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The coefficient of variation, which measures the variability of a distribution from its mean, is not uniquely defined in the multidimensional case, and so is the multidimensional Gini index, which measures the inequality of a distribution in terms of the mean differences among its observations. In this paper, we connect these two notions of sparsity, and propose a multidimensional coefficient of variation based on a multidimensional Gini index. We demonstrate that the proposed coefficient possesses the properties of the univariate coefficient of variation. We also show its connection with the Voinov-Nikulin coefficient of variation, and compare it with the other multivariate coefficients available in the literature., Comment: 23 pages, 4 figures

We study the main properties of the solution of a Fokker-Planck equation characterized by a variable diffusion coefficient and a polynomial superlinear drift, modeling the formation of consensus in a large interacting system of individuals. The Fokker-Planck equation is derived from the kinetic description of the dynamics of a quantum particle system, and in presence of a high nonlinearity in the drift operator, mimicking the effects of the mass in the alignment forces, allows for steady states similar to a Bose-Einstein condensate. The analysis shows that the regularity of the solution is strongly linked to the degree of nonlinearity in the drift, and that finite-time blow-up of the solution can occur when the degree of nonlinearity is sufficiently high. However, the presence of diffusion prevents the solution from forming condensation after the blow-up time.

Measuring distances in a multidimensional setting is a challenging problem, which appears in many fields of science and engineering. In this paper, to measure the distance between two multivariate distributions, we introduce a new measure of discrepancy which is scale invariant and which, in the case of two independent copies of the same distribution, and after normalization, coincides with the scaling invariant multidimensional version of the Gini index recently proposed in [34]. A byproduct of the analysis is an easy-to-handle discrepancy metric, obtained by application of the theory to a pair of Gaussian multidimensional densities. The obtained metric does improve the standard metrics, based on the mean squared error, as it is scale invariant. The importance of this theoretical finding is illustrated by means of a real problem that concerns measuring the importance of Environmental, Social and Governance factors for the growth of small and medium enterprises.

Measuring the degree of inequality expressed by a multivariate statistical distribution is a challenging problem, which appears in many fields of science and engineering. In this paper, we propose to extend the well known univariate Gini coefficient to multivariate distributions, by maintaining most of its properties. Our extension is based on the application of whitening processes that possess the property of scale stability., Comment: 12 pages, 3 tables

In this work, we define a class of models to understand the impact of population size on opinion formation dynamics, a phenomenon usually related to group conformity. To this end, we introduce a new kinetic model in which the interaction frequency is weighted by the kinetic density. In the quasi-invariant regime, this model reduces to a Kaniadakis-Quarati-type equation with nonlinear drift, originally introduced for the dynamics of bosons in a spatially homogeneous setting. From the obtained PDE for the evolution of the opinion density, we determine the regime of parameters for which a critical mass exists and triggers blow-up of the solution. Therefore, the model is capable of describing strong conformity phenomena in cases where the total density of individuals holding a given opinion exceeds a fixed critical size. In the final part, several numerical experiments demonstrate the features of the introduced class of models and the related consensus effects.

This paper presents a critical analysis of the literature and perspective research ideas for modeling the epidemics caused by the SARS-CoV-2 virus. It goes beyond deterministic population dynamics to consider several key complexity features of the system under consideration. In particular, the multiscale features of the dynamics from contagion to the subsequent dynamics of competition between the immune system and the proliferating virus. Other topics addressed in this work include the propagation of epidemics in a territory, taking into account local transportation networks, the heterogeneity of the population, and the study of social and economic problems in populations involved in the spread of epidemics. The overall content aims to show how new mathematical tools can be developed to address the above topics and how mathematical models and simulations can contribute to the decision making of crisis managers.

