• We can still talk about reversibility if the chain is only irreducible and positive-recurrent. • If one assumes that the chain is in stationary distribution from the start, then the backwards chain Xn, Xn−1, . . . has the same transition probabilities as the original chain, hence the name “reversible”. • If π∗ satisfies the detailed balance equation, then π∗ = π∗P . • The reciprocal statement...

‎in this paper‎, ‎we obtain exact solutions involving parameters of some nonlinear pdes in mathmatical physics; namely the one-‎dimensional modified complex ginzburg-landau equation by using the $ (g^{'}/g) $ expansion method‎, homogeneous balance method, extended f-expansion method‎. ‎by ‎using homogeneous balance principle and the extended f-expansion, more periodic wave solutions expres...

Since the 1940s there has been an interest in the question of why social networks often give rise to two antagonistic factions. Recently a dynamical model of how and why such a balance might occur was developed. This note provides an introduction to the notion of social balance and a new (and simplified) analysis of that model. This new analysis allows us to choose general initial conditions, a...

Based on the recently introduced fractional Taylor’s formula, a fractional heat conduction constitutive equation is formulated by expanding the single-phase lag model using the fractional Taylor’s formula. Combining with the energy balance equation, the derived fractional heat conduction equation has been shown to be capable of modeling diffusion-to-Thermal wave behavior of heat propagation by ...

Aggregation kinetics are often described by the population balance (or Smoluchowski coagulation) equation. The population balance equation is a mean-field equation for an aggregation process. An upper critical dimension d, calculated for a given aggregation frequency has been used by other investigators to characterize the validity of this mean-field equation. It is shown that the upper critica...

Particulate processes are often encountered in industry. The polydispersity of some key properties characterizes particulate processes and makes them ill-suited to be modeled within the framework of conventional conservation equations only. In the mid 60s a tool called Population Balance Equation was developed to quantify the dynamics of particulate processes. This paper aims to give an introdu...