Sierpinski-type fractals are differentiably trivial

In nature, objects are not single fractal sets but are a collection of complex multiple fractals that characterise the multi-fractal space, a generalisation of fractal space. While fractal space includes a fractal set, a multi-fractal space includes the union of fractals. A fuzzy fractal space is a fuzzy metric space and is an approach for the construction, analysis, and approximation of sets and images that exhibit fractal characteristics. The finite Cartesian product of fuzzy fractal spaces is called the multi-fuzzy fractal space. We propose in this paper, a theoretical proof to define the multi-fractal dimensions FD of a multi-fuzzy fractal attractor of n objects for the self-similar fractals sets A=∏i=1nAi=(A1,A2,…An) of the contraction mapping W**:∏i=1nH(F(Xi)→∏i=1nH(F(Xi)) with contractivity factor r = max{ri, i = 1, 2,…n} where H(F(Xi) is a fuzzy fractal space for each i = 1, 2,=, n; over a complete metric space (∏i=1nH(F(Xi)),D*) then for all Bi that belong toH(F(Xi)), there exists B* belonging to (∏i=1nH(F(Xi))) such that W**(B*=∏i=1nBi)=∏i=1n(∪j=1n∪k=1k(i,j)ωij*k(Bj)=∏i=1nWi(B*)). By supposing that M(t)=(∑k(rij*k)FD)n×n is the matrix associated with the the contraction mapping ωij*k with contraction factor rij*k, ∀i, j=1, 2,…, n, ∀k=1, 2, …, k(i, j), for all t≥0, and h (t)=det(M (t)-I). Then, we prove that if there exists a FD such that; h(FD)=0, then FD is the multi fractal dimension for the multi fuzzy-fractal sets of IFS; and M(FD) has a fixed point in Rn.In nature, objects are not single fractal sets but are a collection of complex multiple fractals that characterise the multi-fractal space, a generalisation of fractal space. While fractal space includes a fractal set, a multi-fractal space includes the union of fractals. A fuzzy fractal space is a fuzzy metric space and is an approach for the construction, analysis, and approximation of sets and images that exhibit fractal characteristics. The finite Cartesian product of fuzzy fractal spaces is called the multi-fuzzy fractal space. We propose in this paper, a theoretical proof to define the multi-fractal dimensions FD of a multi-fuzzy fractal attractor of n objects for the self-similar fractals sets A=∏i=1nAi=(A1,A2,…An) of the contraction mapping W**:∏i=1nH(F(Xi)→∏i=1nH(F(Xi)) with contractivity factor r = max{ri, i = 1, 2,…n} where H(F(Xi) is a fuzzy fractal space for each i = 1, 2,=, n; over a complete metric space (∏i=1nH(F(Xi)),D*) then for all Bi that belong toH(F(Xi)), there exists B* belonging to...

In modern era, the different variety of antennas are available that can used in the field of communication but some systems require ultra wideband and multiband antennas like commercial and military telecommunication system. Microstrip antennas have limitation of narrow bandwidth. Fractal geometries have marvellous kind of distinct properties from other antennas. For achieving the desired miniaturization and multiband properties, different shapes of fractals are used. Fractal slots can be made in microstrip patch so that its limitations can be evicted. In this paper, we studied about the different fractal slot geometries that enhance its use for different kinds of applications.

Chaos and fractal theory has its particular characteristic in posting essences of things and it's sensitive to complex machine system forepart faults. The fault diagnosis and prediction of complicated mechanical systems are researched by using the fractal and chaos theory. Judging by the kinetic characteristics of mechanical equipment have changed to judge whether the fault exists or not. The frequency spectrum is analyzed to determine the reason and the locality of the fault, further, in high-speed wire rolling mill of fault detection and diagnosis of achieved good effect.

Exponential demand for broadband applications has elicited renewed interest in characterizing noise in power line communication (PLC) networks for optimal error performance analysis. PLC is economical and ubiquitous, however due to multipath phenomena and nonstationary impulsive noise, the PLC channel is very harsh to signal transmission. Recent studies reveal that PLC noise is cyclostationary and modelling its characteristic should exploit the spectral and temporal redundancy. In addition, preliminary research shows that PLC noise exhibits self-similarity and can be predicted by fractal theory. In modelling noise employing fractal theory, the Hausdorff dimension (D) needs to be approximated. D is related to Hurst parameter (H), which is a measure of self-similarity. Hence, it is important to estimate H for accurate noise prediction using fractal theory. H parameter estimators assume stationarity in time series data. Therefore, preprocessing is required for the case of PLC noise before estimating H to make it stationary as periodicity and nonstatioanrity affects the H estimators. In this study, we remove the periodicity hidden in the PLC noise obtained from measurement campaign within several university offices, workshops and laboratories. Then long-range dependence is estimated from this pre-processed noise data by employing three estimation algorithms: R/S (rescaled adjusted range) method, aggregated variance method and absolute values of aggregated series method. Results from the said estimation methods though not the same, show that the Hurst parameters obtained from various noise measurements are greater than 0.5, implying presence of self-similarity with long-term memory. Results from this study will find valuable applications in PLC noise modelling using fractal and chaos theory and in error performance analysis in both narrowband and broadband PLC systems.

