Koch Fractal Research Articles

Distribution of unit mass on one fractal self-similar web-type curve

In the article, we study structural, spectral, topological, metric and fractal properties of distribution of complex-valued random variable$\tau=\sum\nolimits_{n=1}^{\infty}\frac{2\varepsilon_{\tau}}{3^n}\equiv\Delta^g_{\tau_1...\tau_n...}$, where $(\tau_n)$ is a~sequence of independent random variables taking the values $0,1,\cdots,6$ with the probabilities $p_{0n}$, $p_{1n},\cdots,p_{6n}$; $\varepsilon_{6}=0$, $\varepsilon_0$, $\varepsilon_1,\cdots,\varepsilon_5$ are 6th roots of unity. We prove that the set of values of random variable $\tau$ is self-similar six petal snowflake which is a fractal curve $G$ of spider web type with dimension $\log_37$. Its outline is the Koch snowflake. We establish that $\tau$ has either a discrete or a singularly continuous distribution with respect to two-dimensional Lebesgue measure. The criterion of discreteness for the distribution is found and its point spectrum (set of atoms) is described. It is proved that the point spectrum is a countable everywhere dense set of values of the random variable $\tau$, which is the tail set of the seven-symbol representation of the points of the curve $G$. In the case of identical distribution of the random variables $\tau_n$ (namely: $p_{kn}=p_k$) we establish that the spectrum of distribution $\tau$ is a self-similar fractal and that the essential support of density is the fractal Besicovitch-Eggleston type set. The set is defined by terms digits frequencies and has the fractal dimension $\alpha_0(E)=\frac{\ln {p_0^{p_0}\cdots p_6^{p_6}}}{-\ln 7}$ with respect to the Hausdorff-Billingsley $\alpha$-measure. The measure is a probabilistic generalization of the Hausdorff $\alpha$-measure. In this case, the random variables $\tau=\Delta^g_{\tau_1\cdots\tau_n\cdots}$ and $\tau'=\Delta^g_{\tau_1'...\tau_n'...}$ defined by different probability vectors $(p_0,\cdots,p_6)$ and $(p'_0,\cdots,p'_6)$ have mutually orthogonal distributions.

Multilayer multiband hybrid fractal antenna for public safety and 5G Sub-6 GHz bands

In this work, a multiband hybrid fractal antenna is developed for public safety and 5G Sub-6 GHz bands. Proposed antenna design is designed, analyzed, and optimized using HFSS simulator. Radiating element of the proposed antenna consists of Hybrid Fractal shape which is designed using Koch and Hilbert curve fractal geometry. Hybrid Fractal Antenna (HFA) is designed on a Multilayer FR4 substrate which has an overall size of 0.22 × 0.47 × 0.044 ƛ3. Parametric analysis of the proposed design is done to get the optimum results. Proposed HFA resonates at 2.64 GHz (2.34–2.80 GHz), 3.87, 4.54, 5.02, 5.35 GHz (3.73–5.55 GHz), and 6.42 GHz (5.70–6.72 GHz) with a gain of 4.2, 0.9, 2.9, 3.3, 4.6, and 3.7 dBi respectively. The proposed HFA is useful for 5G bands such as: N7 band, N38 band, N40 band, N41 band, N47 band, N53 band, N79 band, N90 band and N93 band, and also useful for public safety band (4.9 GHz). Prototype of the proposed HFA is fabricated and tested in lab for the verification of simulated results. The results show good performance in terms of reflection coefficient, gain, and bandwidth. Measured results are in good agreement with the simulated results.

Koch Curves and Hexagonal Ring-Shaped Geometry Based Ultra-Wideband Fractal Antenna

Koch Curves and Hexagonal Ring-Shaped Geometry Based Ultra-Wideband Fractal Antenna

A Hybrid Approach Combining the Lie Method and Long Short-Term Memory (LSTM) Network for Predicting the Bitcoin Return

