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In this paper, the almost surely globally asymptotical stability and the almost surely exponential stability for dual switching continuous-time nonlinear system are investigated by using the probability analysis method and stochastic Multi-Lyapunov function, respectively. Different from the previous research results, it is the first time that dual switching continuous-time nonlinear system is used as a study object to investigate its switching stability. Then, the probability analysis method is used to overcome the deficiency that the ergodicity no longer holds due to the variable transition rate of Markov process. Some sufficient conditions for the globally asymptomatic stability almost surely and the almost surely exponential stability of dual switching continuous-time nonlinear system are given under the pre-designed deterministic switching strategy. Finally, two numerical examples are provided to verify the effectiveness of the proposed approach.

Convergence of real sequences, as well as complex sequences are studied by B. Liu and X. Chen respectively in uncertain environment. In this treatise, we extend the study of almost convergence by introducing double sequences of complex uncertain variable. Almost convergence with respect to almost surely, mean, measure, distribution and uniformly almost surely are presented and interrelationships among them are studied and depicted in the form of a diagram. We also define almost Cauchy sequence in the same format and establish some results. Conventionally we have, every convergent sequence is a Cauchy sequence and the converse case is not true in general. But taking complex uncertain variable in a double sequence, we find that a complex uncertain double sequence is a almost Cauchy sequence if and only if it is almost convergent. Some suitable examples and counter examples are properly placed to make the paper self suffcient.

The almost surely (a.s.) exponential stability is studied for semi-Markovian switched stochastic systems with randomly impulsive jumps. We start from the case that switches and impulses occur synchronously, in which the impulsive switching signal is a semi-Markovian process. For the case that switches and impulses occur asynchronously, the impulsive arrival time sequence and the types of jump maps are driven by a renewal process and a Markov chain, respectively. By applying the multiple Lyapunov function approach, sufficient conditions of exponential stability a.s. are obtained based upon the ergodic property of semi-Markovian process. The validity of the proposed theoretical results is demonstrated by a numerical example.

The pth ( $p\\geq 1$ ) moment exponential stability, almost surely exponential stability and stability in distribution for stochastic McKean–Vlasov equation are derived based on some distribution-dependent Lyapunov function techniques.

In this paper, we introduce the concept of convergence of complex uncertain series. We initiate matrix transformation of complex uncertain sequence and extend the study via linearity and boundedness. In this context, we prove Silverman-Toeplitz theorem and Kojima-Schur theorem considering complex uncertain sequences. Finally, we establish some results on co-regular matrices.

Mathematical modeling is critical to our understanding of how infectious diseases spread at the individual and population levels. This book gives readers the necessary skills to correctly formulate and analyze mathematical models in infectious disease epidemiology, and is the first treatment of the subject to integrate deterministic and stochastic models and methods. Mathematical Tools for Understanding Infectious Disease Dynamicsfully explains how to translate biological assumptions into mathematics to construct useful and consistent models, and how to use the biological interpretation and mathematical reasoning to analyze these models. It shows how to relate models to data through statistical inference, and how to gain important insights into infectious disease dynamics by translating mathematical results back to biology. This comprehensive and accessible book also features numerous detailed exercises throughout; full elaborations to all exercises are provided. Covers the latest research in mathematical modeling of infectious disease epidemiologyIntegrates deterministic and stochastic approachesTeaches skills in model construction, analysis, inference, and interpretationFeatures numerous exercises and their detailed elaborationsMotivated by real-world applications throughout

We establish three criteria of hyperbolicity of a graph in terms of “average width of geodesic bigons”. In particular we prove that if the ratio of the Van Kampen area of a geodesic bigon β and the length of β in the Cayley graph of a finitely presented group G is bounded above then G is hyperbolic.

The modeling of stochastic dependence is fundamental for understanding random systems evolving in time. When measured through linear correlation, many of these systems exhibit a slow correlation decay--a phenomenon often referred to as long-memory or long-range dependence. An example of this is the absolute returns of equity data in finance. Selfsimilar stochastic processes (particularly fractional Brownian motion) have long been postulated as a means to model this behavior, and the concept of selfsimilarity for a stochastic process is now proving to be extraordinarily useful. Selfsimilarity translates into the equality in distribution between the process under a linear time change and the same process properly scaled in space, a simple scaling property that yields a remarkably rich theory with far-flung applications. After a short historical overview, this book describes the current state of knowledge about selfsimilar processes and their applications. Concepts, definitions and basic properties are emphasized, giving the reader a road map of the realm of selfsimilarity that allows for further exploration. Such topics as noncentral limit theory, long-range dependence, and operator selfsimilarity are covered alongside statistical estimation, simulation, sample path properties, and stochastic differential equations driven by selfsimilar processes. Numerous references point the reader to current applications. Though the text uses the mathematical language of the theory of stochastic processes, researchers and end-users from such diverse fields as mathematics, physics, biology, telecommunications, finance, econometrics, and environmental science will find it an ideal entry point for studying the already extensive theory and applications of selfsimilarity.

The asymptotic behavior of the almost surely extreme values of the queue length and queueing time for queueing systems is analyzed. First, one general limit theorem on the asymptotics of extreme values of regenerative processes is considered. Further, based on this theorem applied to queueing systems M | G |1 and GI | M |1, the law of the repeated logarithm for lim sup and the law of the triple logarithm for lim inf, as well as some of their refinements, are formulated.

Abstract-This article is devoted to analyze the stability of nonlinear stochastic delayed systems driven by G-Brownian motion. Firstly, we study the existence of global unique solution. Secondly, by using G-Ito formula, G-Lyapunov function, Gronwall's inequality and Borel-Cantelli lemma, we discuss the pth moment exponential stability and almost surely exponential stability of stochastic delayed systems. Finally, we provide an example to verify the results.