It presents a mathematical modeling of a biological system, this is a particular case of two fractal sets, an attractor repeller pair, according to the definition given by the Morse-Smalle theory; these are complementary fractal sets, since the complementary of the fractal repeller set is the attractor set, as we can see in the demonstration. The existence of this model in Nature is found in the flight of starlings. This dynamic system works like a beat of a heart, where the movement of the systole is equivalent to the contraction (attractor set) of the flight trajectory of birds and the diastole movement to its expansion (repeller set).The modeling is presented with differential equations written in Matlab code, and several images of the fractal sets associated with the flight pattern of the starlings are generated, so that we can see this model more clearly. We will begin with a definition and its subsequent mathematical demonstration to study this complex system of complementary fractals.

# Structure of Genetic Regulatory Networks: Evidence for Scale Free Networks (L S Liebovitch et al.) # Modelling Fractal Dynamics (B J West) # Fractional Relaxation of Distributed Order (R Gorenflo & F Mainardi) # Fractional Time: Dishomogeneous Poisson Processes vs. Homogeneous Non-Poisson Processes (P Allegrini et al.) # Markov Memory in Multifractal Natural Processes (N Papasimakis & F Pallikari) # Description of Complex Systems in Terms of Self-Organization Processes of Prime Integer Relations (V Korotkikh & G Korotkikh) # Dynamical Decomposition of Multifractal Time Series as Fractal Evolution and Long-Term Cycles: Applications to Foreign Currency Exchange Market (A Turiel & C Perez-Vicente) # Fractal Sets from Noninvertible Maps (Ch Mira) # A Generative Construction and Visualization of 3D Fractal Measures (T Martyn) # Complexity in Nature and Society: Complexity Management in the Age of Globalization (K Mainzer) # Fractals, Complexity and Chaos in Supply Chain Networks (M Pearson) # The Effects of Different Competition Rules on the Power-Law Exponent of High Income Distribution (K Yamamoto et al.) # Synergetics on Its Way to the Life Sciences (H Haken) # Complexity, Fractals, Nature and Industrial Design: Some Connections (N Sala) # Modelling Pattern Formation Upon Laser-Induced Etching (A Mora et al.) # Fractal Properties of Some Machined Surfaces (T R Thomas & B-G Rosen) # Fractals, Morphological Spectrum and Complexity of Interfacial Patterns in Non-Equilibrium Solidification (P K Galenko & D M Herlach) # Competition of Doublon Structure in the Phase-Field Model (S Tokunaga & H Sakaguchi) # Study of Thermal Field in Composite Materials (P Stefkova et al.) # Simulation of Geochemical Banding in Acidization-Precipitation Experiments In Situ (M Msharrafieh et al.) # Bulk Mediated Surface Diffusion: Non Markovian and Biased Behavior (J A Revelli et al) # Multifractal Formalism for Remote Sensing: A Contribution to the Description and the Understanding of Meteorological Phenomena in Satellite Images (J Grazzini et al.) # The Distance Ratio Fractal Image (X-Z Zhang et al.) # Iterated Function Systems in Mixed Euclidean and p-Adic Spaces (B Sing) # Multifractality of the Multiplicative Autogressive Point Processes (B Kaulakys et al.) # Analysis of Geographical Distribution Patterns in Plants Using Fractals (A Bari et al.) # Hierarchy of Cellular Automata in Relation to Control of Chaos or Anticontrol (M Markus et al.) # Fractal Analysis of the Images Using Wavelet Transformation (P Jerabkova et al.) # Clustering Phenomena in the Time Distribution of Lightning (L Telesca et al.) # A Cornucopia of Connections: Finding Four Familiar Fractals in the Tower of Hanoi (D R Camp) # Monitoring the Depth of Anaesthesia Using Fractal Complexity Method (W Klonowski et al.) # The Complex Couplings and Gompertzian Dynamics (P Waliszewski & J Konarski)