Inequality measures are quantitative measures that take values in the unit interval, with a zero value characterizing perfect equality. Although originally proposed to measure economic inequalities, they can be applied to several other situations, in which one is interested in the mutual variability between a set of observations, rather than in their deviations from the mean. While unidimensional measures of inequality, such as the Gini index, are widely known and employed, multidimensional measures, such as Lorenz Zonoids, are difficult to interpret and computationally expensive and, for these reasons, are not much well known. To overcome the problem, in this paper we propose a new scaling invariant multidimensional inequality index, based on the Fourier transform, which exhibits a number of interesting properties, and whose application to the multidimensional case is rather straightforward to calculate and interpret., Comment: arXiv admin note: text overlap with arXiv:2310.20483

The temporal evolution of a contagious viral disease is modelled as the dynamic progression of different classes of population with individuals interacting pairwise. This interaction follows a binary mechanism typical of kinetic theory, wherein agents aim to improve their condition with respect to a mutual utility target. To this end, we introduce kinetic equations of Boltzmann-type to describe the time evolution of the probability distributions of the multi-agent system. The interactions between agents are defined using principles from price theory, specifically employing Cobb-Douglas utility functions for binary exchange and the Edgeworth box to depict the common exchange area where utility increases for both agents. Several numerical experiments presented in the paper highlight the significance of this mechanism in driving the phenomenon toward endemicity.

Measuring inequalities in a multidimensional framework is a challenging problem which is common to most field of science and engineering. Nevertheless, despite the enormous amount of researches illustrating the fields of application of inequality indices, and of the Gini index in particular, very few consider the case of a multidimensional variable. In this paper, we consider in some details a new inequality index, based on the Fourier transform, that can be fruitfully applied to measure the degree of inhomogeneity of multivariate probability distributions. This index exhibits a number of interesting properties that make it very promising in quantifying the degree of inequality in data sets of complex and multifaceted social phenomena.

We propose a kinetic model for understanding the link between opinion formation phenomena and epidemic dynamics. The recent pandemic has brought to light that vaccine hesitancy can present different phases and temporal and spatial variations, presumably due to the different social features of individuals. The emergence of patterns in societal reactions permits to design and predict the trends of a pandemic. This suggests that the problem of vaccine hesitancy can be described in mathematical terms, by suitably coupling a kinetic compartmental model for the spreading of an infectious disease with the evolution of the personal opinion of individuals, in the presence of leaders. The resulting model makes it possible to predict the collective compliance with vaccination campaigns as the pandemic evolves and to highlight the best strategy to set up for maximizing the vaccination coverage. We conduct numerical investigations which confirm the ability of the model to describe different phenomena related to the spread of an epidemic.

We study the large time behavior of a system of interacting agents modeling the relaxation of a large swarm of robots, whose task is to uniformly cover a portion of the domain by communicating with each other in terms of their distance. To this end, we generalize a related result for a Fokker-Planck-type model with a nonlocal discontinuous drift and constant diffusion, recently introduced by three of the authors, of which the steady distribution is explicitly computable. For this new nonlocal Fokker-Planck equation, existence, uniqueness and positivity of a global solution are proven, together with precise equilibration rates of the solution towards its quasi-stationary distribution. Numerical experiments are designed to verify the theoretical findings and explore possible extensions to more complex scenarios.

We obtain equilibration rates for a one-dimensional nonlocal Fokker-Planck equation with time-dependent diffusion coefficient and drift, modeling the relaxation of a large swarm of robots, feeling each other in terms of their distance, towards the steady profile characterized by uniform spreading over a finite interval of the line. The result follows by combining entropy methods for quantifying the decay of the solution towards its quasi-stationary distribution, with the properties of the quasi-stationary profile.

Owing to the analogies between the problem of wealth redistribution with taxation in a multi-agent society, we introduce and discuss a kinetic model describing the statistical distributions in time of the sizes of groups of biological systems with prey-predator dynamic. While the evolution of the mean values is shown to be driven by a classical Lotka-Volterra system of differential equations, it is shown that the time evolution of the probability distributions of the size of groups of the two interacting species is heavily dependent both on a kinetic redistribution operator and the degree of randomness present in the system. Numerical experiments are given to clarify the time-behavior of the distributions of groups of the species.

The Luria--Delbr\"uck mutation model is a cornerstone of evolution theory and has been mathematically formulated in a number of ways. In this paper we illustrate how this model of mutation rates can be derived by means of classical statistical mechanics tools, in particular by modeling the phenomenon resorting to methodologies borrowed from classical kinetic theory of rarefied gases. The aim is to construct a linear kinetic model that can reproduce the Luria--Delbr\"uck distribution starting from the elementary interactions that qualitatively and quantitatively describe the variation of mutated cells. The kinetic description is easily adaptable to different situations and makes it possible to clearly identify the differences between the elementary variations leading to the formulations of Luria--Delbr\"uck, Lea--Coulson, and Kendall, respectively. The kinetic approach additionally emphasizes basic principles which not only help to unify existing results but also allow for useful extensions.

