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The Stochastic Schrödinger–Hirota equation plays a crucial role in describing the quantum behavior of physical systems under the influence of stochastic processes. In this manuscript, we present a comprehensive study on the exact solutions of the Stochastic Schrödinger–Hirota equation employing a novel approach, namely the variable coefficient third-degree generalized Abel equation method. This method extends the applicability of Abel's equation with variable coefficients, providing a powerful tool for solving complex nonlinear equations arising in stochastic quantum mechanics. The manuscript findings hold potential implications for diverse fields, including quantum physics, statistical mechanics, and mathematical physics. Moreover, the developed methodology could find applications in solving other nonlinear equations arising in different branches of science and engineering, broadening its scope and impact. • The paper introduces a novel approach, the variable coefficient third-degree generalized Abel equation method, to comprehensively study and solve the Stochastic Schrödinger–Hirota equation. • This innovative methodology extends the applicability of Abel's equation with variable coefficients, providing a powerful tool for addressing complex nonlinear equations in the realm of stochastic quantum mechanics. • The developed methodology introduces a systematic and effective approach that could enhance our ability to tackle complex nonlinear equations across various scientific disciplines. [ABSTRACT FROM AUTHOR]

Nonlinearity and implicitness are common degradation features of the stochastic degradation equipment for prognostics. These features have an uncertain effect on the remaining useful life (RUL) prediction of the equipment. The current data-driven RUL prediction method has not systematically studied the nonlinear hidden degradation modeling and the RUL distribution function. This paper uses the nonlinear Wiener process to build a dual nonlinear implicit degradation model. Based on the historical measured data of similar equipment, the maximum likelihood estimation algorithm is used to estimate the fixed coefficients and the prior distribution of a random coefficient. Using the on-site measured data of the target equipment, the posterior distribution of a random coefficient and actual degradation state are step-by-step updated based on Bayesian inference and the extended Kalman filtering algorithm. The analytical form of the RUL distribution function is derived based on the first hitting time distribution. Combined with the two case studies, the proposed method is verified to have certain advantages over the existing methods in the accuracy of prediction. [ABSTRACT FROM AUTHOR]

Products are often subject to dynamic environmental conditions in field use. When stress transition occurs, products may be exposed to instantaneous shocks that result in shock damages to the products, causing a permanent change of the degradation signals. Meanwhile, under some conditions, instantaneous shocks also lead to stress drift, causing a temporary change of the degradation signals. In this paper, a degradation model is proposed to assess the reliability and predict the residual lifetime of products operating in a dynamic environment considering shock damage and stress drift. The model is established based on a Wiener process which combines a stress-dependent degradation rate function, a shock damage function and a stress drift function in response to the dynamic environment. The shock damage function is established as a linear function of the stress transition start level and the stress level increment. The stress drift function is established as the difference value of a specified function at the stress transition start and end levels. A simulation study is presented to demonstrate the application of the model, and a case study for miniature light bulbs is used to validate the effectiveness of the proposed model. [ABSTRACT FROM AUTHOR]

Abstract In this paper, two new estimators based on the spirit of the best linear unbiased estimators are separately developed for homogeneous gamma process. Both estimators can be computed recursively, and have high efficiency. We compare the two new estimators with those of Paroissin (2017) by simulations, and find that ours have smaller biases and mean squared errors. Highlights • Two new recursive estimators are developed for homogeneous gamma process. • We show that the two estimators are weakly consistent. • We compare the two new estimators with those of Paroissin (2017) by simulations. [ABSTRACT FROM AUTHOR]

For the large number of nonlinear degradation devices existing in a project, the existing methods have not systematically studied the effects of random effect on the remaining lifetime (RL), the accuracy and efficiency of the parameters estimation are not high, and the current degradation state of the target device is not accurately estimated. In this paper, a nonlinear Wiener degradation model with random effect is proposed and the corresponding probability density function (PDF) of the first hitting time (FHT) is deduced. A parameter estimation method based on modified expectation maximum (EM) algorithm is proposed to obtain the estimated value of fixed coefficient and the priori value of random coefficient in the model. The posterior value of the random coefficient and the current degradation state of target device are updated synchronously by the state space model (SSM) and the Kalman filter algorithm. The PDF of RL with random effect is deduced. A simulation example is analyzed to verify that the proposed method has the obvious advantage over the existing methods in parameter estimation error and RL prediction accuracy. [ABSTRACT FROM AUTHOR]

