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Search results for: fractals
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Crystallization of space: Space-time fractals from fractal arithmetic
PublicationFractals such as the Cantor set can be equipped with intrinsic arithmetic operations (addition, subtraction, multiplication, division) that map the fractal into itself. The arithmetics allows one to define calculus and algebra intrinsic to the fractal in question, and one can formulate classical and quantum physics within the fractal set. In particular, fractals in space-time can be generated by means of homogeneous spaces associated...
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CHAOS SOLITONS & FRACTALS
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Simple Fractal Calculus from Fractal Arithmetic
PublicationNon-Newtonian calculus that starts with elementary non-Diophantine arithmetic operations of a Burgin type is applicable to all fractals whose cardinality is continuum. The resulting definitions of derivatives and integrals are simpler from what one finds in the more traditional literature of the subject, and they often work in the cases where the standard methods fail. As an illustration, we perform a Fourier transform of a real-valued...
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Waves Along Fractal Coastlines: From Fractal Arithmetic to Wave Equations
PublicationBeginning with addition and multiplication intrinsic to a Koch-type curve, we formulate and solve wave equation describing wave propagation along a fractal coastline. As opposed to examples known from the literature, we do not replace the fractal by the continuum in which it is embedded. This seems to be the first example of a truly intrinsic description of wave propagation along a fractal curve. The theory is relativistically...
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Chaos, Solitons and Fractals: X
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Journal of Fractal Geometry
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University of the future: a fractal oganization of knowledge
PublicationW opracowaniu przedstawiono wizję uniwersytetu przyszłości jako organizacji fraktalnej. Pokazano cechy wspólne organizacji fraktalnej i uniwersytetu ewoluującego w kierunku organizacji podporzadkowanej wiedzy.
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Fractal local fields in random composites
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Fractal analysis and modelling of gas bubble formation.
PublicationPrzedstawiono wyniki badań eksperymentalnych generowania pęcherzyków powietrznych z dysz o różnej średnicy, umieszczonych pod powierzchnią wody. Sygnał rejestrowany przez układ laser-fotodioda poddano analizie fraktalnej.
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Statistical and fractal analysis of human heart dynamics.
PublicationPrzedstawiono sposób obliczania podstawowych i bardziej zaawansowanych (dyskrepancja) wielkości statystycznych do analizy dynamiki ludzkiego serca. Oprócz analizy statystycznej pokazano elementy analizy fraktalnej (wykresy Poincare). Przedstawiona analiza umożliwia wykrywanie patologicznych sytuacji i jest pomocna lekarzom w diagnostyce.
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Gas bubble dynamics - experiment and fractal analysis.
PublicationPrzedstawiono nieliniowe aspekty generowania pęcherzyków gazowych z dysz szklanych umieszczonych pod powierzchnią cieczy. W badaniach eksperymentalnych zastosowano układ laser diodowy- fotodetektor oraz hydrofon. Sygnały z fotodetektora i hydrofonu były wzmacniane, rejestrowane za pomocą karty dźwiękowej, a następnie poddane analzie fraktalnej. W wyniku obliczeń ustalono mapy występowania chaosu w funkcji strumienia objętości gazu.
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Generating fractal tiles using Voronoi diagrams
PublicationPraca opisuje szczególną klasę podziałów powierzchni n-wymiarowego torusa na komórki o fraktalnym brzegu. Zbiór komórek przejawia nietypowe własności samopodobieństwa, może zostać użyty do wypełnienia przestrzeni R^n w sposób periodyczny lub aperiodyczny ze zmienną gęstością podziałów. Zaproponowany został algorytm do generowania takich podziałów używając diagramów Woronoja. Opisana metoda może mieć zastosowania w grafice komputerowej.
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Fractal and Fractional
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Compact CPW-fed Sierpinski fractal monopole antenna.