Originally developed for measuring the heterogeneity of wealth measures, inequality indices are quantitative scores that take values in the unit interval, with the zero score characterizing perfect equality. In this paper, we draw attention to a new inequality index, based on the Fourier transform, which exhibits a number of interesting properties that make it very promising in applications. As a by-product, it is shown that other inequality measures, including Gini and Pietra indices can be fruitfully expressed in terms of the Fourier transform, which allows to enlighten properties in a new and simple way.

In this paper, by resorting to classical methods of statistical mechanics, we build a kinetic model able to reproduce the observed statistical weight distribution of many diverse species. The kinetic description of the time variations of the weight distribution is based on elementary interactions that describe in a qualitative and quantitative way successive evolutionary updates, and determine explicit equilibrium distributions. Numerical fittings on mammalian eutherians of the order Chiroptera population illustrates the effectiveness of the approach.

In this paper we study a novel Fokker-Planck-type model that is designed to mimic manufacturing processes through the dynamics characterizing a large set of agents. In particular, we describe a many-agent system interacting with a target domain in such a way that each agent/particle is attracted by the center of mass of the target domain with the aim to uniformly cover this zone. To this end, we first introduce a mean-field model with discontinuous flux whose large time behavior is such that the steady state is globally continuous and uniform over a connected portion of the domain. We prove that a diffusion coefficient that guarantees that a given portion of mass enters in the target domain exists and that it is unique. Furthermore, convergence to equilibrium in 1D is provided through a reformulation of the initial problem involving a nonconstant diffusion function. The extension to 2D is explored numerically by means of recently introduced structure preserving methods for Fokker-Planck equations.

The spreading of Covid-19 pandemic has highlighted the close link between economics and health in the context of emergency management. A widespread vaccination campaign is considered the main tool to contain the economic consequences. This paper will focus, at the level of wealth distribution modelling, on the economic improvements induced by the vaccination campaign in terms of its effectiveness rate. The economic trend during the pandemic is evaluated resorting to a mathematical model joining a classical compartmental model including vaccinated individuals with a kinetic model of wealth distribution based on binary wealth exchanges. The interplay between wealth exchanges and the progress of the infectious disease is realized by assuming on the one hand that individuals in different compartments act differently in the economic process and on the other hand that the epidemic affects risk in economic transactions. Using the mathematical tools of kinetic theory, it is possible to identify the equilibrium states of the system and the formation of inequalities due to the pandemic in the wealth distribution of the population. Numerical experiments highlight the importance of the vaccination campaign and its positive effects in reducing economic inequalities in the multi-agent society., Comment: 22 pages, 20 figures

We introduce and discuss a system of one-dimensional kinetic equations describing the influence of higher education in the social stratification of a multi-agent society. The system is obtained by coupling a model for knowledge formation with a kinetic description of the social climbing in which the parameters characterizing the elementary interactions leading to the formation of a social elite are assumed to depend on the degree of knowledge/education of the agents. In addition, we discuss the case in which the education level of an individual is function of the position occupied in the social ranking. With this last assumption we obtain a fully coupled model in which knowledge and social status influence each other. In the last part, we provide several numerical experiments highlighting the role of education in reducing social inequalities and in promoting social mobility.

In this survey we report some recent results in the mathematical modeling of epidemic phenomena through the use of kinetic equations. We initially consider models of interaction between agents in which social characteristics play a key role in the spread of an epidemic, such as the age of individuals, the number of social contacts, and their economic wealth. Subsequently, for such models, we discuss the possibility of containing the epidemic through an appropriate optimal control formulation based on the policy maker's perception of the progress of the epidemic. The role of uncertainty in the data is also discussed and addressed. Finally, the kinetic modeling is extended to spatially dependent settings using multiscale transport models that can characterize the impact of movement dynamics on epidemic advancement on both one-dimensional networks and realistic two-dimensional geographic settings.

We study one-dimensional functional inequalities of the type of Poincar\'e, logarithmic Sobolev and Wirtinger, with weight, for probability densities with polynomial tails. As main examples, we obtain sharp inequalities satisfied by inverse Gamma densities, taking values on $R_+$, and Cauchy-type densities, taking values on $R$. In this last case, we improve the result obtained by Bobkov and Ledoux in 2009 by introducing a better weight function in the logarithmic Sobolev inequality. The results are obtained by resorting to Fokker-Planck type equations which possess these densities as steady states.