This paper explores the influence of measurement errors on lifetime estimation based on Wiener degradation model with the skew normal distribution. The exact and explicit expressions of the lifetime distribution are derived, which subsumes several existing results as special cases. [ABSTRACT FROM AUTHOR]

It is well-known that the sum of elliptical random variables with constant coefficients is also an elliptical random variable. In this note, we prove that the coefficients can also be random variables, such that the sum of elliptical random variables with random coefficients on a circle is also elliptically distributed. We then examine statistical implications by considering the projection pursuit with the underlying random variables. [ABSTRACT FROM AUTHOR]

With the wide application of condition based maintenance (CBM) in aircraft maintenance practice, the joint optimization of maintenance and inventory management, which can take full advantage of CBM and reduce the aircraft operational cost, is receiving increasing attention. In order to optimize the inspection interval, maintenance decision and spare provisioning together for aircraft deteriorating parts, firstly, a joint inventory management strategy is presented, then, a joint optimization of maintenance inspection and spare provisioning for aircraft parts subject to the Wiener degradation process is proposed based on the strategy. Secondly, a combination of the genetic algorithm (GA) and the Monte Carol method is developed to minimize the total cost rate. Finally, a case study is conducted and the proposed joint optimization model is compared with the existing optimization model and the airline real case. The results demonstrate that the proposed model is more beneficial and effective. In addition, the sensitivity analysis of the proposed model shows that the lead time has higher influence on the optimal results than the urgent order cost and the corrective maintenance cost, which is consistent with the actual situation of aircraft maintenance practices and inventory management [ABSTRACT FROM AUTHOR]

Sufficient conditions for relative controllability of stochastic fractional neutral systems with bounded operator and multiple time varying delay in the control are obtained. The result is proved using an equivalent nonlinear integral equation to the system and Banach contraction principle. The controllability results are derived for systems with both Wiener and systems with L e ´ vy noise. [ABSTRACT FROM AUTHOR]

Purpose The purpose of this paper is to determine confidence intervals for the stochastic solutions in RLCG cells with a potential source influenced by coloured noise.Design/methodology/approach The deterministic model of the basic RLCG cell leads to an ordinary differential equation. In this paper, a stochastic model is formulated and the corresponding stochastic differential equation is analysed using the Itô stochastic calculus.Findings Equations for the first and the second moment of the stochastic solution of the coloured noise-affected RLCG cell are obtained, and the corresponding confidence intervals are determined. The moment equations lead to ordinary differential equations, which are solved numerically by an implicit Euler scheme, which turns out to be very effective. For comparison, the confidence intervals are computed statistically by an implementation of the Euler scheme using stochastic differential equations.Practical implications/implications The theoretical results are illustrated by examples. Numerical simulations in the examples are carried out using Matlab. A possible generalization for transmission line models is indicated.Originality/value The Itô-type stochastic differential equation describing the coloured noise RLCG cell is formulated, and equations for the respective moments are derived. Owing to this original approach, the confidence intervals can be found more effectively by solving a system of ordinary differential equations rather than by using statistical methods. [ABSTRACT FROM AUTHOR]

Noise analysis of the ramp reference voltage and its projection at the output of a conventional single-slope ramp analog-to-digital converter (ADC) is presented. This paper gives insight on the reference voltage noise origins during the continuous-time ramping phase of column-parallel CMOS image sensor ADCs, as well as its effect on the final ADC output noise. Theoretical modeling of an occurring random walk noise on continuous-time-based ramp references is presented first. Noise on the ramp does not directly add to the output noise of the ADC, but it is modulated and projected subject to given ramp rise time conditions. Test measurements show that the signal-dependent output noise is consistent with the rate of noise increase by the proposed random walk noise model. [ABSTRACT FROM AUTHOR]

We consider a non-smooth Grushin-type distribution, defined on R n , whose stochastic perturbation defines the admissible stochastic processes. Our main result is a stochastic accessibility theorem on the corresponding Grushin manifold. More specifically, given two points P and Q , we show how to steer an admissible stochastic processes, starting at P , such that it strikes an arbitrarily small ball centered at Q , asymptotically almost surely. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the long time behavior of the mild solution to delayed reaction–diffusion Hopfield neural networks (DRDHNNs) driven by infinite dimensional Wiener processes. We analyze the existence, uniqueness, and stability of this system under the local Lipschitz function by constructing an appropriate Lyapunov–Krasovskii function and utilizing the semigroup theory. Some easy-to-test criteria affecting the well-posedness and stability of the networks, such as infinite dimensional noise and diffusion effect, are obtained. The criteria can be used as theoretic guidance to stabilize DRDHNNs in practical applications when infinite dimensional noise is taken into consideration. Meanwhile, considering the fact that the standard Brownian motion is a special case of infinite dimensional Wiener process, we undertake an analysis of the local Lipschitz condition, which has a wider range than the global Lipschitz condition. Two samples are given to examine the availability of the results in this paper. Simulations are also given using the MATLAB. [ABSTRACT FROM PUBLISHER]