PublicationZaproponowano nową wielozakresową strukturę antenową dla zastosowań wielosystemowych. Jest to zminiaturyzowana antena monopolowa planarna, której obwód zaprojektowano wykorzystując czwartego rzędu iterację trójkąta Sierpińskiego ze współczynnikiem skali d=1.5. Antenę wykonano na podłożu dielektrycznym i zasilono wykorzystując falowód koplanarny. Badania parametrów wykonanego modelu potwierdziły wielozakresową pracę struktury.
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Fractal dimension for bending–torsion fatigue fracture characterisation
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Compact antenna array comprising fractal-shaped microstripradiators
PublicationA design method of antenna array consisting of eight microstrip patches modified with Sierpinski fractal curves has been presented andexperimentally validated in this paper. Method proposed has enabled the achievement of considerable miniaturization of array length (26%),together with multi-band behavior of the antenna, which proves the attractiveness of presented design methodology and its ability to be implemented in more complex...
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Fractal dimension for bending–torsion fatigue fracture characterisation
PublicationFracture surfaces after biaxial fatigue tests were compared using fractal dimension for three types of metallic materials in smooth and notched specimens made of S355J2 and 10HNAP steels and 2017-T4 aluminium alloy, considering both proportional and nonproportional cyclic loading. High-resolution optical 3D measurement studies were performed on the entire fracture surface. A direct correlation between fractal dimension and fatigue...
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Slow-wave fractal-shaped compact microstrip resonant cell
PublicationA novel fractal-shaped compact microstrip resonant cell (CMRC) featuring a strong slow-wave effect has been presented. Its vital usefulness in the process of microstrip line miniaturisation has been proved and experimentally validated on the basis of a compact 3-dB branch-line coupler illustrating the possibilities of the approach. A prototype example structure has been designed to mirror the characteristics of a conventional device,...
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Fractal analysis of the bending-torsion fatigue fracture of aluminium alloy
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Sensivity of echo envelope fractal dimension to bottom type and bottom depth.
PublicationPrzedstawiono wynik badania zależności pomiędzy wymiarem fraktalnym obwiedni echa akustycznego od dna a rodzajem dna na wpływ głębokości dna na tę zależność. Wykorzystano sygnały ech otrzymane w wyniku numerycznych symulacji rozpraszania na granicy woda dno dla różnych głębokości dna.
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Novel microstrip low-pass filters with fractal defected ground structures
PublicationIn this study, three microstrip low-pass filters (LPFs) containing fractal-shaped defected ground structures have been designed and manufactured. To improve the performance of microstrip LPF, cascades of inductively coupled fractal resonators have been applied. Simultaneously, to achieve relatively constant microstrip impedance resulting in essentially low reflection losses, a modification of the strip geometry has been introduced....
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Diffusion equations with spatially dependent coefficients and fractal Cauer-type networks
PublicationIn this article, we formulate and solve the representation problem for diffusion equations: giving a discretization of the Laplace transform of a diffusion equation under a space discretization over a space scale determined by an increment h > 0, can we construct a continuous in h family of Cauer ladder networks whose constitutive equations match for all h > 0 the discretization. It is proved that for a finite differences discretization...
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a novel modified star-triangular fractal (MSTF) monopole antenna for super-wideband applications
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a novel modified star-triangular fractal (MSTF) monopole antenna for super-wideband applications
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Lamb wave based structural damage detection using cointegration and fractal signal processing
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Higuchi fractal dimension analysis of surface EMG signals and determination of active electrode positions
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FRACTALS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY
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Improved selectivity compact band-stop filter with gosper fractal-shaped defected ground structures
PublicationA novel band-stop filter using Gosper fractal-shaped defected ground structures has been designed and manufactured. The improvement in the filter selectivity has been achieved by introducing a multiresonance fractal-shaped defect leading to a higher filter order, simultaneously maintaining its compact size. The experimental results prove the validity of proposed solution and its utility in novel miniaturized and severe requirement...