This paper introduces hybrid models designed to analyze daily and weekly bitcoin return spanning the periods from 18 July 2010 to 28 December 2023 for daily data, and from 18 July 2010 to 24 December 2023 for weekly data. Firstly, the fractal and chaotic structure of the selected variables was explored. Asymmetric Cantor set, Boundary of the Dragon curve, Julia set z2 −1, Boundary of the Lévy C curve, von Koch curve, and Brownian function (Wiener process) tests were applied. The R/S and Mandelbrot–Wallis tests confirmed long-term dependence and fractionality. The largest Lyapunov test, the Rosenstein, Collins and DeLuca, and Kantz methods of Lyapunov exponents, and the HCT and Shannon entropy tests tracked by the Kolmogorov–Sinai (KS) complexity test determined the evidence of chaos, entropy, and complexity. The BDS test of independence test approved nonlinearity, and the TeraesvirtaNW and WhiteNW tests, the Tsay test for nonlinearity, the LR test for threshold nonlinearity, and White’s test and Engle test confirmed nonlinearity and heteroskedasticity, in addition to fractionality and chaos. In the second stage, the standard ARFIMA method was applied, and its results were compared to the LieNLS and LieOLS methods. The results showed that, under conditions of chaos, entropy, and complexity, the ARFIMA method did not yield successful results. Both baseline models, LieNLS and LieOLS, are enhanced by integrating them with deep learning methods. The models, LieLSTMOLS and LieLSTMNLS, leverage manifold-based approaches, opting for matrix representations over traditional differential operator representations of Lie algebras were employed. The parameters and coefficients obtained from LieNLS and LieOLS, and the LieLSTMOLS and LieLSTMNLS methods were compared. And the forecasting capabilities of these hybrid models, particularly LieLSTMOLS and LieLSTMNLS, were compared with those of the main models. The in-sample and out-of-sample analyses demonstrated that the LieLSTMOLS and LieLSTMNLS methods outperform the others in terms of MAE and RMSE, thereby offering a more reliable means of assessing the selected data. Our study underscores the importance of employing the LieLSTM method for analyzing the dynamics of bitcoin. Our findings have significant implications for investors, traders, and policymakers.

Development a novel design of miniaturized heptagonal Koch fractal wide band antenna for 5G mm wave and IoT applications

The object of the study is a compact heptagonal broadband antenna specially designed for use in the 5G millimeter band using Koch fractals to improve performance. As a result of the study, the most important problem of achieving higher gain, improving bandwidth and reducing interference at higher frequencies, which is necessary for the effective functioning of 5G networks, was solved. As a result, the maximum realized gain of 5 dB was obtained at a frequency of 27.58 GHz with an impressive bandwidth in the range from 26.5 to 40 GHz. The use of Koch fractal geometry and defective ground planes significantly improves impedance matching and expands bandwidth, which explains the excellent antenna performance compared to traditional designs. The features and distinguishing features of the results obtained, thanks to which they allowed solving the problem under study, are its compact dimensions (only 9 mm by 9 mm) and the ability to maintain VSWR at a level of less than 2 in the entire frequency spectrum. These features make the antenna particularly suitable for millimeter-band integration and flexible applications such as portable devices and wearable home appliances. The field of practical application of the results includes integration into portable and wearable devices, improving the performance and connectivity of Internet of Things applications. The conditions of practical use require compliance with 5G network standards and compatibility with millimeter-wave technologies. This characterizes the antenna as a significant achievement in antenna technology, demonstrating its potential for widespread adoption in next-generation wireless communication systems and paving the way for more reliable and high-performance wireless networks

Recursively Generated Spirals

Summary When the angles are changed in the usual algorithm to generate the Koch curve, a wide variety of images may be produced, some of which possess rotational symmetry. One family of such images consists of simple spiral-like images. While there are only two segments in each spiral arm, the number of segments that must be drawn to render a complete image of the spiral grows exponentially with the number of arms in the spiral.

