EVENTUAL STABILITY OF IMPULSIVE DIFFERENTIAL SYSTEMS

In this article, criteria of eventual stability are established for impulsive differential systems using piecewise continuous Lyapunov functions. The sufficient conditions that are obtained significantly depend on the moments of impulses. An example is discussed to illustrate the theorem.

RBF-FD Method for Some Dispersive Wave Equations and Their Eventual Periodicity

In this paper,we approximate the solution and also discuss the periodic behavior termed as eventual periodicity of solutions of(IBVPs)for some dispersive wave equations on a bounded domain corresponding to periodic forcing.The constructed numerical scheme is based on radial kernels and local in nature like finite difference method.The temporal variable is executed through RK4 scheme.Due to the local nature and sparse differentiation matrices our numerical scheme efficiently recovers the solution.The results achieved are validated and examined with other methods accessible in the literature.

A Convergence Theorem for Set-valued Eventual Supermartingle

it In this paper, the properties of set-valued Eventual Supermartingle are discussed. The main result is that suppose {Fn, n ≥ 1） Lfc^1（X） be set-valued Eventual Supermartingle, ifsupE（d（0,Fτ）） 〈 ∞, then Fn →KF and SF^1≠φ, here T is the sets of all T bounded stopping times.

Stand Straight and Stand Tall

My grandfather grew up in war-torn Europe.When German soldiers occupied his hometown,the thriving (繁荣的) city of Tarow,Poland,he refused to obey them and eventually joined the Soviet army to fight for his country’s freedom."Stand straight,stand tall,"he told himself.

Relation Between the Eventual Continuity and the E-property for Markov-Feller Semigroups

We investigate some relations between two kinds of semigroup regularities, namely the e-property and the eventual continuity, both of which contribute to the ergodicity for Markov processes on Polish spaces.More precisely, we prove that for Markov-Feller semigroup in discrete time and stochastically continuous MarkovFeller semigroup in continuous time, if there exists an ergodic measure whose support has a nonempty interior,then the e-property is satisfied on the interior of the support. In particular, it implies that, restricted on the support of each ergodic measure, the e-property and the eventual continuity are equivalent for the discrete-time and the stochastically continuous continuous-time Markov-Feller semigroups.

Several Dynamics of Dynamical Systems with the Eventual Shadowing Property

In this article,we provide some sufficient conditions for the dynamical systems with the eventual shadowing property to have positive topological entropy and several equivalent conditions for the dynamical systems with the eventual shadowing property to be mixing.

Quasimonotone random and stochastic functional differential equations with applications

In this paper, we study monotone properties of random and stochastic functional differential equations and their global dynamics. First, we show that random functional differential equations(RFDEs)generate the random dynamical system(RDS) if and only if all the solutions are globally defined, and establish the comparison theorem for RFDEs and the random Riesz representation theorem. These three results lead to the Borel measurability of coefficient functions in the Riesz representation of variational equations for quasimonotone RFDEs, which paves the way following the Smith line to establish eventual strong monotonicity for the RDS under cooperative and irreducible conditions. Then strong comparison principles, strong sublinearity theorems and the existence of random attractors for RFDEs are proved. Finally, criteria are presented for the existence of a unique random equilibrium and its global stability in the universe of all the tempered random closed sets of the positive cone. Applications to typical random or stochastic delay models in monotone dynamical systems,such as biochemical control circuits, cyclic gene models and Hopfield-type neural networks, are given.

Convergence of Nets in Induced I(L)-Topological Spaces with an Application

In this paper we give a characteristic property of convergence of nets in induced I(L)-topological spaces and a simplified proof for the N-compactness being an I(L)-'good extension'.

