Generalized fractal Jensen and Jensen–Mercer inequalities for harmonic convex function with applications

In this paper, we present a generalized Jensen-type inequality for generalized harmonically convex function on the fractal sets, and a generalized Jensen–Mercer inequality involving local fractional integrals is obtained. Moreover, we establish some generalized Jensen–Mercer-type local fractional integral inequalities for harmonically convex function. Also, we obtain some generalized related results using these inequalities on the fractal space. Finally, we give applications of generalized means and probability density function.

Fractal Analysis of Interval Series of Prime Numbers

The paper presents a fractal analysis of interval series, the members of which are consecutive deviations of natural prime numbers. Fractal analysis, which has been in full force since the end of the last century, has made it possible to identify new, unusual properties of geometric and physical objects and processes, including predicting the behavior of time and spatial series. The combination of two structural blocks - the spatial interval series of increasing power and the fractal set made it possible to apply the fractal technique to the study of the sequence of intervals. With its help, the idea of the phenomenon of intervals of prime numbers as a structure that does not contradict the nature of most natural phenomena is expanded. Using the Hurst method and scaling it is established that the appearance of intervals of prime numbers is not random. With restrictions on the available computer memory, criteria-based estimates of interval series of different capacities were carried out and it was found that they have the properties of scale invariance, multifractality and self-similarity. The performed estimates confirm that the continuum of primes at all scale levels belongs to fractal sets.

JENSEN–MERCER INEQUALITY AND RELATED RESULTS IN THE FRACTAL SENSE WITH APPLICATIONS

The most notable inequality pertaining convex functions is Jensen’s inequality which has tremendous applications in several fields. Mercer introduced an important variant of Jensen’s inequality called as Jensen–Mercer’s inequality. Fractal sets are useful tools for describing the accuracy of inequalities in convex functions. The purpose of this paper is to establish a generalized Jensen–Mercer inequality for a generalized convex function on a real linear fractal set [Formula: see text] ([Formula: see text]. Further, we also demonstrate some generalized Jensen–Mercer-type inequalities by employing local fractional calculus. Lastly, some applications related to Jensen–Mercer inequality and [Formula: see text]-type special means are given. The present approach is efficient, reliable, and may motivate further research in this area.

Sierpinski Carpet and Chaos in the Periodic Table of the Elements

Fractals are self-similar geometric pattern which can be found in nature. They have applications in mathematic, electronic, architecture. Fractal sets also can be used to create chaotic systems. This work is about applying Sierpinski carpet order on the periodic table of the elements to create a new pattern for the chemical elements. Fibonacci numbers and Math lab software are used to transform a linear system to three spiral systems. This new pattern which is consisted of three layers shows that the flows among chemical elements are based on Archimedes spiral equation The purpose of this study is to show Sierpinski carpet order in the periodic table of the chemical elements and also there can be a chaos even in chemical elements.

The Complex-Order Fractional Laplacian: Stable Operators, Spectral Methods, and Applications to Heterogeneous Media

Fractional differential equations have become a fundamental modelling approach for understanding and simulating the many aspects of nonlocality and spatial heterogeneity of complex materials and systems. Yet, while real-order fractional operators are nowadays widely adopted, little progress has been made in extending such operators to complex-order counterparts. In this work, we introduce new definitions for the complex-order fractional Laplacian, fully consistent with the theory of averaging of smooth functions over fractal sets, and present tailored spectral methods for their numerical treatment. The proposed complex-order operators exhibit a dual particle-wave behaviour, with solutions manifesting wave-like features depending on the magnitude of the imaginary part of the fractional order. Reaction-diffusion systems driven by the complex-order fractional Laplacian exhibit unique spatio-temporal dynamics, such as equilibrium of diffusion in random materials by interference of scattered waves, conduction block and highly fractionated propagation, or the generation of completely novel Turing patterns. Taken together, our results support that the proposed complex-order operators hold unparalleled capabilities to advance the description of multiscale transport phenomena in physical and biological processes highly influenced by the heterogeneity of complex media.

Visualization of Mandelbrot and Julia Sets of Möbius Transformations

This work reports on a study of the Mandelbrot set and Julia set for a generalization of the well-explored function η(z)=z2+λ. The generalization consists of composing with a fixed Möbius transformation at each iteration step. In particular, affine and inverse Möbius transformations are explored. This work offers a new way of visualizing the Mandelbrot and filled-in Julia sets. An interesting and unexpected appearance of hyperbolic triangles occurs in the structure of the Mandelbrot sets for the case of inverse Möbius transforms. Several lemmas and theorems associated with these types of fractal sets are presented.