Asymptotics of quasi-stationary distributions of small noise stochastic dynamical systems in unbounded domains

We consider a collection of Markov chains that model the evolution of multitype biological populations. The state space of the chains is the positive orthant, and the boundary of the orthant is the absorbing state for the Markov chain and represents the extinction states of different population types. We are interested in the long-term behavior of the Markov chain away from extinction, under a small noise scaling. Under this scaling, the trajectory of the Markov process over any compact interval converges in distribution to the solution of an ordinary differential equation (ODE) evolving in the positive orthant. We study the asymptotic behavior of the quasi-stationary distributions (QSD) in this scaling regime. Our main result shows that, under conditions, the limit points of the QSD are supported on the union of interior attractors of the flow determined by the ODE. We also give lower bounds on expected extinction times which scale exponentially with the system size. Results of this type when the deterministic dynamical system obtained under the scaling limit is given by a discrete-time evolution equation and the dynamics are essentially in a compact space (namely, the one-step map is a bounded function) have been studied by Faure and Schreiber (2014). Our results extend these to a setting of an unbounded state space and continuous-time dynamics. The proofs rely on uniform large deviation results for small noise stochastic dynamical systems and methods from the theory of continuous-time dynamical systems. In general, QSD for Markov chains with absorbing states and unbounded state spaces may not exist. We study one basic family of binomial-Poisson models in the positive orthant where one can use Lyapunov function methods to establish existence of QSD and also to argue the tightness of the QSD of the scaled sequence of Markov chains. The results from the first part are then used to characterize the support of limit points of this sequence of QSD.

Fluid Limits for Multiclass Many-Server Queues with General Reneging Distributions and Head-of-the-Line Scheduling

We describe a fluid model with time-varying input that approximates a multiclass many-server queue with general reneging distribution and multiple customer classes (specifically, the multiclass G/GI/N+GI queue). The system dynamics depend on the policy, which is a rule for determining when to serve a given customer class. The class of admissible control policies are those that are head-of-the-line (HL) and nonanticipating. For a sequence of many-server queues operating under admissible HL control policies and satisfying some mild asymptotic conditions, we establish a tightness result for the sequence of fluid scaled queue state descriptors and associated processes and show that limit points of such sequences are fluid model solutions almost surely. The tightness result together with the characterization of distributional limit points as fluid model solutions almost surely provides a foundation for the analysis of particular HL control policies of interest. We leverage these results to analyze a set of admissible HL control policies that we introduce, called weighted random buffer selection (WRBS), and an associated WRBS fluid model that allows multiple classes to be partially served in the fluid limit (which is in contrast to previously analyzed static priority policies).

Geometrically nonlinear analysis of the stability of the stiffened plate taking into account the interaction of eigenforms of buckling

The aims of this work are a detailed consideration in a geometrically nonlinear formulation of the stages of the equilibrium behavior of a compressed stiffened plate, taking into account the interaction of the general form of buckling and local forms of wave formation in the plate or in the reinforcing ribs, comparison of the results of the semi-analytical solution of the system of nonlinear equations with the results of the numerical solution on the Patran-Nastran FEM complex of the problem of subcritical and postcritical equilibrium of a compressed stiffened plate. Methods. Geometrically-nonlinear analysis of displacement fields, deformations and stresses, calculation of eigenforms of buckling and construction of bifurcation solutions and solutions for equilibrium curves with limit points depending on the initial imperfections. An original method is proposed for determining critical states and obtaining bilateral estimates of critical loads at limiting points. Results. An algorithm for studying the equilibrium states of a stiffened plate near critical points is described in detail and illustrated by examples, using the first nonlinear (cubic terms) terms of the potential energy expansion, the coordinates of bifurcation points and limit points, as well as the corresponding values of critical loads. The curves of the critical load sensitivity are plotted depending on the value of the initial imperfections of the total deflection. Equilibrium curves with characteristic bifurcation points of local wave formation are constructed using a numerical solution. For the case of action of two initial imperfections, an algorithm is proposed for obtaining two-sided estimates of critical loads at limiting points.

New Results on I 2 -Statistically Limit Points and I 2 -Statistically Cluster Points of Sequences of Fuzzy Numbers

In this paper, some existing theories on convergence of fuzzy number sequences are extended to I 2 -statistical convergence of fuzzy number sequence. Also, we broaden the notions of I -statistical limit points and I -statistical cluster points of a sequence of fuzzy numbers to I 2 -statistical limit points and I 2 -statistical cluster points of a double sequence of fuzzy numbers. Also, the researchers focus on important fundamental features of the set of all I 2 -statistical cluster points and the set of all I 2 -statistical limit points of a double sequence of fuzzy numbers and examine the relationship between them.

Further aspects of I K-convergence in topological spaces

In this paper, we obtain some results on the relationships between different ideal convergence modes namely, I K, I K∗ , I, K, I ∪ K and (I ∪K) ∗ . We introduce a topological space namely I K-sequential space and show that the class of I K-sequential spaces contain the sequential spaces. Further I K-notions of cluster points and limit points of a function are also introduced here. For a given sequence in a topological space X, we characterize the set of I K-cluster points of the sequence as closed subsets of X.

SOME PROPERTIES OF EQUIDISTRIBUTED AND WELL DISTRIBUTED SEQUENCES

Many authors studied properties related to distribution and summability of sequences of real numbers. In these studies, different types of limit points of a sequence were introduced and studied including statistical and uniform statistical cluster points of a sequence. In this paper, we aim to prove some new results about the nature of different types of limit points, this time connected to equidistributed and well distributed sequences.