Fractional Domination Number of the Fractal Graphs

Background/Objectives: A fractal is a collection of clearly defined structures that are uneven or irregular and can be divided into smaller pieces. Each of these smaller pieces has a relationship to the overall structure at random ranges that are analogous or identical. Fractal graphs are a highly effective way to construct well-defined patterns. The fractional domination number in a graph refers to the weight of a minimum fractional dominating function. The study is to find the bounds of the fractional domination and their related parameters in the fractal graphs. Methods: The sharp bounds and the constant ratio of fractional domination and their related parameters were calculated using an iterative process in all iterations of a self-similarity fractal graph. Findings: The study established the relation between the fractional domination number and other relevant parameters as fractional independence domination numbers for each iteration in the fractal graphs and explained how the concept of fractional domination works on fractal graphs such as the Koch snowflake and the Sierpinski triangle. Novelty: Previous researchers have studied bounds on maximum matching and dominating sets in fractal graphs such as the Koch snowflake and the Sierpinski triangle. The present study enhances the idea of finding the bounds of fractional domination parameters in a fractal graph. The construction of new beautiful ornaments, locating damaged cells for drug injection in medical science, placing stones in jewels, plumping work, and designing fractal antennas all can be made possible by this work in self-similarity fractal graphs. Keywords: Fractals, Fractional Dominating Functions, Fractional Domination Number, Fractional Independence Domination Number, Fractional Domination Chain

Effect of Using Fractals as Grain Configuration on the Performance of Solid Rocket Motors

Grain design is a very crucial consideration for determining the performance of solid propellant rocket motors. A rocket’s performance and combustion characteristics (thrust profile, chamber pressure variation, propellant mass flow rate, and sliver among other important performance parameters) depend very strongly on the grain configuration. We propose fractal shapes (Koch star, Koch snowflake, Gosper, and Cesaro) as new port geometries and theoretically predict their performance using simulations based on the fast-marching method implemented in the open-source software Open-Motor. The initial thrust for fractal geometry grains was found to be significantly higher than the commonly used industry standard BATES and Finosyl ports. The fractal grains also showed a significant decrease in sliver as compared to the Finocyl port that is generally used to increase performance as compared to the circular BATES ports. Especially, The Koch snowflake port had a drastically lower sliver. These new grain geometries proposed in this work can be used for applications in which low initiation and ignition times along with high initial thrust are crucial. Moreover, these geometries can improve the performance of the solid rocket motor (SRM) owing to their better regression characteristics which promise complete utilization of the entire grain web.

Impact of detour on traffic flow in branching Koch curve network with bottleneck

There are various routes with and without detours in city traffic network with a complex connectivity. Branching Koch curve fractal has such complex connectivity with singly, doubly, and multiply connecting links. We consider branching Koch curve as a city traffic network. We study the effect of detour routes (bypasses) on macroscopic traffic flow in branching Koch curve network with a bottleneck. The bottleneck effect depends on whether or not there are detour routes. We propose a mathematical dynamic model for the traffic flow with a bottleneck in the directed Koch curve network. The macroscopic dynamic equations of vehicular densities are derived in the Koch curve network. Vehicular densities and currents are obtained at all roads in the network by solving the density equations numerically. The fundamental diagrams (flow-density relations) are derived. The traffic currents (flows) depend highly on the position of the bottleneck due to the connectivity of the fractal network. It is shown where and how traffic congestion is induced by the bottleneck on the directed fractal network.

Multi band multi polarised fractal antenna for white space TV band

ABSTRACT Designing a fractal antenna for white space TV band frequencies is the main goal of the suggested work. In this design 4 iteration Koch snowflakes patch is used to get multiband. As the number of iterations increased size of the antenna is going to reduce. Further 4th iteration Koch snowflakes slots have been created in the patch. By introducing a fourth iteration cross slot into the ground plane, three bands have been created. Circular polarisation is attained through the incorporation of cross slots in the ground plane, inducing a phase shift between vertically and horizontally polarised electromagnetic waves. The antenna primarily operates within the last frequency bands (686–806 MHz) with linear polarisation. However, it can also function in the initial frequency band (487–540 MHz) with left-hand circular polarisation. The physical size if antenna is reduced by 38 mm with a gain of 4.5dBi. Axial ratio bandwidth (481–552 MHz) is 71 MHz. Axial ratio bandwidth is 18 MHz more as compared to impedance bandwidth. The same antenna can work in linear polarisation as well as circular polarisation. Multi-polarised, multiband antenna with impedance bandwidth of 10.3% (1st band) and 16.08% (last band) is designed for white space TV band frequencies.

A Koch Fractal Octagonal Antenna with a Compact Design and Defected Ground Structure (DGS) for Ultra-Wideband (UWB) Wireless Usage.