We introduce a class of one-dimensional linear kinetic equations of Boltzmann and Fokker--Planck type, describing the dynamics of individuals of a multi-agent society questing for high status in the social hierarchy. At the Boltzmann level, the microscopic variation of the status of agents around a universal desired target, is built up introducing as main criterion for the change of status a suitable value function in the spirit of the prospect theory of Kahneman and Twersky. In the asymptotics of grazing interactions, the solution density of the Boltzmann type kinetic equation is shown to converge towards the solution of a Fokker--Planck type equation with variable coefficients of diffusion and drift, characterized by the mathematical properties of the value function. The steady states of the statistical distribution of the social status predicted by the Fokker--Planck equations belong to the class of Amoroso distributions with Pareto tails, which correspond to the emergence of a \emph{social elite}. The details of the microscopic kinetic interaction allow to clarify the meaning of the various parameters characterizing the resulting equilibrium. Numerical results then show that the steady state of the underlying kinetic equation is close to Amoroso distribution even in an intermediate regime in which interactions are not grazing.

We investigate properties of some extensions of a class of Fourier-based probability metrics, originally introduced to study convergence to equilibrium for the solution to the spatially homogeneous Boltzmann equation. At difference with the original one, the new Fourier-based metrics are well-defined also for probability distributions with different centers of mass, and for discrete probability measures supported over a regular grid. Among other properties, it is shown that, in the discrete setting, these new Fourier-based metrics are equivalent either to the Euclidean-Wasserstein distance $W_2$, or to the Kantorovich-Wasserstein distance $W_1$, with explicit constants of equivalence. Numerical results then show that in benchmark problems of image processing, Fourier metrics provide a better runtime with respect to Wasserstein ones., Comment: 18 pages, 2 figures, 1 table

Unlike the classical kinetic theory of rarefied gases, where microscopic interactions among gas molecules are described as binary collisions, the modelling of socio-economic phenomena in a multi-agent system naturally requires to consider, in various situations, multiple interactions among the individuals. In this paper, we collect and discuss some examples related to economic and gambling activities. In particular, we focus on a linearisation strategy of the multiple interactions, which greatly simplifies the kinetic description of such systems while maintaining all their essential aggregate features, including the equilibrium distributions., Comment: 21 pages, 6 figures

We investigate the relaxation to equilibrium of the solution of a class of one-dimensional linear Fokker--Planck type equations that have been recently considered in connection with the study of addiction phenomena in a system of individuals. The steady states of these equations belong to the class of generalized Gamma densities. As a by-product of the relaxation analysis, we prove new weighted Poincar\'e and logarithmic Sobolev type inequalities for this class of densities.

We study the evolution in time of the statistical distribution of some addiction phenomena in a system of individuals. The kinetic approach leads to build up a novel class of Fokker--Planck equations describing relaxation of the probability density solution towards a generalized Gamma density. A qualitative analysis reveals that the relaxation process is very stable, and does not depend on the parameters that measure the main microscopic features of the addiction phenomenon., Comment: 16 pages

We introduce a class of new one-dimensional linear Fokker--Planck type equations describing the evolution in time of the wealth in a multi-agent society. The equations are obtained, via a standard limiting procedure, by introducing an economically relevant variant to the kinetic model introduced in 2005 by Cordier, Pareschi and Toscani according to previous studies by Bouchaud and M\'ezard. The steady state of wealth predicted by these new Fokker--Planck equations remains unchanged with respect to the steady state of the original Fokker--Planck equation. However, unlike the original equation, it is proven by a new logarithmic Sobolev inequality with weight and classical entropy methods that the solution converges exponentially fast to equilibrium.

We study the rate of convergence to equilibrium of the solution of a Fokker--Planck type equation introduced by one of the authors in 2006 to describe opinion formation in a multi-agent system. The main feature of this Fokker--Planck equation is the presence of a variable diffusion coefficient and boundaries, which introduce new challenging mathematical problems in the study of its long-time behavior.

