Special Issue: Current Progressions of Fixed Points and Fractals with Mathematical Applications
Fixed point theory plays an important role in many branches of mathematics. Fractal Analysis is an application of fixed point theory by using Hutchinson-Barnsley theory. A set together with a finite family of contraction mappings in Hausdorff metric space generates a fractal. Fractal theory plays a vital part in studying non-regular geometric shapes which are irregular and complex structures found in nature, using the fractal dimension. A self-similar set is a mathematical object, often a geometric shape or structure, that exhibits the property of self-similarity. This property means that the set is composed of smaller copies of itself, either exactly or approximately, at various scales. Self-similar sets are characterized by their recursive construction, scale invariance, fractal dimension, and hierarchical organization. These features make them a fundamental concept in fractal geometry, illustrating how complex and detailed structures can emerge from simple iterative processes. Also, self-similar sets typically have a fractal dimension, which is a non-integer dimension that quantifies their "roughness" or complexity. The fractal dimension often exceeds the topological dimension of the set. Fractal has a wide range of applications in diverse fields such as medical analysis, stock market analysis, and dynamical systems. Fractals, with their intricate self-similar structures, help visualize the long-term behavior of the complex systems. Their applications span diverse fields, including network science, data science, image processing, and artificial intelligence, contributing to advancements in both theoretical understanding and practical solutions.
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