• Journal of the Korean Mathematical Society
• /
• v.40 no.2
• /
• pp.251-272
• /
• 2003

Multidimensional integrals are expressed in terms of lower dimensional integrals and function evaluations. An iterative process is used where a trapezoidal and three point identities are used as generators for higher dimensional identities. Bounds are obtained utilising the resulting identities. It is demonstrated that earlier Ostrowski type results are obtained as particular instances of the current work.

• Korean Journal of Mathematics
• /
• v.22 no.1
• /
• pp.71-84
• /
• 2014

By using the multidimensional normal approximation of functionals of Gaussian fields, we prove that functionals of Gaussian fields, as functions of t, converge weakly to a standard Brownian motion. As an application, we consider the convergence of the Stratonovich-type Riemann sums, as a function of t, of fractional Brownian motion with Hurst parameter H = 1/4.

• Journal of applied mathematics & informatics
• /
• v.7 no.3
• /
• pp.709-717
• /
• 2000

In addition to some new cubature formulas for the approximation of integrals over the unit disk, we present a survey of all known cubature formulas of algebraic degree for this region.

• Bulletin of the Korean Chemical Society
• /
• v.24 no.8
• /
• pp.1069-1074
• /
• 2003

A theoretical description and experimental demonstration of homodyne-detected two-color transient grating (2-C TG) signal are presented. By treating the coupled bath degrees of freedom as a collection of harmonic oscillators and using a short-time expansion method, approximated nonlinear response functions were obtained. An analytic expression for the two-color transient grating signal was obtained by carrying out relevant Gaussian integrals. The initial rising and decaying parts of the 2-C TG signal is shown to be critically dependent on the ultrafast inertial component of the solvation correlation function. The experimental results confirm the predictions of the theoretical model.

• Communications for Statistical Applications and Methods
• /
• v.26 no.4
• /
• pp.371-383
• /
• 2019

Panel data sets have been developed in various areas, and many recent studies have analyzed panel, or longitudinal data sets. Maximum likelihood (ML) may be the most common statistical method for analyzing panel data models; however, the inference based on the ML estimate will have an inflated Type I error because the ML method tends to give a downwardly biased estimate of variance components when the sample size is small. The under estimation could be severe when data is incomplete. This paper proposes the restricted maximum likelihood (REML) method for a random effects panel data model with a censored dependent variable. Note that the likelihood function of the model is complex in that it includes a multidimensional integral. Many authors proposed to use integral approximation methods for the computation of likelihood function; however, it is well known that integral approximation methods are inadequate for high dimensional integrals in practice. This paper introduces to use the moments of truncated multivariate normal random vector for the calculation of multidimensional integral. In addition, a proper asymptotic standard error of REML estimate is given.

• Journal of the Korean Physical Society
• /
• v.73 no.9
• /
• pp.1269-1278
• /
• 2018

Using the thawed Gaussian wave packets [E. J. Heller, J. Chem. Phys. 62, 1544 (1975)] and the adaptive reinitialization technique employing the frame operator [L. M. Andersson et al., J. Phys. A: Math. Gen. 35, 7787 (2002)], a trajectory-based Gaussian wave packet method is introduced that can be applied to scattering and time-dependent problems. This method does not require either the numerical multidimensional integrals for potential operators or the inversion of nearly-singular matrices representing the overlap of overcomplete Gaussian basis functions. We demonstrate a possibility that the method can be a promising candidate for the time-dependent $Schr{\ddot{o}}dinger$ equation solver by applying to tunneling, high-order harmonic generation, and above-threshold ionization problems in one-dimensional model systems. Although the efficiency of the method is confirmed in one-dimensional systems, it can be easily extended to higher dimensional systems.