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This research primarily focuses on the volatility of cryptocurrency prices. In this study, the top five cryptocurrencies based on their market capitalization and data for the financial year 2022–2023 are considered for analysis via fractal dimensions and fractal functions, well-known tools in the subject of Fractal Geometry for measuring and modeling irregular and non-smooth phenomena. By integrating abnormal return with the capital asset pricing model, we ascertain the response of the acquiring cryptocurrency’s price. The study finds that BTC has consistency in its fluctuations, TETH has shown greater volatility over time, and ETH has shown some consistency; at $\alpha =0.5$, XRP is riskier than the others. While BNB is easily predictable at $\alpha =0.5$, it does not fluctuate in a systematic manner at $\alpha =0.3$. AAR indicates a short-term profit, while CAAR is decreasing, suggesting a long-term lack of profit. These reasons stem from investors’ increased expectations of specific strongholds in the cryptocurrency market, as well as increased investment.

A new method for constructing recurrent bivariate fractal interpolation surfaces through points sampled on rectangular lattices is proposed. This offers the advantage of a more flexible fractal modeling compared to previous fractal techniques that used affine transformations. The compression ratio for the above mentioned fractal scheme as applied to real images is higher than other fractal methods or JPEG, though not as high as JPEG2000. Theory, implementation and analytical study are also presented.

In the present paper, the smoothness of a Coalescence Hidden-variable Fractal Interpolation Surface (CHFIS), as described by its Lipschitz exponent, is investigated. This is achieved by considering the simulation of a generally uneven surface using CHFIS. The influence of free variables and Lipschitz exponent on the smoothness of CHFIS is demonstrated by considering interpolation data generated from a sample surface.

We consider the theory and applications of bivariate fractal interpolation surfaces constructed as attractors of iterated function systems. Specifically, such kind of surfaces constructed on rectangular domains have been used to demonstrate their efficiency in computer graphics and image processing. The methodology followed is based on the labeling used for the vertices of the rectangular domain rather than on the constraints satisfied by the contractivity factors or the boundary data.

We construct hidden variable bivariate fractal interpolation surfaces (FIS). The vector valued iterated function system (IFS) is constructed in ℝ⁴ and its projection in ℝ³ is taken. The extra degree of freedom coming from ℝ⁴ provides hidden variable, which is an important factor for flexibility and diversity in the interpolated surface. In the present paper, we construct an IFS that generates both self-similar and non-self-similar FIS simultaneously and show that the hidden variable fractal surface may be self-similar under certain conditions.

We present a method of construction of vector valued bivariate fractal interpolation functions on random grids in ℝ². Examples and applications are also included.

Two methods for representing discrete image data on rectangular lattices using fractal surfaces are proposed. They offer the advantage of a more general fractal modeling compared to previous one-dimensional fractal interpolation techniques resulting in higher compression ratios. Theory, implementation and analytical study of the proposed methods are also presented.

In this work, we propose a new method to describe fractal shapes using parametric L-systems. This method consists of introducing scaling factors in the production rules of the parametric L-system grammars. We present a turtle monoid on which we base our calculations to show the exact mathematical relation between L-systems and iterated function systems (IFS); we then establish the conditions for the scaling factors to produce plants' and curves' fractal shapes from parametric L-systems. We demonstrate that with specific values of the scaling factors, we find the exact relationship established by Prusinkiewicz and Hammel between L-systems and IFS. Finally, we present some examples of fractal plant forms and curves created using parametric L-systems with scaling factors.

The decision problem of satisfiability of Boolean expression in k-conjunctive normal form (kSAT) is a typical NP-complete problem. In this paper, by mapping the whole Boolean expressions in k-conjunctive normal form onto a unit hypercube, we visualize the problem space of kSAT. The pattern of kSAT is shown to have self-similarity which can be deciphered in terms of graph directed iterated function system. We provide that the Hausdorff dimension of the pattern of kSAT is equal to the box-counting dimension and increases with k. This suggests that the time complexity of kSAT increases with k.

This paper presents two self-similar models that allow the control of curves and surfaces. The first model is based on IFS (Iterated Function Systems) theory and the second on subdivision curve and surface theory. Both of these methods employ the detail concept as in the wavelet transform, and allow the multiresolution control of objects with control points at any resolution level.

In the first model, the detail is inserted independently of control points, requiring it to be rotated when applying deformations. In contrast, the second method describes details relative to control points, allowing free control point deformations.

Modeling examples of curves and surfaces are presented, showing manipulation facilities of the models.

In the present paper, the bounds on fractal dimension of Coalescence Hidden-variable Fractal Interpolation Surface (CHFIS) in ℝ³ on a equispaced mesh are found. These bounds determine the conditions on the free parameters for fractal dimension of the constructed CHFIS to become close to 3. The results derived here are tested on a tsunami wave surface by computing the lower and upper bounds of the fractal dimension of its CHFIS simulation.

A new fractal interpolation method called PPA (Pointed Point Algorithm) based on IFS is proposed to interpolate the self-affine signals with the expected interpolation error, solving the problem that the ordinary fractal interpolation can't get the value of any arbitrary point directly, which has not been found in the existing literatures. At the same time, a new method to calculate the vertical scaling factors is proposed based on the genetic algorithm, which works together with the PPA algorithm to get the better interpolation performance. Experiments on the theoretical data and real field seismic data show that the proposed interpolation schemes can not only get the expected point's value, but also get a great accuracy in reconstruction of the seismic profile, leading to a significant improvement over other trace interpolation methods.

IFS fractals — the attractors of Iterated Function Systems — have motivated plenty of research to date, partly due to their simplicity and applicability in various fields, such as the modeling of plants in computer graphics, and the design of fractal antennas. The statement and resolution of the Fractal-Line Intersection Problem is imperative for a more efficient treatment of certain applications. This paper intends to take further steps towards this resolution, building on the literature. For the broad class of hyperdense fractals, a verifiable condition guaranteeing intersection with any line passing through the convex hull of a planar IFS fractal is shown, in general ℝ^d for hyperplanes. The condition also implies a constructive algorithm for finding the points of intersection. Under certain conditions, an infinite number of approximate intersections are guaranteed, if there is at least one. Quantification of the intersection is done via an explicit formula for the invariant measure of IFS.

Let $S$ be the attractor (fractal) of a contractive iterated function system (IFS) with place-dependent probabilities. An IFS with place-dependent probabilities is a random map

To construct symmetrical patterns on the unit sphere from the planar iterative function systems (IFSs), we present a method of constructing IFSs with $D_3$ symmetry which is composed of three-fold rotational symmetries together with reflections. An algorithm is developed to generate strange attractors with $D_3$ symmetry on a triangular face and then project it onto the surface of the unit sphere to form aesthetics patterns with spherical symmetry. As an illustrative example, we consider the regular inscribed icosahedron in the unit sphere which contains 20 triangular faces. This method is valid to randomly generate aesthetic spherical patterns using planar IFSs.

In this paper, we define the notion of affine curvatures on a discrete planar curve. By the moving frame method, they are in fact the discrete Maurer–Cartan invariants. It shows that two curves with the same curvature sequences are affinely equivalent. Conditions for the curves with some obvious geometric properties are obtained and examples with constant curvatures are considered. On the other hand, by using the affine invariants and optimization methods, it becomes possible to collect the IFSs of some self-affine fractals with desired geometrical or topological properties inside a fixed area. In order to estimate their Hausdorff dimensions, GPUs can be used to accelerate parallel computing tasks. Furthermore, the method could be used to a much broader class.