Exponential Mean Research Articles

Study of the electron transfer reaction between [Co(EDTA)]− and [Ru(NH3 )5 py]2+ in methanol-water mixtures

The title reaction was studied in different water–cosolvent (methanol) mixtures. The results have been rationalized employing the Marcus–Hush treatment. To apply this treatment, the true, unimolecular, electron-transfer rate constants (ket) were obtained from the experimentally measured rate constants after calculation of the equilibrium constant for the processes of formation of the encounter complex. This calculation was carried out using Eigen–Fuoss (EF) and exponential mean spherical (EMSA) approaches employing effective values of the solvent dielectric constant. These effective values were obtained from the measured association constants corresponding to other ion pairs. The results reveal that in these media there is an additional component of reorganization energy, absent in neat solvents. An explanation of the origin of this component is given. © 2009 Wiley Periodicals, Inc. Int J Chem Kinet 41: 658–666, 2009

Expected Posterior Priors in Selecting the Largest Mean of the Exponential Distributions

In reliability theory or survival analysis, selecting the largest mean among many exponential distributions is an important issue. Such a problem can also be viewed as a model selection problem via the Bayesian approach. It is well known that Bayes factors under proper priors have been very successful in Bayesian model selection or testing problems. However, Bayes factors are typically invalid with respect to improper noninformative priors. Objective Bayesian criteria are thus desired. In this work, we consider to use the expected posterior priors originally proposed by Pérez and Berger (2002) to select the largest exponential mean. Specific expected posterior priors are derived in recursive formulas. Some simulation results are also given to illustrate the method.

Exponential stabilization of switched stochastic dynamical networks

A switched stochastic dynamical network (SSDN) model will firstly be formulated in this paper, which considers two mutually independent switching signals including individual node switching and network topology switching. In the proposed SSDN model, the stochastic perturbation is described by multi-dimensional Brownian motion, and the switching signals are arbitrary under the constraint of the average dwell time. A multiple Lyapunov function is utilized to cope with the switching problem. A single controller is then designed for the exponential mean square stabilization for SSDNs. The coupling matrix of the SSDN can be assumed to be irreducible symmetric or irreducible asymmetric. The obtained criteria are given in terms of linear matrix inequalities, which can be solved efficiently by standard software packages. Numerical examples, including small-world and scale-free networks, are exploited to illustrate the effectiveness of the theoretical results. It can be observed from the examples that our results are also applicable to large-scale dynamical networks.

Dynamic Analysis of Stochastic Reaction-Diffusion Cohen-Grossberg Neural Networks with Delays

Stochastic effects on convergence dynamics of reaction-diffusion Cohen-Grossberg neural networks (CGNNs) with delays are studied. By utilizing Poincare inequality, constructing suitable Lyapunov functionals, and employing the method of stochastic analysis and nonnegative semimartingale convergence theorem, some sufficient conditions ensuring almost sure exponential stability and mean square exponential stability are derived. Diffusion term has played an important role in the sufficient conditions, which is a preeminent feature that distinguishes the present research from the previous. Two numerical examples and comparison are given to illustrate our results.

Multipliers in the space of analytic functions with exponential mean growth

Multipliers in the space of analytic functions with exponential mean growth

Stability of Stochastic Differential Equations Under Discretization

Long time behavior of stochastic differential equations (SDE) involves two instances of exponential mean square stability (EMS-stability). First deals with stability of the original continuous time system while the second is concerned with stability after the time step discretization. By considering a linear operator S associated with SDE, we show that the discrete system is EMS-stable if and only if S is a positive contraction on the set of symmetric positive definite matrices.

Effect of small time delay in a predator-prey model within random environment

We consider a non-Kolmogorov type predator-prey model with and without delay. The local stability and Hopf bifurcation results are stated for both the cases of the deterministic system. The effect of stochastic perturbation in the form of parametric white and colored noise are considered. The exponential mean square stability of the trivial solutions for the stochastic differential equations and stochastic delay differential equations are studied under the Ito interpretation. To study the exponential mean square stability of the interior equilibrium point for the delayed model system in the presence of parametric white and colored noise, an approximation procedure is followed for the system of stochastic delay differential equation model system under the assumption that the magnitude of time delay is small. We have obtained threshold magnitude of discrete time delay. Critical magnitude for noise intensities are obtained for stochastic stability aspects. Numerical simulation results are provided in support of our analytical findings.