Streffer, Friedrich Theoretical Investigation of Cooperative Effects of H-Bonds in Biomolecular Systems The present work investigates signatures of cooperative effects of H-bonds especially in DNA with statistical and quantum chemical methods. Part I investigates results of information-theoretical analyses of DNA sequences of living organisms, which are tested qualitatively and quantitatively with respect to their biological aspects and/or implications. The results concern ’long-range correlations’ and ’fractals’ in intron-containing DNA sequences, their possible ’linguistic’ structure, and other related aspects. The investigations demonstrate that the findings a ’fractal’ structure in DNA are trivially equivalent to variations of the base pair composition, or patchiness, of different regions in a natural DNA sequence. It is explicitly shown that neither a well-defined ’scaling’ or ’fractal’ exponent, nor a well-defined Zipf exponent of the ’linguistic’ test does exist. The biological origins of such variations are discussed. Quantitative comparisons of natural DNAs with computer-generated, artificial sequences are made. But the present work shows that certain natural DNA sequences (especially those with compact genomes) do have certain stochastic characteristics (say, pseudo-fractal exponents, averaged Zipf slopes, etc.) which are intrinsically different from artificial sequences. To shed more light on this point, investigations concerning the short range correlation of base pair, which may be caused by short-lived quantum entanglement of protons, are presented. The most striking finding is that quantum entanglement appears preferably between the third base of a codon and the first base of the following one. Results on a large number of DNA sequences of various types and from widely different taxa are reported. Additionally, results of current investigations concerning the so-called ’detrended fluctuation analysis’ and the ’Kullback information measure’ of DNA sequences are reported. Part II deals with quantum chemical investigations of the AT, GC and the artificial κχ base pair. Different levels of theory have been applied with full geometry optimization and the ’frozen-core’ approximation; among them are B3LYP/6-31G** and MP2/6-31G**. The calculations in the ’frozen-core’ approximation confirm the double well character of all investigated potential energy surfaces, with a decreased energy of the transition states on the inclusion of the electron correlation effects. Couplings of the different proton transfer reactions are discussed, testing quantitavely the assumptions of a quantum mechanical Jordan-Block structure found in the GC pase pair. The geometry optimization of the relevant stationary points of the double proton transfer reactions in the AT and GC base pair has been performed at the B3LYP/6-31G** level of theory. The most striking finding is found during the normal mode analysis of the vibrations. A vibrational mode involving the relevant protons for the double proton transfer does exist in the GC base pair and all tautomers, whereas the AT base pair does not have such a vibrational mode. Implications are discussed. Finally, two experiments are proposed to test the findings.

The fractal dimension is quantitatively to define the irregular characteristic of the shape in natural. It can be useful in describing morphological characteristics of various wear particles. This paper was undertaken to diagnose failure condition for sliding members in lubrication by fractal dimension. It will be possible to diagnose wear mechanism, friction and damage state of machines through analysis of shape characteristics for wear particle on driving condition by fractal parameters. In this study, the calculating and analyzing methods of fractal dimensions were constructed for the condition monitoring and wear particle analysis in lubricant condition. So, we carried out the Friction and wear test with the ball on disk type tester, and the fractal parameters of wear particle in lubricated conditions were calculated. Fractal parameters were defined as texture fractal dimension (), structure fractal dimension () and total fractal dimension (D).

Euclidian geometry pertained only to the artificial realities of the first, second and third dimensions. Fractal geometry is a new branch of mathematics that proves useful in representing natural phenomena whose dimensions (fractal dimensions) are non-integer values. Fractal geometry was conceived in the 1970s, and mainly developed by Benoit Mandelbrot. In fractal geometry fractals are normally the results of an iterative or recursive construction using corresponding algorithm. Fractal analysis is a nontraditional mathematical and experimental method derived from Mandelbrot’s Fractal Geometry of Nature, Euclidean geometry and calculus. The main aims of the present study are: 1) to address the dimensional imbalances in some texts on fractal geometry, proving that logarithm of a physical quantity (e.g. length of a segment) is senseless; 2) to define the modified capacity dimension, calculate its value for Koch fractal set and show that such definition satisfies basic demands of physics, before all the dimensional balance; and 3) to calculate theoretically the fractal dimension of a circle of unit radius. A quantitative determination of the similarity using the set of Koch fractals is carried out. An important result is the relationship between the modified capacity dimension and fractal dimension obtained using the log-log method. The text includes some important modifications and advances in fractal theory. It is important to notice that these modifications and quantifications do not affect already known facts in fractal geometry and fractal analysis.

Proper understanding of offshore wind speed variability is of essential importance in practice, which provides useful information to a wide range of coastal and marine activities. In this paper, long-term wind speed data recorded at various offshore stations are analyzed in the framework of fractal dimension analysis. Fractal analysis is a well-established data analysis tool, which is particularly suitable to determine the complexity in time series from a quantitative point of view. The fractal dimension is estimated using the conventional box-counting method. The results suggest that the wind speed data are generally fractals, which are likely to exhibit a persistent nature. The mean fractal dimension varies from 1.31 at an offshore weather station to 1.43 at an urban station, which is mainly associated with surface roughness condition. Monthly variability of fractal dimension at offshore stations is well-defined, which often possess larger values during hotter months and lower values during winter. This is partly attributed to the effect of thermal instability. In addition, with an increase in measurement interval, the mean and minimum fractal dimension decrease, whereas the maximum and coefficient of variation increase in parallel.

