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We define a contact metric structure on the manifold corresponding to a second order ordinary differential equation $d^2y/dx^2=f(x,y,y')$ and show that the contact metric structure is Sasakian if and only if the 1-form $\frac{1}{2}(dp-fdx)$ defines a Poisson structure. We consider a Hamiltonian dynamical system defined by this Poisson structure and show that the Hamiltonian vector field, which is a multiple of the Reeb vector field, admits a compatible bi-Hamiltonian structure for which $f$ can be regarded as a Hamiltonian function. As a particular case, we give a compatible bi-Hamiltonian structure of the Reeb vector field such that the structure equations correspond to the Maurer-Cartan equations of an invariant coframe on the Heisenberg group and the independent variable plays the role of a Hamiltonian function. We also show that the first Chern class of the normal bundle of an integral curve of a multiple of the Reeb vector field vanishes iff $f_x+ff_p = \Psi (x)$ for some $\Psi$.

Multilayer networks represent multiple types of connections between the same set of nodes. Clearly, a multilayer description of a system adds value only if the multiplex does not merely consist of independent layers, i.e. if the inter-layer overlap is nontrivial. On real-world multiplexes, it is expected that the observed overlap may partly result from spurious correlations arising from the heterogeneity of nodes and partly from true interdependencies. However, no rigorous way to disentangle these two effects has been developed. In this paper we introduce an unbiased maximum-entropy model of multiplexes with controllable node degrees and controllable overlap. The model can be mapped to a generalized Ising model where the combination of node heterogeneity and inter-layer coupling leads to the possibility of local phase transitions. In particular, we find that an increased heterogeneity in the network results in different critical points for different pairs of nodes, which in turn leads to local phase transitions that may ultimately increase the overlap. The model allows us to quantify how the overlap can be increased by either increasing the heterogeneity of the network (spurious correlation) or the strength of the inter-layer coupling (true correlation), thereby disentangling the two effects. As an application, we show that the empirical overlap in the International Trade Multiplex is not merely a spurious result of the correlation between node degrees across different layers, but requires a non-zero inter-layer coupling in its modeling.

Nowadays, in our globalized world,the local and intercountry movements of population have been increased. This situation makes it important for host countries to do right predictions for the future population of their native people as well as immigrant people. The knowledge of the attained number of accumulated population is necessary for future planning, concerning to education,health, job, housing, safety requirements, etc. In this work, for updating historically well known formulas of population dynamics of a community are revisited in the framework of compound growth and fractional calculus to get more realistic relations. Within this context, for a time t, the population evolution of a society which owns two different components is calculated. Concomitant relations have been developed to provide a comparison between the native population and the immigrant population that come into existence where at each time interval a colonial population is joined. Eventually at time t, the case where the native population becomes equal to immigrant population is discussed. Moreover, the differential equation of a probability function for the equilibrium state of accumulative population is obtained. It is seen that the equilibrium solution can be written as a series in terms of Bernoulli numbers. In the calculation of population dynamics, Mittag- Leffler(M-L) function replaces the exponential function., Comment: 21 pages,4figures,research manuscript, biodynamics, cumulative approaches, fractional calculus

In this study the general formula for differential and integral operations of fractional calculus via fractal operators by the method of cumulative diminution and cumulative growth is obtained. The under lying mechanism in the success of traditional fractional calculus for describing complex systems is uncovered. The connection between complex physics with fractional differentiation and integration operations is established., Comment: 18 pages