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Self-assembly of vertically oriented graphene nanostructures: multivariate characterisation by Minkowski functionals and fractal geometry
PublicationThe enormous self-assembly potential that graphene and its derived layered materials offer for responding to the contemporary environmental challenges has made it one of the most investigated materials. Hence, tuning its extraordinary properties and understanding the effect at all scales is crucial to tailoring highly customised electrodes. Vertically oriented graphene nanostructures, also known as carbon nanowalls (CNWs), due...
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Correlation between Fractal Dimension and Areal Surface Parameters for Fracture Analysis after Bending-Torsion Fatigue
PublicationThis paper investigates the fracture surface topography of two steel and aluminum alloys subject to bending-torsion fatigue loadings, as well as their susceptibility to fatigue performance and failure mechanisms. Using fracture surface topography data analysis, elements with different geometries were elaborated. A correlation between the fractal dimension, other selected parameters of surface topography such as areal Sx, and...
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The impact of surface slope and calculation resolution on the fractal dimension for fractures of steels after bending-torsion fatigue
PublicationThe article presents the results of the fractal dimension measurements on the fatigue fracture surfaces of 10HNAP and S355J2 steels specimens after combined bending-torsion fatigue. For smooth and ring-notched specimens, three loading conditions were analyzed: (1) bending; (2) bending-torsion; and (3) torsion fatigue. Post-failure surface topography measurements were carried out on the entire fracture surfaces using an optical...
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Fractographic-fractal dimension correlation with crack initiation and fatigue life for notched aluminium alloys under bending load
PublicationIn this study, fatigue fracture surfaces of aluminium alloy 2017-T4 notched specimens were investigated under cyclic bending to find an alternative failure loading index.. The surface topographies were measured on the entire fracture area with an optical profilometer for different loading conditions. Fatigue crack initiation life Ni and total fatigue life Nf were examined using standard surface topography parameters (such as, root...
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Application of Wavelet Transform and Fractal Analysis for Esophageal pH-Metry to Determine a New Method to Diagnose Gastroesophageal Reflux Disease
PublicationIn this paper, a new method for analysing gastroesophageal reflux disease (GERD) is shown. This novel method uses wavelet transform (WT) and wavelet-based fractal analysis (WBFA) on esophageal pH-metry measurements. The esophageal pH-metry is an important diagnostic tool supporting the physician’s work in diagnosing some forms of reflux diseases. Interpreting the results of 24-h pH-metry monitoring is time-consuming, and the conclusions...
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Proteome changes in renal cortex and medulla induced by dietary supplementation with inulin‐type fructans in growing pigs
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The correlation of fractal dimension to fracture surface slope for fatigue crack initiation analysis under bending-torsion loading in high-strength steels
PublicationIn this study, the fractal dimension of fatigue fracture surfaces is investigated in order to find an alternative failure loading indicator. Some of many metrological factors are generalized by reducing the fracture surface structure to one factor and develop an entire fracture surface procedure by analyzing the impact of surface slope and calculation resolution. Three notched geometries are studied under cyclic bending-torsion:...
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Inulin-type fructans trigger changes in iron concentration and activity of bone metabolism biomarkers in blood plasma of growing pigs
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Fourier transforms on Cantor sets: A study in non-Diophantine arithmetic and calculus
PublicationFractals equipped with intrinsic arithmetic lead to a natural definition of differentiation, integration, and complex structure. Applying the formalism to the problem of a Fourier transform on fractals we show that the resulting transform has all the required basic properties. As an example we discuss a sawtooth signal on the ternary middle-third Cantor set. The formalism works also for fractals that are not self-similar.
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Marek Czachor prof. dr hab.
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Integrated model for the fast assessment of flood volume: Modelling – management, uncertainty and sensitivity analysis
PublicationThe specific flood volume is an important criterion for assessing the performance of sewage networks. It has been shown that its value is greatly influenced by the layout of the sewers in the catchment area, which is usually expressed by a fractal dimension. Currently, only mechanistic models (such as SWMM) enable the determination of the impact of the layout of the sewers on flooding volume, but they require additional and robust...
