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The particles in condensed matter physics are almost characterized by Brownian motion. This phenomenon is the basis for a very important understanding of the particles motion in condensed matter. For our previous research, there is already applied and confirmed the complex fractal correction which includes influence of parameters from grains and pores surface and also effects based on particles’ Brownian motion. As a chaotic structure of these motions, we have very complex research results regarding the particles’ trajectories in three-dimension (3D). In our research paper, we applied fractal interpolation within the idea to reconstruct the above mentioned trajectories in two dimensions at this stage. Because of the very complex fractional mathematics on Brownian motion, we found and developed much simpler and effective mathematization. The starting point is within linear interpolation. In our previous research, we presented very original line fractalization based on tensor product. But, in this paper, we applied and successfully confirmed that by fractal interpolation (Akimo polynomial method) that is possible to reconstruct the chaotical trajectories lines structures by several fractalized intervals and involved intervals. This novelty is very important because of the much more effective procedure that we can reconstruct and in that way control the particles’ trajectories. This is very important for further advanced research in microelectronics, especially inter-granular micro capacitors. 

The materials’ consolidation, especially ceramics, is very important in advanced research development and industrial technologies. Science of sintering with all incoming novelties is the base of all these processes. A very important question in all of this is how to get the more precise structure parameters within the morphology of different ceramic materials. In that sense, the advanced procedure in collecting precise data in submicro-processes is also in direction of advanced miniaturization. Our research, based on different electrophysical parameters, like relative capacitance, breakdown voltage, and [Formula: see text], has been used in neural networks and graph theory successful applications. We extended furthermore our neural network back propagation (BP) on sintering parameters’ data. Prognosed mapping we can succeed if we use the coefficients, implemented by the training procedure. In this paper, we continue to apply the novelty from the previous research, where the error is calculated as a difference between the designed and actual network output. So, the weight coefficients contribute in error generation. We used the experimental data of sintered materials’ density, measured and calculated in the bulk, and developed possibility to calculate the materials’ density inside of consolidated structures. The BP procedure here is like a tool to come down between the layers, with much more precise materials’ density, in the points on morphology, which are interesting for different microstructure developments and applications. We practically replaced the errors’ network by density values, from ceramic consolidation. Our neural networks’ application novelty is successfully applied within the experimental ceramic material density [Formula: see text] [kg/m 3 ], confirming the direction way to implement this procedure in other density cases. There are many different mathematical tools or tools from the field of artificial intelligence that can be used in such or similar applications. We choose to use artificial neural networks because of their simplicity and their self-improvement process, through BP error control. All of this contributes to the great improvement in the whole research and science of sintering technology, which is important for collecting more efficient and faster results. 

Many recently published research papers examine the representation of nanostructures and biomimetic materials, especially using mathematical methods. For this purpose, it is important that the mathematical method is simple and powerful. Theory of fractals, artificial neural networks and graph theory are most commonly used in such papers. These methods are useful tools for applying mathematics in nanostructures, especially given the diversity of the methods, as well as their compatibility and complementarity. The purpose of this paper is to provide an overview of existing results in the field of electrochemical and magnetic nanostructures parameter modeling by applying the three methods that are “easy to use”: theory of fractals, artificial neural networks and graph theory. We also give some new conclusions about applicability, advantages and disadvantages in various different circumstances. 

The world energy crisis necessitated the cause of the research interest into new, renewable and alternative energy sources. From this point of view, there is research on phenomena and different synthetic methods and on structure and electronic property optimization expressed by important material and device advancement. Efficiency and electricity generation (batteries, fuel cells, hydrogen energy) are nowadays actual questions. Because of that research, innovations and applications require extended knowledge by fractal nature characterization. The electrochemical energy sources solutions, especially electrolytes, are in fractal nature science focus. Based on the research novelties, especially electronic materials, we presented an investigation on fractal structure influence in electrochemistry. We explore the activation energy and fundamental thermodynamic functions and values, also the electrode surface changed by complex fractal correction through fractal dimension of grains and pores, and Brownian motion of involved particles, as well. At the end, the electrochemical Arrhenius and Butler–Volmer equation fractalization is applied. All of these open new perspectives for electrochemical energy processes, within electrolyte bulk and related electrodes and more precise energy generation. This is important for semiconductor processing in solar cells and devices. So, we included the knowledge of fractal sciences advancement in this field for current–voltage equation. 

Advanced research frontiers are extended from biophysics relations on the Earth upto the discovering any type of alive matter within the whole space. Microorganisms’ motion within the molecular biology processes integrates variety of microorgnisms functions. In continuation of our Brownian motion phenomena research, we consistently build molecular-microorganisms structures hierarchy. We recognize everywhere biomimetic similarities between the particles in alive and nonalive matter. The research data are based on real experiments, without external energy impulses. So, we develop the analysis, inspired by fractal nature Brownian motion, as recognized joint parameter between particles in alive and nonalive biophysical systems. This is also in line with advance trends in hybrid submicroelectronic integrations. The important innovation in this paper is that we introduced approximation of trajectory and error calculations, using discrete mean square approximation, what cumulatively provide much more precise biophysical systems parameters. By this paper, we continue to generate new knowledge in direction to get complex relations between the particles clusters in biophysical systems condensed matter.