Sequential sampling models are widely used in modeling the empirical data obtained from different decision making experiments. Since 1960s, several instantiations of these models have been proposed. A common assumption among these models is that the subject accumulates noisy information during the time course of a decision. The decision is made when the accumulated information favoring one of the responses reaches a decision boundary. Different models, however, make different assumptions about the information accumulation process and the implementation of the decision boundaries. Comparison among these models has proven to be challenging. In this paper we investigate the relationship between several of these models using a theoretical framework called the inverse first passage time problem. This framework has been used in the literature of applied probability theory in investigating the range of the first passage time distributions that can be produced by a stochastic process. In this paper, we use this framework to prove that any Wiener process model with two time-constant boundaries can be mimicked by an independent race model with time-varying boundaries. We also examine the numerical computation of the mimicking boundaries. We show that the mimicking boundaries of the race model are not symmetric. We then propose an equivalent race model in which the boundaries are symmetric and time-constant but the drift coefficients are time-varying. [ABSTRACT FROM AUTHOR]

This paper proposes a numerical method based on the dual reciprocity boundary elements method (DRBEM) to solve the stochastic partial differential equations (SPDEs). The concept of dual reciprocity method is used to convert the domain integral to the boundary. The conventional DRBEM starts with approximation of the source term of the original PDEs with radial basis functions (RBFs). Due to the fact that the nonhomogeneous term of SPDEs considered in this paper involves Wiener process, the traditional DRBEM cannot be applied. So a modification of it is suggested that has some advantages in comparison with the traditional DRBEM and can be developed for solving the SPDEs. The time evolution is discretized by using the finite difference method, while the modified DRBEM is proposed for spatial variations of field variables. The noise term is approximated at the collocation points at each time step. We employ the generalized inverse multiquadrics (GIMQ) RBFs to approximate functions in the presented technique. To confirm the accuracy of the new approach, several examples are employed and simulation results are reported. Also the convergence of the new technique is studied numerically. [ABSTRACT FROM AUTHOR]

With the advances in biochemistry, molecular biology, and neurochemistry there has been impressive progress in understanding the molecular properties of anesthetic agents. However, there has been little focus on how the molecular properties of anesthetic agents lead to the observed macroscopic property that defines the anesthetic state, that is, lack of responsiveness to noxious stimuli. In this paper, we develop a mean field synaptic drive firing rate cortical neuronal model and demonstrate how the induction of general anesthesia can be explained using multistability; the property whereby the solutions of a dynamical system exhibit multiple attracting equilibria under asymptotically slowly changing inputs or system parameters. In particular, we demonstrate multistability in the mean when the system initial conditions or the system coefficients of the neuronal connectivity matrix are random variables. Uncertainty in the system coefficients is captured by representing system uncertain parameters by a multiplicative white noise model wherein stochastic integration is interpreted in the sense of Itô. Modeling a priori system parameter uncertainty using a multiplicative white noise model is motivated by means of the maximum entropy principle of Jaynes and statistical analysis. [ABSTRACT FROM AUTHOR]

Abstract: For a mixed stochastic differential equation driven by independent fractional Brownian motions and Wiener processes, the existence and integrability of the Malliavin derivative of the solution are established. It is also proved that the solution possesses exponential moments. [Copyright &y& Elsevier]

Highlights: [•] The stochastic reaction–diffusion equations with general forcing function as well as multiplicative are considered. [•] The existence of a pullback random attractors is established for the generated RDS. [•] The unique random equilibria exist for a wide range of coefficients. [ABSTRACT FROM AUTHOR]

Highlights: [•] We model the spiking activity of a pair of coupled neurons. [•] The model considers a bivariate Wiener process with jumps. [•] We study the joint distribution of the forward and inter-spike times. [•] Closed form expression and bounds of the joint distribution are determined. [Copyright &y& Elsevier]

