• East Asian mathematical journal
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• v.33 no.3
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• pp.271-289
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• 2017

It is our purpose in this paper to introduce the concept of best random proximity pair for subsets A and B of a separable Banach space E. We prove some best random approximation and best random proximity pair theorems of certain classes of random operators, which is the stochastic verse of the deterministic results of Eldred et al. [22], Eldred et al. [18] and Eldred and Veeramani [19]. Furthermore, our results generalize and extend recent results of Okeke and Abbas [42] and Okeke and Kim [43]. Moreover, we shall apply our results to study nonlinear stochastic integral equations of the Hammerstein type.

• Journal of applied mathematics & informatics
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• v.23 no.1_2
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• pp.455-460
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• 2007

In allelic model $X=(x_1,\;x_2,\;{\cdots},\;x_d)$, $$M_f(t)=f(p(t))-{\int}_0^t\;Lf(p(t))ds$$ is a P-martingale for diffusion operator L under the certain conditions. In this note, we try to apply diffusion processes for countable-allelic model in population genetic model and we can define a new diffusion operator $L^*$. Since the martingale problem for this operator $L^*$ is related to diffusion processes, we can define a integral which is combined with operator $L^*$ and a bilinar form $<{\cdot},{\cdot}>$. We can find properties for this integral using maximum principle.

• Structural Engineering and Mechanics
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• v.28 no.4
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• pp.373-386
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• 2008

In this study stochastic analysis of non-linear dynamical systems under ${\alpha}$-stable, multiplicative white noise has been conducted. The analysis has dealt with a special class of ${\alpha}$-stable stochastic processes namely sub-Gaussian white noises. In this setting the governing equation either of the probability density function or of the characteristic function of the dynamical response may be obtained considering the dynamical system forced by a Gaussian white noise with an uncertain factor with ${\alpha}/2$- stable distribution. This consideration yields the probability density function or the characteristic function of the response by means of a simple integral involving the probability density function of the system under Gaussian white noise and the probability density function of the ${\alpha}/2$-stable random parameter. Some numerical applications have been reported assessing the reliability of the proposed formulation. Moreover a proper way to perform digital simulation of the sub-Gaussian ${\alpha}$-stable random process preventing dynamical systems from numerical overflows has been reported and discussed in detail.

• Journal of applied mathematics & informatics
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• v.41 no.5
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• pp.923-935
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• 2023

In this paper, we will discuss existence of solution of square-mean pseudo almost automorphic solution for fractional stochastic evolution equations driven by G-Brownian motion which is given as ^c₀D^𝛼_𝜌 Ψ_𝜌 = 𝒜(𝜌)Ψ_𝜌d𝜌 + 𝚽(𝜌, Ψ_𝜌)d𝜌 + ϒ(𝜌, Ψ_𝜌)d ⟨ℵ⟩_𝜌 + χ(𝜌, Ψ_𝜌)dℵ_𝜌, 𝜌 ∈ R. Furthermore, we also prove that solution of the above equation is unique by using Lipschitz conditions and Cauchy-Schwartz inequality. Moreover, examples demonstrate the validity of the obtained main result and we obtain the solution for an equation, and proved that this solution is unique.

• Proceedings of the Computational Structural Engineering Institute Conference
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• 2002.04a
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• pp.369-376
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• 2002

The stochastic analysis of semi-infinite domain is presented using the weighted integral method, which is improved to include the higher order terms in expanding the displacement vector. To improve the weighted integral method, the Lagrangian remainder is taken into account in the expansion of the status variable with respect to the mean value of the random variables. In the resulting formulae only the 'proportionality coefficients' are introduced in the resulting equation, therefore no additional computation time and memory requirement is needed. The equations are applied in analyzing the semi-infinite domain. The results obtained by the improved weighted integral method are reasonable and are in good agreement with those of the Monte Carlo simulation. To model the semi-infinite domain, the Bettess's infinite element is adopted, where the theoretical decomposition of the strain-displacement matrix to calculate the deviatoric stiffness of the semi-infinite domains is introduced. The calculated value of mean and the covariance of the displacement are revealed to be larger than those given by the finite domain assumptions which is thought to be rational and should be considered in the design of structures on semi-infinite domains.