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Bifractal receiver operating characteristic curves: a formula for generating receiver operating characteristic curves in credit-scoring contexts
PublicationThis paper formulates a mathematical model for generating receiver operating characteristic (ROC) curves without underlying data. Credit scoring practitioners know that the Gini coefficient usually drops if it is only calculated on cases above the cutoff. This fact is not a mathematical necessity, however, as it is theoretically possible to get an ROC curve that keeps the same Gini coefficient no matter how big a share of lowest...
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Analytical and numerical solution of a coupled KdV - MKdV system.
PublicationTransformację Darboux zastosowano do całkowania układów równań KdV - MKdV.Reprezentacja Laxa używa 2x2 macierzowe zagadnienie spektralne drugiego rzędu. Numeryczną metodę wprowadzono razem z dowodem zbieżności.
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Theoretical and computational analysis of nonlinear fractional integro-differential equations via collocation method
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Nilpotent singularities and chaos: Tritrophic food chains
PublicationLocal bifurcation theory is used to prove the existence of chaotic dynamics in two well-known models of tritrophic food chains. To the best of our knowledge, the simplest technique to guarantee the emergence of strange attractors in a given family of vector fields consists of finding a 3-dimensional nilpotent singularity of codimension 3 and verifying some generic algebraic conditions. We provide the essential background regarding...
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Advancement of Non-Newtonian Fluid with Hybrid Nanoparticles in a Convective Channel and Prabhakar’s Fractional Derivative—Analytical Solution
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Sensitive Demonstration of the Twin-Core Couplers including Kerr Law Non-Linearity via Beta Derivative Evolution
PublicationTo obtain new solitary wave solutions for non-linear directional couplers using optical meta-materials, a new extended direct algebraic technique (EDAT) is used. This model investigates solitary wave propagation inside a fiber. As a result, twin couplers are the subject of this study. Kerr law is the sort of non-linearity addressed there. Because it offers solutions to problems with large tails or infinite fluctuations, the resulting...
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Signal Propagation in Electromagnetic Media Modelled by the Two-Sided Fractional Derivative
PublicationIn this paper, wave propagation is considered in a medium described by a fractional-order model, which is formulated with the use of the two-sided fractional derivative of Ortigueira and Machado. Although the relation of the derivative to causality is clearly specified in its definition, there is no obvious relation between causality of the derivative and causality of the transfer function induced by this derivative. Hence, causality...
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Experimental verification and comparison of mode shape-based damage detection methods
PublicationThis paper presents experimental verification and comparison of damage detection methods based on changes in mode shapes such as: mode shape curvature (MSC), modal assurance criterion (MAC), strain energy (SE), modified Laplacian operator (MLO), generalized fractal dimension (GFD) and Wavelet Transform (WT).
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Parameter values for topological chaos in the reduced Chialvo model
Open Research DataThe following dataset is connected with a map-based neuron model introduced by D. Chialvo (Chaos, Solitons & Fractals, 5 (3-4) 1995). The reduced version of this model is a one dimensional discrete system which describes the evolution of the membrane voltage when the value of the second variable, the recovery variable, is fixed. We have recently...
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FRAKTALE 22/23
e-Learning CoursesFraktale: wykład (30 godz.), ćwiczenia (15 godz.), projekt (15 godz.)Semestr zimowy dla studentów I roku II stopnia na kierunku Matematyka, specjalności GiGK.
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Fracture Surface Behavior of 34CrNiMo6 High-Strength Steel Bars with Blind Holes under Bending-Torsion Fatigue
PublicationThe present study evaluates the fracture surface response of fatigued 34CrNiMo6 steel bars with transverse blind holes subjected to bending with torsion loading. The analysis of the geometric product specification was performed by means of height parameters Sx, functional volume parameters Vx, and fractal dimension Df. Surface topography measurements were carried out using an optical profilometer with focus variation technology....