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Many scientists have struggled to uncover the elusive origin of "complexity", and its many equivalent jargons, such as emergence, self-organization, synergetics, collective behaviors, nonequilibrium phenomena, etc. They have provided some qualitative, but not quantitative, characterizations of numerous fascinating examples from many disciplines. For example, Schrödinger had identified "the exchange of energy" from open systems as a necessary condition for complexity. Prigogine has argued for the need to introduce a new principle of nature which he dubbed "the instability of the homogeneous". Turing had proposed "symmetry breaking" as an origin of morphogenesis. Smale had asked what "axiomatic" properties must a reaction–diffusion system possess to make the Turing interacting system oscillate.

The purpose of this paper is to show that all the jargons and issues cited above are mere manifestations of a new fundamental principle called local activity, which is mathematically precise and testable. The local activity theorem provides the quantitative characterization of Prigogine's "instability of the homogeneous" and Smale's quest for an axiomatic principle on Turing instability.

Among other things, a mathematical proof is given which shows none of the complexity-related jargons cited above is possible without local activity. Explicit mathematical criteria are given to identify a relatively small subset of the locally-active parameter region, called the edge of chaos, where most complex phenomena emerge.

The first part of our paper presents a general survey on the modeling, analytic problems, and applications of the dynamics of human crowds, where the specific features of living systems are taken into account in the modeling approach. This critical analysis leads to the second part which is devoted to research perspectives on modeling, analytic problems, multiscale topics which are followed by hints towards possible achievements. Perspectives include the modeling of social dynamics, multiscale problems and a detailed study of the link between crowds and swarms modeling.

This paper is devoted to the multidisciplinary modelling of a pandemic initiated by an aggressive virus, specifically the so-called SARS–CoV–$2$ Severe Acute Respiratory Syndrome, corona virus n.$2$. The study is developed within a multiscale framework accounting for the interaction of different spatial scales, from the small scale of the virus itself and cells, to the large scale of individuals and further up to the collective behaviour of populations. An interdisciplinary vision is developed thanks to the contributions of epidemiologists, immunologists and economists as well as those of mathematical modellers. The first part of the contents is devoted to understanding the complex features of the system and to the design of a modelling rationale. The modelling approach is treated in the second part of the paper by showing both how the virus propagates into infected individuals, successfully and not successfully recovered, and also the spatial patterns, which are subsequently studied by kinetic and lattice models. The third part reports the contribution of research in the fields of virology, epidemiology, immune competition, and economy focussed also on social behaviours. Finally, a critical analysis is proposed looking ahead to research perspectives.

The modeling of living systems composed of many interacting entities is treated in this paper with the aim of describing their collective behaviors. The mathematical approach is developed within the general framework of the kinetic theory of active particles. The presentation is in three parts. First, we derive the mathematical tools, subsequently, we show how the method can be applied to a number of case studies related to well defined living systems, and finally, we look ahead to research perspectives.

Models used in the understanding of complex entities, like organisations, are problematic in several respects. After an introductory discussion of this problem, this paper addresses the problems associated with the boundaries of complex systems, arguing that although boundaries do exist, they have a peculiar nature. Similarly, it is argued that although hierarchies form an important part of the structure of complex systems, they are not clearly defined or "nested" as is often assumed. Hierarchies should also in principle be transformable in a viable system. Finally, the usefulness of network models is investigated. The conclusion is that although network models have a structure similar to that of complex systems, they are subject to the same limitations all models of complexity are faced with. A few implications for our understanding of organisations are mentioned.

This paper shows the action potential (spikes) generated from the Hodgkin–Huxley equations emerges near the edge of chaos consisting of a tiny subset of the locally active regime of the HH equations. The main result proves that the eigenvalues of the 4 × 4 Jacobian matrix associated with the mathematically intractable system of four nonlinear differential equations are identical to the zeros of a scalar complexity function from complexity theory. Moreover, we show the loci of a pair of complex-conjugate zeros migrate continuously as a function of an externally applied DC current excitation emulating the net synaptic excitation current input to the neuron. In particular, the pair of complex-conjugate zeros move from a subcritical Hopf bifurcation point at low excitation current to a super-critical Hopf bifurcation point at high excitation current. The spikes are generated as the excitation current approaches the vicinity of the edge of chaos, which leads to the onset of the subcritical Hopf bifurcation regime. It follows from this in-depth qualitative analysis that local activity is the origin of spikes.