• Journal of the Computational Structural Engineering Institute of Korea
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• v.17 no.1
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• pp.83-91
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• 2004

Nonlinear system responses of an impact system under randomly perturbed harmonic excitations are predicted in the probability domain by adopting the semi-analytical procedure previously developed. The semi-analytical procedure is obtained by solving the Fokker-Planck equation corresponding to the stochastic differential equation of the given impact system by utilizing the path-integral solution. The evolutionary joint probability density functions are generated by using the method, and the characteristics of nonlinear dynamic response behaviors of the system are examined. Noise effects on the responses are also examined. It Is found that the semi-analytical method can provides the accurate information of the responses via the joint probability functions for the impact system. It is found that the noises weaken and eventually terminate the chaos in the responses, but it is also found that the chaotic signatures reside in the presence of the external noise with relatively high intensity. The joint probability density function shows that the ensemble of the system responses are weakly stationary.

• Ocean Systems Engineering
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• v.1 no.3
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• pp.249-261
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• 2011

The stochastic nonlinear dynamic behavior and probability density function of ship rolling are studied using the nonlinear dynamical systems approach and probability theory. The probability density function of the rolling response is evaluated through solving the Fokker Planck Equation using the path integral method based on a Gauss-Legendre interpolation scheme. The time-dependent probability of ship rolling restricted to within the safe domain is provided and capsizing is investigated from the probability point of view. The random differential equation of ships' rolling motion is established considering the nonlinear damping, nonlinear restoring moment, white noise and colored noise wave excitation.

• The Korean Journal of Applied Statistics
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• v.30 no.1
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• pp.1-12
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• 2017

Many stochastic processes satisfy the Markov property exactly or at least approximately. An interested property in the Markov process is the first passage time. Since the sequential analysis by Wald, the approximation of the first passage time has been studied extensively. The Statistical computing technique due to the development of high-speed computers made it possible to calculate the values of the properties close to the true ones. This article introduces an exponentially weighted moving average (EWMA) control chart as an example of the Markov process, and studied how to calculate the average run length with problematic issues that should be cautioned for correct calculation. The results derived for approximation of the first passage time in this research can be applied to any of the Markov processes. Especially the approximation of the continuous time Markov process to the discrete time Markov chain is useful for the studies of the properties of the stochastic process and makes computational approaches easy.

• Proceedings of the Korea Water Resources Association Conference
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• 2020.06a
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• pp.125-125
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• 2020

European Space Agency's Sentinel-1 has an improved spatial and temporal resolution, as compared to previous satellite data such as Envisat Advanced SAR (ASAR) or Advanced Scatterometer (ASCAT). Thus, the assumption used for low-resolution retrieval algorithms used by ENVISAT ASAR or ASCAT is not applicable to Sentinel-1, because a higher degree of land surface heterogeneity should be considered for retrieval. The assumption of homogeneity over land surface is not valid any more. In this study, considering that soil roughness is one of the key parameters sensitive to soil moisture retrievals, various approaches are discussed. First, soil roughness is spatially inverted from Sentinel-1 backscattering over Yanco sites in Australia. Based upon this, Artificial Neural Networks data (feedforward multiplayer perception, MLP, Levenberg-Marquadt algorithm) are compared with Fractal approach (brownian fractal, Hurst exponent of 0.5). When using ANNs, training data are achieved from theoretical forward scattering models, Integral Equation Model (IEM). and Sentinel-1 measurements. The network is trained by 20 neurons and one hidden layer, and one input layer. On the other hand, fractal surface roughness is generated by fitting 1D power spectrum model with roughness spectra. Fractal roughness profile is produced by a stochastic process describing probability between two points, and Hurst exponent, as well as rms heights (a standard deviation of surface height). Main interest of this study is to estimate a spatial variability of roughness without the need of local measurements. This non-local approach is significant, because we operationally have to be independent from local stations, due to its few spatial coverage at the global level. More fundamentally, SAR roughness is much different from local measurements, Remote sensing data are influenced by incidence angle, large scale topography, or a mixing regime of sensors, although probe deployed in the field indicate point data. Finally, demerit and merit of these approaches will be discussed.