Suppose that an equilibrium is asymptotically stable when external inputs vanish. Then, every bounded trajectory which corresponds to a control which approaches zero and which lies in the domain of attraction of the unforced system, must also converge to the equilibrium. This “well-known” but hard-to-cite fact is proved and slightly generalized here.

We obtain a strong renewal theorem with infinite mean beyond regular variation, when the underlying distribution belongs to domain of geometric partial attraction semistable law index $$\alpha \in (1/2,1]$$ . In process we local limit theorems for both finite and mean, that is, whole range (0,2)$$ also derive asymptotics function (0,1]$$

In a number of applications, an admissible domain of the perturbed motion is associated with the domain of attraction of a stable equilibrium of the unperturbed system. If noise is weak, escape from the reference domain is a rare event associated with large deviations in the system. Despite the well developed large deviations theory, estimation of the statistical parameters for the multidimensi...

The Casimir attraction can significantly interfere the physical response of nanoactuators. The intensity of the Casimir force depends on the geometries of interacting bodies. The present paper is dedicated to model the influence of the Casimir attraction on the electrostatic stability of nanoactuators made of cylindrical conductive nanowire/nanotube. An asymptotic solution, based on path-integr...

Higher order Markov chains are useful tools for modeling local dependence in time series. In this dissertation, we first deal with extremal behavior of a higher order (k-th order, k ~ 1) stationary Markov chain {Xn }. Assuming that the joint distribution of k + 1 consecutive variables belongs to the domain of attraction of some multivariate extreme value distribution, the limiting distributions...

In a mathematical model involving a set of ordinary differential equations, a particular solution of the system is completely determined by the initial condition. When the omega limit set reduces to a point, the solution settles at steady state. The possible steady states of the system are completely determined by its parameters. However, it is possible that more steady states can originate fro...

Growing evidence suggests that sensory neuron synapses not merely pass, but actively encode sensory information and convey it to the central nervous system. The chemosensory preferences of Caenorhabditis elegans, as manifested in the direction of chemotaxis, are reversibly regulated by prior experience at the level of sensory neurons; the attractive drive is promoted by diacylglycerol (DAG) sig...