This paper presents a survey and critical analysis of the mathematical literature on modeling and simulation of human crowds taking into account behavioral dynamics. The main focus is on research papers published after the review [N. Bellomo and C. Dogbè, On the modeling of traffic and crowds: A survey of models, speculations, and perspectives, SIAM Rev. 53 (2011) 409–463], thus providing important research perspectives related to new, emerging trends. The presentation addresses the scaling problem corresponding to microscopic (individual-based), mesoscopic (kinetic), and macroscopic (hydrodynamic) modeling and analysis. A multiscale vision guides the overall content of the paper. The critical analysis of the overall content naturally leads to research perspectives. A selection of them is brought to the attention of the interested reader together with hints on how to deal with them.

We introduce a generic framework of dynamical complexity to understand and quantify fluctuations of physiologic time series. In particular, we discuss the importance of applying adaptive data analysis techniques, such as the empirical mode decomposition algorithm, to address the challenges of nonlinearity and nonstationarity that are typically exhibited in biological fluctuations.

We propose a simple and effective method of characterizing complexity of EEG-signals for monitoring the depth of anaesthesia using Higuchi's fractal dimension method. We demonstrate that the proposed method may compete with the widely used BIS monitoring method.

This paper focuses on Herbert A. Simon’s visionary theory of the Artificial World. The artificial world evolves over time as a result of various actions, including interactions with the external world as well as interactions among its internal components. This paper proposes a mathematical theory of the conceptual framework of the artificial world. This goal requires the development of new mathematical tools, inspired in some way by statistical physics and stochastic game theory. The mathematical theory is applied in particular to the study of the dynamics of organizational learning, where cooperation and competition evolve through decomposition and recombination of organizational structures; the effectiveness of the evolutionary changes depends on the dynamic prevalence of cooperative over selfish behaviors, showing features common to the evolution of all living systems.

A complex network approach is combined with time dynamics in order to conduct a space–time analysis applicable to longitudinal studies aimed to characterize the progression of Alzheimer's disease (AD) in individual patients. The network analysis reveals how patient-specific patterns are associated with disease progression, also capturing the widespread effect of local disruptions. This longitudinal study is carried out on resting electroence phalography (EEGs) of seven AD patients. The test is repeated after a three months' period. The proposed methodology allows to extract some averaged information and regularities on the patients' cohort and to quantify concisely the disease evolution. From the functional viewpoint, the progression of AD is shown to be characterized by a loss of connected areas here measured in terms of network parameters (characteristic path length, clustering coefficient, global efficiency, degree of connectivity and connectivity density). The differences found between baseline and at follow-up are statistically significant. Finally, an original topographic multiscale approach is proposed that yields additional results.

Evaluation of the correlation among the activities of various organs is an important research area in physiology. In this paper, we analyzed the correlation between the brain and skin reactions in response to various auditory stimuli. We played three different music (relaxing, pop, and rock music) to eleven subjects (4 M and 7 F, 18–22 years old) and accordingly analyzed the changes in the complexity of Electroencephalogram (EEG) and Galvanic Skin Response (GSR) signals by calculating their fractal exponent and sample entropy. A strong correlation was observed among the alterations of the complexity of GSR and EEG signals in the case of fractal dimension ($r=0.9971$) and also sample entropy ($r=0.8120$), which indicates the correlation between the activities of skin and brain. This analysis method could be further applied to investigate the correlation among the activities of the brain and other organs of the human body.

This paper analyzed the coupling among the reactions of eyes and brain in response to visual stimuli. Since eye movements and electroencephalography (EEG) signals as the features of eye and brain activities have complex patterns, we utilized fractal theory and sample entropy to decode the correlation between them. In the experiment, subjects looked at a dot that moved on different random paths (dynamic visual stimuli) on the screen of a computer in front of them while we recorded their EEG signals and eye movements simultaneously. The results indicated that the changes in the complexity of eye movements and EEG signals are coupled ($r=-0.8043$ in case of fractal dimension and $r=-0.9259$ in case of sample entropy), which reflects the coupling between the brain and eye activities. This analysis could be extended to evaluate the correlation between the activities of other organs versus the brain.