In recent years, there has been a proliferation of online gambling sites, which made gambling more accessible with a consequent rise in related problems, such as addiction. Hence, the analysis of the gambling behaviour at both the individual and the aggregate levels has become the object of several investigations. In this paper, resorting to classical methods of the kinetic theory, we describe the behaviour of a multi-agent system of gamblers participating in lottery-type games on a virtual-item gambling market. The comparison with previous, often empirical, results highlights the ability of the kinetic approach to explain how the simple microscopic rules of a gambling-type game produce complex collective trends, which might be difficult to interpret precisely by looking only at the available data.

In most countries, alcohol consumption distributions have been shown to possess universal features. Their unimodal right-skewed shape is usually modeled in terms of the Lognormal distribution, which is easy to fit, test, and modify. However, empirical distributions often deviate considerably from the Lognormal model, and both Gamma and Weibull distributions appear to better describe the survey data. In this paper we explain the appearance of these distributions by means of classical methods of kinetic theory of multi-agent systems. The microscopic variation of alcohol consumption of agents around a universal \emph{social} accepted value of consumption, is built up introducing as main criterion for consumption a suitable value function in the spirit of the prospect theory of Kahneman and Twersky. The mathematical properties of the value function then determine the unique macroscopic equilibrium which results to be a generalized Gamma distribution. The modeling of the microscopic kinetic interaction allows to clarify the meaning of the various parameters characterizing the generalized Gamma equilibrium., Comment: arXiv admin note: text overlap with arXiv:1809.01365

In this paper, we discuss the passage to hydrodynamic equations for kinetic models of opinion formation. The considered kinetic models feature an opinion density depending on an additional microscopic variable, identified with the personal preference. This variable describes an opinion-driven polarisation process, leading finally to a choice among some possible options, as it happens e.g. in referendums or elections. Like in the kinetic theory of rarefied gases, the derivation of hydrodynamic equations is essentially based on the computation of the local equilibrium distribution of the opinions from the underlying kinetic model. Several numerical examples validate the resulting model, shedding light on the crucial role played by the distinction between opinion and preference formation on the choice processes in multi-agent societies., Comment: 30 pages, 15 figures

Gagliardo--Nirenberg inequalities are interpolation inequalities which were proved independently by Gagliardo and Nirenberg in the late fifties. In recent years, their connections with theoretic aspects of information theory and nonlinear diffusion equations allowed to obtain some of them in optimal form, by recovering both the sharp constants and the explicit form of the optimizers. In this note, at the light of these recent researches, we review the main connections between Shannon-type entropies, diffusion equations and a class of these inequalities.

In recent years it has been increasing evidence that lognormal distributions are widespread in physical and biological sciences, as well as in various phenomena of economics and social sciences. In social sciences, the appearance of lognormal distribution has been noticed, among others, when looking at body weight, and at women's age at first marriage. Likewise, in economics, lognormal distribution appears when looking at consumption in a western society, at call-center service times, and others. The common feature of these situations, which describe the distribution of a certain people's hallmark, is the presence of a desired target to be reached by repeated choices. In this paper we discuss a possible explanation of lognormal distribution forming in human activities by resorting to classical methods of statistical mechanics of multi-agent systems. The microscopic variation of the hallmark around its target value, leading to a linear Fokker--Planck type equation with lognormal equilibrium density, is built up introducing as main criterion for decision a suitable value function in the spirit of the prospect theory of Kahneman and Twersky.

We present a kinetic approach to the formation of urban agglomerations which is based on simple rules of immigration and emigration. In most cases, the Boltzmann-type kinetic description allows to obtain, within an asymptotic procedure, a Fokker--Planck equation with variable coefficients of diffusion and drift, which describes the evolution in time of some probability density of the city size. It is shown that, in dependence of the microscopic rules of migration, the equilibrium density can follow both a power law for large values of the size variable, which contains as particular case a Zipf's law behavior, and a lognormal law for middle and low values of the size variable. In particular, connections between the value of Pareto index of the power law at equilibrium and the disposal of the population to emigration are outlined. The theoretical findings are tested with recent data of the populations of Italy and Switzerland.

