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This paper aims to explain, first of all, signal modeling steps using Poincaré, and then considering the occurred events, concept of information applying Poincaré section and information approach, the brain pattern variations in autism spectrum disorder (ASD) cases will be diagnosed. A kind of representation of electroencephalogram (EEG) signal, namely, complementary plot, in which the main characteristic is special attention to asymmetry and symmetry coexist in natural and human processes, is introduced. In this paper, a new model is provided whose variations of patterns are similar to EEG’s when the transformation parameter is changed. A significant difference between ASD and healthy cases was also observed, which could be used to distinguish between various types of systems.

Complementary plot method is one of the most proper representations for Poincaré section of complex dynamics, because, as it was said about its characteristics, it has a qualitative approach toward signal (Sabelli, 2000, 2001, 2003, 2008, 2005, Sabelli et al., 2011). Considering the special conditions of this representation, here, intersection with a circle y² + x² = r² will be used; the important fact is, on the contrary to previous representations in which circular section had energy concept, here circular section considers phases. For finding trajectory intersection points, after calculating the sin and cosine of each term of EEG, plotting them in XY plane and drawing a chord between successive points of presentation transitions, then its intersections with the assumed circle are determined. But considering the sampling frequency, chords and Poincaré section, in this space, a minimum error – as the threshold – should be assumed in the program.

Natural and human processes are biotic (life-like) and creative (Sabelli and Galilei), and studying coexisting opposites by calculating the sine and cosine of each term in heartbeat intervals, weather variables and integer biotic series or random walk reveals an astonishingly regular mandala pattern; these patterns are not generated by random, periodic or chaotic series (Sabelli, 2005). This paper shows that in EEG of ASD children, mandala-like patterns of concentric rings are emergent in all situations (baseline – watching animation with voice and without voice) and electrode site (C3 and C4), but not in healthy individuals. The authors take the relation between sine and cosine functions as a mathematical model for complementary opposition, because it involves reciprocity and orthogonality sine and cosine are natural models for information. In fact, trigonometric analyses of empirical data to be described in this paper suggest expanding the concept of co-creative opposition to include uncorrelated opposites and partial opposites, i.e. partial agonists and partial antagonists that are neither linear nor orthogonal. Using Poincaré sections, it is shown that the difference in information and creativity of the data is the distinctive characteristic in ASD and healthy cases. Creation is the generation of novelty, diversity and complexity in complex systems.

This paper is an original paper based on cybernetic approaches for studying the variations of ASD children.

The purpose of this paper is to evaluate a European option using the fractional version of the Black-Scholes model.

In this paper, the authors employ the block-pulse operational matrix algorithm to approximate the solution of the fractional Black-Scholes equation with the initial condition for a European option pricing problem.

The fractional derivative will be described in the Caputo sense in this paper. The authors show the accuracy and computational efficiency of the proposed algorithm through some numerical examples.

This is the first paper that considers an alternative algorithm for pricing a European option using the fractional Black-Scholes model.