We investigated the correlation among brain and leg muscle activations by analyzing Electroencephalogram (EEG) and Electromyogram (EMG) signals in different conditions. Twelve subjects performed four tasks, including (1) quarter turns, (2) U-turns, (3) bypass obstacles, and (4) repeating quarter turns and U-turns two times. Then, we quantified the alterations of the complexity of these signals by computing the fractal dimension and sample entropy. The results showed that EEG and EMG signals in the case of the first task are more complex than the second task, in which they are more complex than the third task. Furthermore, the brain and muscle signals show the least complexity in the case of the fourth task. Moreover, we found strong correlations in the variations of fractal dimension ($r=0.9835$) and sample entropy ($r=0.9168$) between EEG and EMG signals in various tasks. Therefore, brain and muscle activations are strongly correlated in different tasks. Similar analyses can be conducted in the case of other organs to decode their correlations.

Legs are the contact point of humans during walking. In fact, leg muscles react when we walk in different conditions (such as different speeds and paths). In this research, we analyze how walking path affects leg muscles’ reaction. In fact, we investigate how the complexity of muscle reaction is related to the complexity of path of movement. For this purpose, we employ fractal theory. In the experiment, subjects walk on different paths that have different fractal dimensions and then we calculate the fractal dimension of Electromyography (EMG) signals obtained from both legs. The result of our analysis showed that the complexity of EMG signal increases with the increment of complexity of path of movement. The conducted statistical analysis also supported the result of analysis. The method of analysis used in this research can be further applied to find the relation between complexity of path of movement and other physiological signals of humans such as respiration and Electroencephalography (EEG) signal.

We consider the problem of minimizing the Euclidean distance function on Rⁿ subject to m equality constraints and upper and lower bounds (box constraints). We provide a parametric characterization in R^m of the family of solutions to this problem, thereby showing equivalence with a problem of search in an arrangement of hyperplanes in R^m. We use this characterization and the technique for constructing arrangements due to Edelsbrunner, O'Rourke and Seidel to develop an exact algorithm for the problem. The algorithm is strongly polynomial running in time Θ(n^m) for each fixed m. We further develop an algorithm for the problem which uses the search scheme of Megiddo and Dyer to give a running time of Θ(n) for each fixed m.

The past history of chess and its past variants is examined based on a new measure of game refinement and complexity.The history of chess variants can be divided into the three stages: the origin with moderate complexity and refinement, many diverging variants with high complexity and low refinement, and the modern sophisticated chess with low complexity and high refinement.

In this work, we analyze the appearance of cracking/overturning of a family of self-gravitating spheres. The models under consideration correspond to four anisotropic solutions with like-Tolman IV complexity factor obtained in the context of gravitational decoupling. The models present both cracking/overturning depending on the values of the free parameter involved. A detailed analysis and physical interpretation of the results obtained for each models are shown.

An important category of studies in vision science is related to the analysis of the influence of environmental changes on human eye movement. In this way, scientists analyze human eye movement in different conditions using different methods. An important category of works is devoted to the decoding of eye reaction to color tonality. In this research for the first time, we examined the application of fractal theory for decoding of eye reaction to variations in color intensity of visual stimuli. Three green visual stimuli with different color intensities have been applied to subjects and accordingly the fractal dimension of their eye movements has been analyzed. We also tested the eye movement in non-stimulation condition (rest). Based on the obtained results, increasing the color intensity of visual stimuli caused a lower complexity in subject’s eye movement. We also observed that eye movement is less complex in case of non-stimulation compared to different stimulation conditions. The application of fractal theory in analysis of eye movement can be extended to analyze the effect of other stimulation conditions on eye movement to investigate about the decoding behavior of human eye, which is very important in vision science.

COVID-19 is a pandemic disease, which massively affected human lives in more than 200 countries. Caused by the coronavirus SARS-CoV-2, this acute respiratory illness affects the human lungs and can easily spread from person to person. Since the disease heavily affects human lungs, analyzing the X-ray images of the lungs may prove to be a powerful tool for disease investigation. In this research, we use the information contained within the complex structures of X-ray images between the cases of COVID-19 and other respiratory diseases, whereas the case of healthy lungs is taken as the reference point. To analyze X-ray images, we benefit from the concept of Shannon’s entropy and fractal theory. Shannon’s entropy is directly related to the amount of information contained within the X-ray images in question, whereas fractal theory is used to analyze the complexity of these images. The results, obtained in this study, show that the method of fractal analysis can detect the level of infection among different respiratory diseases and that COVID-19 has the worst effect on the human lungs. In other words, the complexity of X-ray images is proportional to the severity of the respiratory disease. The method of analysis, employed in this study, can be used even further to analyze how COVID-19 progresses in affected patients.