This paper introduces a Koch fractal octagonal antenna designed for the Ultrawideband (UWB) Frequency range. The utilization of the Koch fractal in the antenna design contributes to size reduction as well as provides compactness in the antenna for UWB. Additionally, it has been noted that the Koch fractal offers wideband operation. The antenna employs copper as the conductor material, and Flame Retardant-4 (FR4) serves as the substrate. The substrate has a dielectric constant ϵr=4.4, a loss tangent tanδ=0.02, and a thickness (h) of 1.6 mm. The overall antenna’s dimensions are 34.3 × 26.5 × 1.6 or electrical dimensions represent as 0.68λ0 × 0.53λ0 × 0.032λ0 . It achieves a maximum gain of 8.94 dBi at a frequency of 12.25 GHz, offering a broad bandwidth ranging from 2 GHz to 12.1 GHz.This antenna exhibits resonance at three distinct frequencies, namely 3.3 GHz, 6 GHz, and 8.6 GHz, making it highly efficient with an overall efficiency of 96.8%. The time domain characteristic is acceptable for UWB, group delay is 1.1 ns, the The proposed antenna has a high-fidelity factor of 90.2, and the correlation coefficient is 0.9. which makes the antenna a good candidate for UWB. Due to its performance characteristics, this proposed antenna is well suited for short-range wireless personal area networks (WPANs), supporting high-data-rate applications like Bluetooth and wireless USB and Wireless Body Area Network (WBAN) applications. Its proficiency also extends to industrial settings, where it helps with precision in control systems, asset tracking, and short-range sensing and radar systems.

A Detailed Design Procedure for Printed Log-Periodic Antennas with Koch Fractal Dipoles

This paper presents a complete procedure for the design of printed log-periodic antennas. An investigation on the coaxial-to-parallel-strip transition usually present in this kind of antenna is discussed in detail and it is demonstrated to be very critical for the achievement of the desired impedance bandwidth. Furthermore, in order to reduce the overall antenna size, the dipoles have been implemented based on Koch fractals. It has been demonstrated that the use of Koch fractals allowed reducing the antenna area by 25 % in comparison to the classical printed log-periodic structure. Both designs have been optimized for the frequency range 2.3 – 6 GHz, so as to cover several wireless services, such as WLAN, mobile communications and WiMAX, among other applications. Experimental results validate the proposed design strategy.

Numerical evaluation of singular integrals on non-disjoint self-similar fractal sets

We consider the numerical evaluation of a class of double integrals with respect to a pair of self-similar measures over a self-similar fractal set (the attractor of an iterated function system), with a weakly singular integrand of logarithmic or algebraic type. In a recent paper (Gibbs et al. Numer. Algorithms 92, 2071–2124 2023), it was shown that when the fractal set is “disjoint” in a certain sense (an example being the Cantor set), the self-similarity of the measures, combined with the homogeneity properties of the integrand, can be exploited to express the singular integral exactly in terms of regular integrals, which can be readily approximated numerically. In this paper, we present a methodology for extending these results to cases where the fractal is non-disjoint but non-overlapping (in the sense that the open set condition holds). Our approach applies to many well-known examples including the Sierpinski triangle, the Vicsek fractal, the Sierpinski carpet, and the Koch snowflake.

УМОВИ ВИКОРИСТАННЯ КОНФОРМНИХ ВІДОБРАЖЕНЬ ДЛЯ ВИВЧЕННЯ ФРАКТАЛЬНОГО СТИСНЕННЯ ІНФОРМАЦІЇ

To describe the properties of self-similarity and invariance observed in various physical situations, the theory of fractals and multifractals is actively used in modern science. We consider a conformal mapping of the first kind, which is given by the entire linear function w = az + b, where w, z are complex variables, a, b are complex constants, a ≠ 0 . With its help, affine transformations are performed to construct the Sierpinski triangle and the Koch curve by specifying the functions w3(z) and finding the coefficients Re аі, Im аі, Re bi, Im bi, і = 1,2,3 or і = 1,2,3,4. These coefficients encode the image of the object and it can be unambiguously reconstructed by them. It is determined that the method in which self-similar regions in an object are detected and conformal mapping coefficients are found for them works under the condition that each such mapping is compressive. Only then will Banach’s fixed point theorem ensure image collection during decompression. This method of compressing graphic information is called fractal.

