaculaires.La TRE occupe encore, en 2006, une position marginale dans le champ de la physique. Son statut diffère toutefois radicalement des 'théories' produites à l'extérieur du champ, sans correspondre pour autant à celui de la science stabilisée et sanctionnée: comme nous allons le montrer par une analyse bibliométrique détaillée, sa diffusion au sein du champ scientifique est relativement modeste mais réelle, et ses résultats, quand ils reçoivent la sanction d'une publication scientifique, sont rarement pris en compte par les chercheurs qui n'y collaborent pas déjà. Cette

Introduction The porous composite material of the fractal structure (bio polimineral composite material) formed from organic and inorganic substances and provides a spatial arrangement of the material and the various aspects of the practical use, and relates to the field of ecology home and industrial environment, preferred field of use of this group of inventions will be a construction, primarily finishing etc. works made in order to protect the living environment of people and quality of life. The use of meta-analysis and conducted a systematic review of publications in various disciplines and areas of basic and applied sciences allowed to form and reinforce the understanding and purpose of experimental and research work. Performed experimental and research work based on the principles of bionics / biomimetics, that is, the scientific direction which regards the study of the structure and functions of biological systems and objects as models for the creation of engineering solutions. It is known that the microorganisms and living cells have their own energy and constantly use a large amount of protein and minerals to build (create) self-organizing way cellular materials micro and nano-meter size. Thus, the examples from the world of nature are the basis for the creation of engineering solutions. Based on what is currently known composite materials with the structures of micro and nano levels produced, both of synthetic and of biological materials (proteins, collagen, etc.), Natural minerals, produce membranes like membranes for reverse osmosis (i.e., delay and neutralize organic compounds) and that the proteins have a natural ability to form a regular structure in the form of crystalline lattices, and the ability of bacteria to absorb (leaching) metals, it becomes possible to offer bio polimineral composite material, which is a hierarchical structure from the pores. Proposed bio polimineral material is a composite material made of proteins, microorganisms, and collection of the desired, compounds based on inorganic components (i.e., inorganic and / or organometallic compounds). Construction bio polimineral structure of the composite material is carried out by self-assembly with the participation of microorganisms in a managed and controlled impact on the process. Construction of composite structures by self-assembly is carried out with the participation of microorganisms in a managed and controlled impact on the process. The possibility of practical use of the composite material is based on the phenomenon of biomineralization, which will be a preparatory step to the main biochemical process technology, which forms a film coating structural pores (cells carrying the appointment of photo electrochemical cells) and for bonding them in the formation of the porous structure. The term biomineralization (induced biomineralization), that is producing minerals under the influence of microorganisms introduced in a scientific revolution in palaeontology. As an example of induced biomineralization may be brought mineralogical thiol bacteria side of life (oxidizing) and sulphate-reducing bacteria. It is known that the microorganisms and living cells have their own energy and constantly use a large amount of protein and minerals to build (create) self-organizing way cellular materials micro and nano-meter size. Thus, the examples from the world of nature are the basis for the creation of engineering solutions.

The purpose of the present work is to study the fractal behaviour of prime Indian stock exchanges, namely Bombay Stock Exchange Sensitivity Index (BSE Sensex) and National Stock Exchange (NSE). To analyze the monofractality of these indices we have used Higuchi method and Katz method separately. By applying Mutifractal Detrended Fluctuation Analysis (MFDFA) technique we have calculated the generalized Hurst exponents, multifractal scaling exponents and generalized multifractal dimensions for the present indices. We have deduced Hölder exponents as well as singularity spectra for BSE and NSE. It has been observed that both the stock exchanges are possessing self-similarity at different small ranges separately and inhomogeneously. By comparing the multifractal behaviour of the BSE and NSE indices, we have found that the second one exhibits a richer multifractal feature than the first one.