We introduce and discuss optimal control strategies for kinetic models for wealth distribution in a simple market economy, acting to minimize the variance of the wealth density among the population. Our analysis is based on a finite time horizon approximation, or model predictive control, of the corresponding control problem for the microscopic agents' dynamic and results in an alternative theoretical approach to the taxation and redistribution policy at a global level. It is shown that in general the control is able to modify the Pareto index of the stationary solution of the corresponding Boltzmann kinetic equation, and that this modification can be exactly quantified. Connections between previous Fokker-Planck based models and taxation-redistribution policies and the present approach are also discussed., Comment: 21 pages, 6 figures

Call centers are service networks in which agents provide telephone-based services. An important part of call center operations is represented by service durations. In recent statistical analysis of real data, it has been noticed that the distribution of service times reveals a remarkable fit to the lognormal distribution. In this paper we discuss a possible source of this behavior by resorting to classical methods of statistical mechanics of multi-agent systems. The microscopic service time variation leading to a linear kinetic equation with lognormal equilibrium density is built up introducing as main criterion for decision a suitable value function in the spirit of the prospect theory of Kahneman and Twersky.

We consider here a Fokker--Planck equation with variable coefficient of diffusion which appears in the modeling of the wealth distribution in a multi-agent society. At difference with previous studies, to describe a society in which agents can have debts, we allow the wealth variable to be negative. It is shown that, even starting with debts, if the initial mean wealth is assumed positive, the solution of the Fokker--Planck equation is such that debts are absorbed in time, and a unique equilibrium density located in the positive part of the real axis will be reached.

We introduce a system of kinetic equations describing an exchange market consisting of two populations of agents (dealers and speculators) expressing the same preferences for two goods, but applying different strategies in their exchanges. We describe the trading of the goods by means of some fundamental rules in price theory, in particular by using Cobb-Douglas utility functions for the exchange. The strategy of the speculators is to recover maximal utility from the trade by suitably acting on the percentage of goods which are exchanged. This microscopic description leads to a system of linear Boltzmann-type equations for the probability distributions of the goods on the two populations, in which the post-interaction variables depend from the pre-interaction ones in terms of the mean quantities of the goods present in the market. In this case, it is shown analytically that the strategy of the speculators can drive the price of the two goods towards a zone in which there is a marked utility for their group. Also, the general system of nonlinear kinetic equations of Boltzmann type for the probability distributions of the goods on the two populations is described in details. Numerical experiments then show how the policy of speculators can modify the final price of goods in this nonlinear setting.

We survey some recent developments on the mathematical modeling of opinion dynamics. After an introduction on opinion modeling through interacting multi-agent systems described by partial differential equations of kinetic type, we focus our attention on two major advancements: optimal control of opinion formation and influence of additional social aspects, like conviction and number of connections in social networks, which modify the agents' role in the opinion exchange process., Comment: Chapter of the Birkhauser-Springer book (to appear) by N. Bellomo, P. Degond, and E. Tadmor Eds., "Active Particles Volume 1, Theory, Methods, and Applications"

We introduce and analyze a linear kinetic model that describes the evolution of the probability density of the number of firms in a society, in which the microscopic rate of change obeys to the so-called law of proportional effect proposed by Gibrat. Despite its apparent simplicity, the possible mean field limits of the kinetic model are varied. In some cases, the asymptotic limit can be described by a first-order partial differential equation. In other cases, the mean field equation is a linear diffusion with a non constant diffusion coefficient that models also the geometric Brownian motion and can be studied analytically. In this case, it is shown that the large-time behavior of the solution is represented, for a large class of initial data, by a lognormal distribution with constant mean value and variance increasing exponentially in time at a precise rate. The relationship between the kinetic and the diffusion models allow to introduce an easy-to- implement expression for computing the Fourier transform of the lognormal distribution.

We consider the central limit theorem for stable laws in the case of the standardized sum of independent and identically distributed random variables with regular probability density function. By showing decay of different entropy functionals along the sequence we prove convergence with explicit rate in various norms to a L\'evy centered density of parameter $\lambda >1$ . This introduces a new information-theoretic approach to the central limit theorem for stable laws, in which the main argument is shown to be the relative fractional Fisher information, recently introduced by the same author (cf. arXiv 1504.07057). In particular, it is proven that, with respect to the relative fractional Fisher information, the L\'evy density satisfies an analogous of the logarithmic Sobolev inequality, which allows to pass from the monotonicity and decay to zero of the relative fractional Fisher information in the standardized sum to the decay to zero in relative entropy with an explicit decay rate., Comment: arXiv admin note: substantial text overlap with arXiv:1504.07057