A SCALING LAW RELATING THE RATE OF DESTRUCTION OF A SOLID TUMOR AND THE FRACTAL DIMENSION OF ITS BOUNDARY

In this paper, we investigate the scaling law relating the size of the boundary of a solid tumor and the rate at which it is lysed by a cell population of non-infiltrating cytotoxic lymphocytes. We do it in the context of enzyme kinetics through geometrical, analytical and numerical arguments. Following the Koch island fractal model, a scale-dependent function that describes the constant rate of the decay process and the fractal dimension is obtained. Then, in silico experiments are accomplished by means of a stochastic hybrid cellular automaton model. This model is used to grow several tumors with varying morphology and to test the power decay law when the cell-mediated immune response is effective, confirming its validity.

Hausdorff dimension method reduces the error of box counting

The Hausdorff dimension of an object is a topological measure of the size of its covering properties. To compute the Hausdorff dimension of an object, this report reviews the method of box counting, a method of gathering data for analyzing complex patterns by breaking a dataset, object, image, etc. into smaller and smaller pieces, typically "box"-shaped, and analyzing the pieces at each smaller scale. However, in this process, errors are always generated because of the extreme cases in which boxes are just covered with a small corner of the pattern but still should be counted. As a result, the number of boxes counted becomes larger than usual resulting in inaccurate results. In order to reduce the error in the final result, this report applies an improvement to this method to reduce the errors in the result and provides the example of Koch snowflake, and also examines the effect of the improvement.

Exact Self-Similar Patterns of Atomic-Layer Boron Nitride with Koch Fractal Complexity

Exact Self-Similar Patterns of Atomic-Layer Boron Nitride with Koch Fractal Complexity

Wideband 4-port MIMO antenna array using fractal and complimentary split-ring structure for Ku-band appliances

This article presents the wideband compact 4-port fractal MIMO antenna array for Ku-band applications. It incorporates the Koch snowflake fractal in combination with a complementary split-ring resonator (CSRR) to achieve both miniaturization and a broad bandwidth. Further, to improve the effectiveness of the single-driven element, the 2-element linear array is constructed, fabricated, and tested. The single-driven element and the constructed array give good performance results. For pattern diversity, a 4-port MIMO configuration is developed and tested. The proposed MIMO array geometry realizes a wide bandwidth of 13.6−17.5 GHz, fractional bandwidth of 25.08%, a maximum gain of 6.95 dBi, and radiation efficiency of beyond 90% with superior isolation (>−30 dB). It provides better diversity performance parameters (Envelope correlation coefficient <0.001) over the resonating band. Also, channel capacity loss (CCL) <0.11 bit/s/Hz and the total active reflection coefficient (TARC) <−10 dB, parameters are perceived to be within adequate range. Therefore, the designed compact fractal antenna, 2-element array, and 4-port MIMO array are applicable for wide Ku-band applications.

New method based on genetic algorithm and Minkowski fractal for multiband antenna designs

Abstract In this paper, a new method based on a genetic algorithm and Minkowski Island fractal is proposed for multiband antennas. Three-antenna configurations are chosen to validate the proposed optimization procedure. The first configuration is a wide-band antenna, operating in the WLAN (wireless local area network) UNII-2C band. The second configuration is a dual-band antenna, operating in the WLAN UNII-2 and UNII-2C bands. In contrast, the third is a tri-band antenna operating in the UNII-2, UNII-2C, and UNII-3 bands. The optimization process is accelerated by using the Computer Simulation Technology (CST) Application Programming Interface which allows all genetic operators to be performed in MATLAB while the numerical calculations are running in the internal CST Finite-Difference Time-Domain -solver using parallel computing with GPU acceleration. All three designed configurations are manufactured using a $\textstyle0.8\;\text{mm}$ thick FR4 epoxy substrate with a relative dielectric constant of $4.8$ . The return loss and the radiation pattern’s measurements agree well with the simulation results. Further, the methodology presented can be very effective in terms of size reduction; the designed antennas are $24 \times 24 \times 0.8\;{\textrm{m}}{{\textrm{m}}^3}$ ( $460\;{\textrm{m}}{{\textrm{m}}^3}$ ).

Statement of Retraction: Internet of Thing based Koch Fractal Curve Fractal Antennas for Wireless Applications

Statement of Retraction: Internet of Thing based Koch Fractal Curve Fractal Antennas for Wireless Applications