Fractal antennas have been shown to demonstrate repetitive multi-band or log-periodic behavior that has been attributed to the self-similar scale factor of the antenna's geometry. Additionally, fractal loop antennas have been shown to demonstrate improved impedance and SWR performance in a reduced physical area when compared to non-fractal Euclidean geometries. The purpose of this paper is to numerically model fractal and non-fractal loop antennas occupying the same physical area in an effort to understand whether the fractal geometry of the loop is the significant parameter in determining the loop performance. It is demonstrated through numerical modeling that the fractal geometry is not the critical factor in determining the loop performance and in fact, non-fractal geometries offer similar and in some cases, improved performance over their fractal counterparts.

The heavy-duty machine tool is usually assumed in the concrete foundation, in which the machine tool-foundation joints have a critical effect on the working accuracy and life of heavy-duty machine tool. This paper proposed a novel contact stiffness model of concrete–steel joint based on the fractal theory. The topography of contact surface exist in concrete–steel joint has a fractal feature and can be described by fractal parameters. Asperities are considered as elastic, plastic deformation in micro-scale. However, the asperities of concrete surface will be crushed when the stress is larger than their yield limit. Then, the force balance of contact surfaces will be broken. Here, an iteration model is proposed to describe the contact state of concrete–steel joint. Because the contact asperities cover a very small proportion (less than 1%), the load on crushed asperities is assumed carried by other no contact asperities. This process will be repeated again and again until the crushed asperities are not being produced under external load. After that, the real contact area, contact stiffness of the concrete–steel joint can be calculated by integrating the asperities of contact surfaces. Nonlinear relationships between contact stiffness and load, fractal roughness parameter G, fractal dimension D can be revealed based on the presented model. An experimental setup with concrete–steel test-specimens is designed to validate the proposed model. Results indicate that the theoretical vibration mode shapes agree well with the experimental variation mode shapes. The errors between theoretical and experimental natural frequencies are less than 4.18%. The presented model can be used to predict the contact stiffness of concrete–steel joint, which will provide a theoretical basis for optimizing the connection characteristic of machine tool-concrete foundation.

A novel fault diagnosis method for turbo-generator set based on fractal exponent theory and wavelet network is presented. When faults occur, they usually produce nonstationary vibration signals. The wavelet transform is used to localizes the characteristics of vibration signal in the time frequency domains and in a view of the inter relationship of wavelet transform between fractal theory, the whole and local fractal exponents obtained from wavelet transform coefficients as features are presented for extracting fault signals, which are inputted into radial basis function for fault pattern recognition. The improved Levenberg-Marquardt (LM) optimization technique is used to complete the network structure parameters. By means of choosing enough samples to train the fault diagnosis network and the information representing the faults is input into the trained wavelet network, and according to the output result the type of fault can be determined. The practical diagnosis for stator temperature fluctuation and rotor vibration demonstrates that the wavelet fractal network can provide an effective way to diagnosis faults for turbo-generator set in power system.

Aramid nanofiber aerogels (ANFAs) have been considered as ideal thermal insulation materials due to their low bulk density, hierarchical porous structure, high thermal insulation, and thermal stability. Thermal conductivity of ANFAs is highly determined by their complicated porous structure. Fractal geometry theory can effectively simulate pore size distribution, but it is rarely used in the study of the thermal conductivity of aerogels. In this work, the Sierpinski carpet model and two modified Sierpinski carpet models were established to study the pore size distribution of ANFAs, and then the equivalent resistance method was used to transform the models into series and parallel fractal resistance networks, which were applied for the calculation of the effective thermal conductivity of ANFAs. Results show that the calculated results are closer to the experimental data, with a minimum error of 0.78%, indicating the thermal conductivity prediction model is effective and reliable.

There is an increasing demand for the simulation of large-scale "free-flow" urban environments in games or other virtual reality worlds. Currently most 3D cityscapes are generated by time consuming manually pre-scripted "room-to-room" approaches. Although the effectiveness of fractal modelling of rural landscape or architecture has been studied extensively, there is a relative dearth of research on fractal modelling of cityscapes. Inspired by modern fractal city theory, we intend to develop an automatic fractal-based method to address this problem. In particular, we will investigate how fractal theory can be used to mathematically simulate the multi-level hierarchical urban structure corresponding to various land use property probabilities of city regions. Tentative findings and results are presented in this paper