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The thermodynamical formalism has been developed by the authors for a very general class of transcendental meromorphic functions. A function In the present manuscript we first improve upon our earlier paper in providing a systematic account of the thermodynamical formalism for such a meromorphic function Then we provide various, mainly geometric, applications of this theory. Indeed, we examine the finer fractal structure of the radial (in fact non-escaping) Julia set by developing the multifractal analysis of Gibbs states. In particular, the Bowen’s formula for the Hausdorff dimension of the radial Julia set from our earlier paper is reproved. Moreover, the multifractal spectrum function is proved to be convex, real-analytic and to be the Legendre transform conjugate to the temperature function. In the last chapter we went even further by showing that, for a analytic family satisfying a symmetric version of the growth condition (1.1) in a uniform way, the multifractal spectrum function is real-analytic also with respect to the parameter. Such a fact, up to our knowledge, has not been so far proved even for hyperbolic rational functions nor even for the quadratic family

We study a family of renormalization transformations of generalized diamond hierarchical Potts models through complex dynamical systems. We prove that the Julia set (unstable set) of a renormalization transformation, when it is treated as a complex dynamical system, is the set of complex singularities of the free energy in statistical mechanics. We give a sufficient and necessary condition for the Julia sets to be disconnected. Furthermore, we prove that all Fatou components (components of the stable sets) of this family of renormalization transformations are Jordan domains with at most one exception which is completely invariant. In view of the problem in physics about the distribution of these complex singularities, we prove here a new type of distribution: the set of these complex singularities in the real temperature domain could contain an interval. Finally, we study the boundary behavior of the first derivative and second derivative of the free energy on the Fatou component containing the infinity. We also give an explicit value of the second order critical exponent of the free energy for almost every boundary point.

The thoughts and ideas of artists and craftsmen, according to the way they perceive the manifestations of the world, are preserved in the form of works of visual arts, architecture and decorative crafts which are exhibited, as souvenirs of culture human. Aesthetics and art sciences introduce these works according to the change and transformation of human experiences, thus the precision and elegance “creativity” of artists throughout history, cultural value, dignity and validity of works of art are measured and determined. Today, given the new dimensions that have changed with technological progress and the production of new technical tools, the way of seeing the world and material effects has transformed and led to the creation of works of art more detailed and more complete. Paying more attention to (space) and creating new precise and comprehensive artistic dimensions is considered one of the modern achievements. The characteristics of design and creation of forms, which occupied the artist’s mind throughout the ages to see the world better and create better especially when he used “geometry”, he went through the exact terms of the world of senses and reason. Recently, some artists have paid special attention to the new common branch between mathematics, geometry and art called fractal geometry, which provides information about the common secret and harmony of the universe. Among them, modern architects and visual arts have created significant works in this regard.

The phenomenon of intermittency has remained a theoretical concept without any attempts to approach it geometrically with the use of a simple visualization. In this paper, a particular geometric model of point clustering approaching the Cantor shape in 2D, with a symmetry scale θ being an intermittency parameter, is proposed. To verify its ability to describe intermittency, to this model, we applied the entropic skin theory concept. This allowed us to obtain a conceptual validation. We observed that the intermittency phenomenon in our model was adequately described with the multiscale dynamics proposed by the entropic skin theory, coupling the fluctuation levels that extended between two extremes: the bulk and the crest. We calculated the reversibility efficiency γ with two different methods: statistical and geometrical analyses. Both efficiency values, γstat and γgeo, showed equality with a low relative error margin, which actually validated our suggested fractal model for intermittency. In addition, we applied the extended self-similarity (E.S.S.) to the model. This highlighted the intermittency phenomenon as a deviation from the homogeneity assumed by Kolmogorov in turbulence.

Porous structures exhibiting randomly sized and distributed pores are required in biomedical applications (producing implants), materials science (developing cermet-based materials with desired properties), engineering applications (objects having controlled mass and energy transfer properties), and smart agriculture (devices for soilless cultivation). In most cases, a scaffold-based method is used to design porous structures. This approach fails to produce randomly sized and distributed pores, which is a pressing need as far as the aforementioned application areas are concerned. Thus, more effective porous structure design methods are required. This article presents how to utilize fractal geometry to model porous structures and then print them using 3D printing technology. A mathematical procedure was developed to create stochastic point clouds using the affine maps of a predefined Iterative Function Systems (IFS)-based fractal. In addition, a method is developed to modify a given IFS fractal-generated point cloud. The modification process controls the self-similarity levels of the fractal and ultimately results in a model of porous structure exhibiting randomly sized and distributed pores. The model can be transformed into a 3D Computer-Aided Design (CAD) model using voxel-based modeling or other means for digitization and 3D printing. The efficacy of the proposed method is demonstrated by transforming the Sierpinski Carpet (an IFS-based fractal) into 3D-printed porous structures with randomly sized and distributed pores. Other IFS-based fractals than the Sierpinski Carpet can be used to model and fabricate porous structures effectively. This issue remains open for further research.