An efficient and highly available framework of data recency enhancement for eventually consistent data stores

Data items are usually replicated in modem dis- tributed data stores to obtain high performance and avail- ability. However, the availability-consistency and latency- consistency trade-offs exist in data replication, thus system designers intend to choose weak consistency models, such as eventual consistency, which may result in stale reads. Since stale data items may lead to serious application semantic problems, we consider how to increase the probability of data recency which provides a uniform view on recent versions of data items for all clients. In this work, we propose HARP, a framework that can enhance data recency of eventually con- sistent distributed data stores in an efficient and highly avail- able way. Through detecting possible stale reads under fail- ures or not, HARP can perform reread operations to elim- inate stale results only when needed based on our analysis on write/read processes. We also present solutions on how to deal with some practical anomalies in HARP, including de- layed, reordered and dropped messages and clock drift, and show how to extend HARP to multiple datacenters. Finally we implement HARP based on Cassandra, and the experi- ments show that HARP can effectively eliminate stale reads, with a low overhead （less than 6.9%） compared with original eventually consistent Cassandra.

On the Eventual Shadowing Property and Eventually Shadowable Point of Set-valued Dynamical Systems

The authors introduce the concepts of the eventual shadowing property and eventually shadowable point for set-valued dynamical systems and prove that a set-valued dynamical system has the eventual shadowing property if and only if every point in the phase space is eventually shadowable;every chain transitive set-valued dynamical system has either the eventual shadowing property or no eventually shadowable points;and a set-valued dynamical system admits an eventually shadowable point if and only if it admits a minimal eventually shadowable point.Moreover,it is proved that a set-valued dynamical system with the eventual shadowing property is chain mixing if and only if it is mixing and if and only if it has the specification property.

ON THE NONEXISTENCE OF NONTRIVIAL SMALL CYCLES OF THE μ FUNCTION IN 3x+ 1 CONJECTURE

This paper studies the property of the recursive sequences in the 3x ＋ 1 conjecture. The authors introduce the concept of μ function, with which the 3x ＋ 1 conjecture can be transformed into two other conjectures： one is eventually periodic conjecture of the μ function and the other is periodic point conjecture. The authors prove that the 3x ＋ 1 conjecture is equivalent to the two conjectures above. In 2007, J. L. Simons proved the non-existence of nontrivial 2-cycle for the T function. In this paper, the authors prove that the μ function has nol-periodic points for 2 ≤ 1 ≤12. In 2005, J. L. Simons and B. M. M de Weger proved that there is no nontrivial/-cycle for the T function for 1 ≤68, and in this paper, the authors prove that there is no nontrivial l-cycle for the μ function for 2 ≤ 1≤ 102.

Eventual Leader Election with Weak Assumptions on Initial Knowledge,Communication Reliability,and Synchrony

This paper considers the eventual leader election problem in asynchronous message-passing systems where an arbitrary number t of processes can crash（t〈n,where n is the total number of processes）.It considers weak assumptions both on the initial knowledge of the processes and on the network behavior.More precisely,initially,a process knows only its identity and the fact that the process identities are different and totally ordered（it knows neither n nor t）.Two eventual leader election protocols and a lower bound are presented.The first protocol assumes that a process also knows a lower bound α on the number of processes that do not crash.This protocol requires the following behavioral properties from the underlying network：the graph made up of the correct processes and fair lossy links is strongly connected,and there is a correct process connected to（n〈f）-α other correct processes（where f is the actual number of crashes in the considered run） through eventually timely paths（paths made up of correct processes and eventually timely links）.This protocol is not communication-efficient in the sense that each correct process has to send messages forever.The second protocol is communication-efficient：after some time,only the final common leader has to send messages forever.This protocol does not require the processes to know α,but requires stronger properties from the underlying network：each pair of correct processes has to be connected by fair lossy links（one in each direction）,and there is a correct process whose n〈f-1 output links to the rest of correct processes have to be eventually timely.A matching lower bound result shows that any eventual leader election protocol must have runs with this number of eventually timely links,even if all processes know all the processes identities.In addition to being communication-efficient,the second protocol has another noteworthy efficiency property,namely,be the run finite or infinite,all the local variables and message fields have a finite domain in the run.