Robust exponential stability for delayed uncertain Hopfield neural networks with Markovian jumping parameters

This Letter is concerned with delay-dependent robust exponential mean square stability for delayed uncertain Hopfield neural networks with Markovian jumping parameters. Time delays here are discrete and distributed time-varying delays. Based on Lyapunov–Krasovskii stability theory, delay-dependent stability conditions are derived in terms of linear matrix inequalities (LMIs). Finally, numerical examples are given to illustrate the effectiveness of the proposed method.

Delay‐distribution‐dependent robust stability of uncertain systems with time‐varying delay

AbstractBy employing the information of the probability distribution of the time delay, this paper investigates the problem of robust stability for uncertain systems with time‐varying delay satisfying some probabilistic properties. Different from the common assumptions on the time delay in the existing literatures, it is assumed in this paper that the delay is random and its probability distribution is known a priori. In terms of the probability distribution of the delay, a new type of system model with stochastic parameter matrices is proposed. Based on the new system model, sufficient conditions for the exponential mean square stability of the original system are derived by using the Lyapunov functional method and the linear matrix inequality (LMI) technique. The derived criteria, which are expressed in terms of a set of LMIs, are delay‐distribution‐dependent, that is, the solvability of the criteria depends on not only the variation range of the delay but also the probability distribution of it. Finally, three numerical examples are given to illustrate the feasibility and effectiveness of the proposed method. Copyright © 2008 John Wiley & Sons, Ltd.

A class of k-log-convex functions and their applications to some special functions

Let a and b be two real numbers and f be a positive and differentiable function on an interval I. The authors establish the i-log-convex or i-log-concave properties for i∈ℕ of the function [f(bx)] a /[f(ax)] b for ax∈I and bx∈I when the function u k−1[ln f(u)](k) for k∈ℕ is monotonic and apply these properties to deduce some known and new conclusions related to some special functions, such as the gamma function, Riemann's zeta function, complete elliptic integrals, exponential mean, and extended mean values.

Estimation of the exponential mean time to failure under a weighted balanced loss function

This paper considers estimation of a exponential mean time to failure using a loss function that reflects both goodness of fit and precision of estimation. The admissibility and inadmissibility of a class of linear estimators of the form\(c\bar X + d\) are studied.

Sampled-data Networked Control Systems with Random Time Delay

The stability and performance of a networked control system (NCS) strongly depends on the transmission delay. However, the randomness of the transmission delay is an intrinsic property in the network communication, e.g. Ethernet. Aiming at NCS with random transmission delay, a novel control approach is proposed. The transmission delays from sensor-to-controller (SC) and controller-to-actuator (CA) are modelled by two independent Markovian processes. A controller, which is able to monitor the SC delay and synchronously switches according to it, is considered. The resulting closed-loop system is a Markovian jump linear system with randomly piecewise continuous delay. The exponential mean square stability for the given model is established by using a Lyapunov-Krasovskii functional. The performance benefits of the proposed approach are demonstrated in a numerical example.

Optimal Prefix Codes for Infinite Alphabets With Nonlinear Costs

Let $P = \{p(i)\}$ be a measure of strictly positive probabilities on the set of nonnegative integers. Although the countable number of inputs prevents usage of the Huffman algorithm, there are nontrivial $P$ for which known methods find a source code that is optimal in the sense of minimizing expected codeword length. For some applications, however, a source code should instead minimize one of a family of nonlinear objective functions, $\beta$-exponential means, those of the form $\log_a \sum_i p(i) a^{n(i)}$, where $n(i)$ is the length of the $i$th codeword and $a$ is a positive constant. Applications of such minimizations include a novel problem of maximizing the chance of message receipt in single-shot communications ($a<1$) and a previously known problem of minimizing the chance of buffer overflow in a queueing system ($a>1$). This paper introduces methods for finding codes optimal for such exponential means. One method applies to geometric distributions, while another applies to distributions with lighter tails. The latter algorithm is applied to Poisson distributions and both are extended to alphabetic codes, as well as to minimizing maximum pointwise redundancy. The aforementioned application of minimizing the chance of buffer overflow is also considered.