Purpose Antenna miniaturization, multiband operation and wider operational bandwidth are vital to achieve optimal design for modern wireless communication devices. Using fractal geometries is recognized as one of the most promising solutions to attain these characteristics. The purpose of this paper is to present a unique structure of patch antenna using hybrid fractal technique to enhance the performance characteristics for various wireless applications and to achieve better miniaturization. Design/methodology/approach In this paper, the authors propose a novel hybrid fractal antenna by combining Koch and Minkowski (K-M) fractal geometries. A microstrip patch antenna (MPA) operating at 1.8 GHz is incorporated with a novel K-M hybrid fractal geometry. The proposed fractal antenna is designed and simulated in CST Microwave studio and compared with existing Koch fractal geometry. The prototype for the third iteration of the K-M fractal antenna is then fabricated on FR-4 substrate and tested through vector network analyzer for operating band/voltage standing wave ratio. Findings The third iteration of the proposed K-M fractal geometry results in achieving a 20% size reduction as compared to an ordinary MPA for the same resonant frequency with impedance bandwidth of 16.25 MHz and a directional gain of 6.48 dB, respectively. The operating frequency of MPA also lowers down to 1.44 GHz. Originality/value Further testing for the radiation patterns in an anechoic chamber shows good agreement to those of simulated results.

The growing demand for enhanced capacities, broadband services, and high transmission speeds to accommodate speech, image, multimedia, and data communication simultaneously puts a requirement for antenna to operate in multiple frequency bands. A novel compact fractal antenna based on self-similar stair-shaped fractal geometry is proposed in this paper. The fractal antenna is designed by modifying the patch antenna through the iterative process using stair-shaped fractal geometry. The third iteration results in a tri-band response, and the antenna resonate at 3.65, 4.825, and 6.325 GHz with impedance bandwidths of 75.6, 121.2, and 211.4 MHz, respectively. The antenna is designed in CST Microwave studio, and evaluated for operating bands and radiation characteristics. Prototype for the third iteration of the fractal antenna is fabricated on FR-4 substrate which is further tested for measured operating bands and radiation characteristics. The simulated and measured results show good agreement.

A breeder can select a visually appealing phenotype, whether for ornamentation or landscaping. However, the organic vision is not accurate and objective, making it challenging to bring a reliable phenotyping intervention into implementation. Therefore, the objective of this study was to develop an innovative solution to predict the intensity of the flower’s color upon the external shape of the crop. We merged the single linear iterative clustering (SLIC) algorithm and box-counting method (BCM) into a framework to extract useful imagery data for biophysical modeling. Then, we validated our approach by fitting Gompertz function to data on intensity of flower’s color and fractal dimension (SD) of the architecture of white-flower, yellow-flower, and red-flower varieties of Portulaca umbraticola. The SLIC algorithm segmented the images into uniform superpixels, enabling the BCM to precisely capture the SD of the architecture. The SD ranged from 1.938315 to 1.941630, which corresponded to pixel-wise intensities of 220.85 and 47.15. Thus, the more compact the architecture the more intensive the color of the flower. The sigmoid Gompertz function predicted such a relationship at radj2 > 0.80. This study can provide further knowledge to progress the field’s prominence in developing breakthrough strategies toward improving the control of visual quality and breeding of ornamentals.

In den letzten Jahrzehnten haben die bildgebenden Moglichkeiten des Computers zum vieldiskutierten Pictorial Turn - der Wende zum Bild - in den Naturwissenschaften gefuhrt. Mit dem offentlichkeitswirksamen Auftritt der Bilder von Chaos und fraktaler Geometrie sowie ihrer breiten Popularisierung ab Mitte der 1980er-Jahre erfasste dieser Trend auch die Mathematik und damit diejenige Disziplin, die als Reich des reinen Denkens traditionell fur ihre Bilderskepsis bekannt war.Die Bilder dieses Forschungsfelds werden in der vorliegenden Studie zum ersten Mal bildtheoretisch reflektiert und diskutiert. Im Zentrum stehen Arbeitsmaterialien aus privaten Bildarchiven von Mathematikern und Physikern. Eine besondere Rolle spielt dabei die Handzeichnung als Denkform, die auf der Schwelle zum digitalen Medienumbruch eine neue Schwungkraft gewinnt.

In this study, we created metamaterials consisting of square unit cells—inspired by fractal geometry—and described the parametric equation necessary for their creation. The area and thus the volume (density) and mass of these metamaterials are constant regardless of the number of cells. They were created with two layout types; one consists solely of compressed rod elements (ordered layout), and in the other layout, due to a geometrical offset, certain regions are exposed to bending (offset layout). In addition to creating new metamaterial structures, our aim was to study their energy absorption and failure. Finite element analysis was performed on their expected behavior and deformation when subjected to compression. Specimens were printed from polyamide with additive technology in order to compare and validate the results of the FEM simulations with real compression tests. Based on these results, increasing the number of cells results in a more stable behavior and increased load-bearing capacity. Furthermore, by increasing the number of cells from 4 to 36, the energy absorption capability doubles; however, further increase does not significantly change this capability. As for the effect of layout, the offset structures are 27% softer, on average, but exhibit a more stable deformation behavior.