Abstract: We propose an improved method for calculating the cumulative first-passage time distribution in Wiener diffusion models with two absorbing barriers. This distribution function is frequently used to describe responses and error probabilities in choice reaction time tasks. The present work extends related work on the density of first-passage times [Navarro, D.J., Fuss, I.G. (2009). Fast and accurate calculations for first-passage times in Wiener diffusion models. Journal of Mathematical Psychology, 53, 222–230]. Two representations exist for the distribution, both including infinite series. We derive upper bounds for the approximation error resulting from finite truncation of the series, and we determine the number of iterations required to limit the error below a pre-specified tolerance. For a given set of parameters, the representation can then be chosen which requires the least computational effort. [Copyright &y& Elsevier]

Abstract: This paper considers the stabilization problem of a class of uncertain Itô stochastic fuzzy systems driven by a multidimensional Wiener process. The uncertainty modeled in the systems is of the linear fractional type which includes the norm-bounded uncertainty as a special case. The objective is to design a state-feedback fuzzy controller such that the closed-loop system is robustly asymptotically stable under a stochastic setting. By using a stochastic Lyapunov approach, sufficiency conditions for the stability and stabilization of this class of systems are established based on a novel matrix decomposition technique. The derived stability conditions are then employed to design controllers which stabilize the uncertain Itô stochastic fuzzy systems. Two simulation examples are given to illustrate the effectiveness of the approaches proposed. [ABSTRACT FROM AUTHOR]

Abstract: A continuous spatial model for populations that are not density-regulated is analyzed. The model is a generalization of the geometric Brownian motion often used to describe populations at a single location. The locations are linked by migration and spatial correlation in the noise. At any point of time, the population size at a given location is lognormally distributed so the log population size constitutes a Gaussian field. The model is homogeneous in space but not in time. In particular, we analyze the case when the stochastic growth rate is negative and the total population approaches extinction. The properties of the extinction process is studied by considering local quasi-extinctions. A major conclusion is that migration tends to increase the time to extinction provided that there is no cost of migration. However, as the area occupied by the species starts to decrease, the decrease is faster for populations with larger migration. [Copyright &y& Elsevier]

Abstract: The processes of the form , where K is a constant, and a Brownian bridge, are investigated. We show that and are both Brownian bridges, and establish the independence of and , this implying that the law of coincides with the conditional law of B, given that . We provide the Karhunen–Loève expansion on of , making use of the Bessel functions and . Applications and variants of these results are discussed. In particular, we establish a comparison theorem concerning the supremum distributions of and on . [Copyright &y& Elsevier]

We consider a sequential test procedure which detects possible changes in the mean of observations satisfying a weak invariance principle. Our test statistic is based on weighted CUSUMs of the underlying random variables. In this paper, we study the asymptotic behaviour of the delay time if a change has occurred in the sample after a training period of size m in which the observations stay in control. It turns out that in this situation the limiting distribution of the delay time for m→∞ is normal under a suitable standardization provided the change appeared sufficiently soon after m. [Copyright &y& Elsevier]

Diffusion processes (e.g., Wiener process, Ornstein–Uhlenbeck process) are powerful approaches to model human information processes in a variety of psychological tasks. Lack of mathematical tractability, however, has prevented broad applications of these models to empirical data. This tutorial explains step by step, using a matrix approach, how to construct these models, how to implement them on a computer, and how to calculate the predictions made by these models. In particular, we present models for binaries choices for unidimensional and multiattribute choice alternatives; for simple reaction time tasks; and for three alternatives choice problems. [Copyright &y& Elsevier]

We present an analytic solution for a race model of n stochastic accumulators for multiple choice reaction time. We show that to maintain a constant level of accuracy, the response criterion needs to be increased approximately logarithmically with n, to compensate for the increase with n in the likelihood that an incorrect alternative will be most active after any fixed amount of time accumulating information. Assuming that participants monitor and maintain a constant level of performance can then explain the logarithmic dependency of the response latency on n as specified by Hick''s law. Moreover, we show that for short time intervals, the Shannon information that observers extract from a stimulus, is predicted to increase linearly with processing time. [Copyright &y& Elsevier]

The Wiener diffusion model with two absorbing barriers is often used to describe response times and error probabilities in two-choice decisions. Different representations exist for the density and cumulative distribution of first-passage times, all including infinite series, but with different convergence for small and large times. We present a method that controls the approximation error of the small-time representation that occurs due to finite truncation of these series. Our approach improves and simplifies related work by Navarro and Fuss (2009) and Blurton et al. (2012, both in the Journal of Mathematical Psychology). [ABSTRACT FROM AUTHOR]