Exact likelihood inference for two exponential populations under joint Type-II censoring

Comparative lifetime experiments are of paramount importance when the object of a study is to ascertain the relative merits of two competing products in regard to the duration of their service life. In this paper, we discuss exact inference for two exponential populations when Type-II censoring is implemented on the two samples in a combined manner. We obtain the conditional maximum likelihood estimators (MLEs) of the two exponential mean parameters. We then derive the moment generating functions and the exact distributions of these MLEs along with exact confidence intervals and simultaneous confidence regions. Moreover, simultaneous approximate confidence regions based on the asymptotic normality of the MLEs and simultaneous credible confidence regions from a Bayesian viewpoint are also discussed. A comparison of the exact, approximate, bootstrap and Bayesian intervals is also made in terms of coverage probabilities. Finally, an example is presented in order to illustrate all the methods of inference discussed here.

Stabilization of stochastically perturbed nonlinear oscillations

A problem of stabilization of stochastic oscillatory modes of nonlinear dynamic system is considered. The solution of this problem rests on the spectral criterion of the exponential mean square (root-mean-square) stability of stochastically perturbed limit cycles. The analysis of the stabilizability reduces to the minimization of the spectral radius of a certain positive operator. The efficient possibilities of the obtained stabilizability criterion are illustrated for the case of the cycle on the plane, where the construction of a stabilizing regulator reduces to the minimization of the quadratic functional.

EXPONENTIAL MEAN SQUARE STABILITY OF STOCHASTICALLY FORCED INVARIANT MANIFOLDS FOR NONLINEAR SDEs

An exponential mean square stability for the invariant manifold [Formula: see text] of a nonlinear stochastic system is considered. The stability analysis is based on the [Formula: see text]-quadratic Lyapunov function technique. The local dynamics of the nonlinear system near manifold is described by the stochastic linear extension system. We propose a general notion of the projective stability (P-stability) and prove the following theorem. The smooth compact manifold [Formula: see text] is exponentially mean square stable if and only if the corresponding stochastic linear extension system is P-stable.

Stochastic monotonicity of the MLE of exponential mean under different censoring schemes

In this paper, we present a general method which can be used in order to show that the maximum likelihood estimator (MLE) of an exponential mean θ is stochastically increasing with respect to θ under different censored sampling schemes. This propery is essential for the construction of exact confidence intervals for θ via “pivoting the cdf” as well as for the tests of hypotheses about θ. The method is shown for Type-I censoring, hybrid censoring and generalized hybrid censoring schemes. We also establish the result for the exponential competing risks model with censoring.

Ranked-Set-Sample-Based Tests for Normal and Exponential Means

We consider the problems of testing a normal mean and an exponential mean under the median ranked set sampling (MRSS) scheme. The tests based on MRSS outperform those based on the usual ranked-set samples. In addition, through heuristic arguments we postulate working formulas for computing cut-off points for RSS-based and MRSS-based tests, respectively. Our simulation studies indicate these formulas perform surprisingly well, even for small samples, and solve the task of computation for large samples.

The self-dual core and the anti-self-dual remainder of an aggregation operator

In most decisional models based on pairwise comparison between alternatives, the reciprocity of the individual preference representations expresses a natural assumption of rationality. In those models self-dual aggregation operators play a central role, in so far as they preserve the reciprocity of the preference representations in the aggregation mechanism from individual to collective preferences. In this paper we propose a simple method by which one can associate a self-dual aggregation operator to any aggregation operator on the unit interval. The resulting aggregation operator is said to be the self-dual core of the original one, and inherits most of its properties. Our method constitutes thus a new characterization of self-duality, with some technical advantages relatively to the traditional symmetric sums method due to Silvert. In our framework, moreover, every aggregation operator can be written as a sum of a self-dual core and an anti-self-dual remainder which, in some cases, seems to give some indication on the dispersion of the variables. In order to illustrate the method proposed, we apply it to two important classes of continuous aggregation operators with the properties of idempotency, symmetry, and stability for translations: the OWA operators and the exponential quasiarithmetic means.

Convergence dynamics of stochastic reaction–diffusion recurrent neural networks with continuously distributed delays

Convergence dynamics of reaction–diffusion recurrent neural networks (RNNs) with continuously distributed delays and stochastic influence are considered. Some sufficient conditions to guarantee the almost sure exponential stability, mean value exponential stability and mean square exponential stability of an equilibrium solution are obtained, respectively. Lyapunov functional method, M-matrix properties, some inequality technique and nonnegative semimartingale convergence theorem are used in our approach. These criteria ensuring the different exponential stability show that diffusion and delays are harmless, but random fluctuations are important, in the stochastic continuously distributed delayed reaction–diffusion RNNs with the structure satisfying the criteria. Two examples are also given to demonstrate our results.