• Journal of the Korean Mathematical Society
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• v.61 no.2
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• pp.377-393
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• 2024

In this paper, we study the positive solutions to a discrete harmonic function for a random walk satisfying finite range and ellipticity conditions, killed at the boundary of an unbounded cylinder in ℤ^d. We first prove the existence and uniqueness of positive solutions, and then establish that all the positive solutions are generated by two special solutions, which are exponential growth at one end and exponential decay at the other. Our method is based on maximum principle and a Harnack type inequality.

• Journal of the Korean Mathematical Society
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• v.57 no.6
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• pp.1573-1590
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• 2020

In this paper, we classify all solutions bounded from below to uniformly elliptic equations of second order in the form of Lu(x) = a_ij(x)D_iju(x) + b_i(x)D_iu(x) + c(x)u(x) = f(x) or Lu(x) = D_i(a_ij(x)D_ju(x)) + b_i(x)D_iu(x) + c(x)u(x) = f(x) in unbounded cylinders. After establishing that the Aleksandrov maximum principle and boundary Harnack inequality hold for bounded solutions, we show that all solutions bounded from below are linear comb_inations of solutions, which are sums of two special solutions that exponential growth at one end and exponential decay at the another end, and a bounded solution that corresponds to the inhomogeneous term f of the equation.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.18 no.2
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• pp.107-127
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• 2014

The goal of this note is to describe the asymptotic behaviour of problems set in cylinders when the size of them is becoming infinite. This leads to consider problems in unbounded domains as well as new singular perturbations issues.

• Journal of the Korean Mathematical Society
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• v.34 no.3
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• pp.731-770
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• 1997

We study Diriclet forms and related subjects for the Gibbs measures of classical unbounded sping systems interacting via potentials which are superstable and regular. For any Gibbs measure $\mu$, we construct a Dirichlet form and the associated diffusion process on $L^2(\Omega, d\mu), where \Omega = (R^d)^Z^\nu$. Under appropriate conditions on the potential we show that the Dirichlet operator associated to a Gibbs measure $\mu$ is essentially self-adjoint on the space of smooth bounded cylinder functions. Under the condition of uniform log-concavity, the Gibbs measure exists uniquely and there exists a mass gap in the lower end of the spectrum of the Dirichlet operator. We also show that under the condition of uniform log-concavity, the unique Gibbs measure satisfies the log-Sobolev inequality. We utilize the general scheme of the previous works on the theory in infinite dimensional spaces developed by e.g., Albeverio, Antonjuk, Hoegh-Krohn, Kondratiev, Rockner, and Kusuoka, etc, and also use the equilibrium condition and the regularity of Gibbs measures extensively.

• Interaction and multiscale mechanics
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• v.6 no.2
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• pp.211-235
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• 2013

An incompressible smoothed particle hydrodynamics (ISPH) method based on the incremental pressure projection method is developed in this study. The Rayleigh-B$\acute{e}$nard convection in a square enclosure is used as a validation case and the results obtained by the proposed ISPH model are compared to the benchmark solutions. The comparison shows that the established ISPH method has a good performance in terms of accuracy. Subsequently, the proposed ISPH method is employed to simulate natural convection from a heated cylinder in a square enclosure. It shows that the predictions obtained by the ISPH method are in good agreements with the results obtained by previous studies using alternative numerical methods. A rotating and heated cylinder is also considered to study the effect of the rotation on the heat transfer process in the enclosure space. The numerical results show that for a square enclosure at, the addition of kinetic energy in the form of rotation does not enhance the heat transfer process. The method is also applied to simulate forced convection from a circular cylinder in an unbounded uniform flow. In terms of results, it turns out that the proposed ISPH model is capable to simulate heat transfer problems with the complex and moving boundaries.

• Communications of the Korean Mathematical Society
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• v.22 no.1
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• pp.91-109
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• 2007

In this paper, we derive a change of scale formula for conditional Wiener integrals, as integral transforms, of possibly unbounded functions over Wiener paths in abstract Wiener space. In fact, we derive the change of scale formula for the product of the functions in a Banach algebra which is equivalent to both the Fresnel class and the space of measures of bounded variation over a real separable Hilbert space, and the $L_p-type$cylinder functions over Wiener paths in abstract Wiener space. As an application of the result, we obtain a change of scale formula for the conditional analytic Fourier-Feynman transform of the product of the functions.

• Honam Mathematical Journal
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• v.35 no.1
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• pp.51-71
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• 2013

In the present paper, we evaluate the analytic conditional Fourier-Feynman transforms and convolution products of unbounded function which is the product of the cylinder function and the function in a Banach algebra which is defined on an analogue o Wiener space and useful in the Feynman integration theories and quantum mechanics. We then investigate the inverse transforms of the function with their relationships and finally prove that th analytic conditional Fourier-Feynman transforms of the conditional convolution products for the functions, can be expressed in terms of the product of the conditional Fourier-Feynman transforms of each function.

• Journal of the Society of Naval Architects of Korea
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• v.32 no.3
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• pp.51-61
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• 1995

A numerical method was developed to solve the Navies-Stokes equations for unsteady laminar flow around a hydrofoil. The present method used the finite-difference scheme in the collocated grid system and the pressure-Poisson method was employed to obtain divergence-free velocity field each time step. The numerical method was applied at first to laminar flow around a circular cylinder to confirm capability of the code. In the next, calculations were carried out for a hydrofoil in an unbounded fluid at the Reynolds number of $10^4$ in order to investigate unsteady phenomena with vortex shedding. The calculate results showed reasonable features about laminar vortex shedding around a streamlined body.

• Proceedings of the KSME Conference
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• 2002.08a
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• pp.197-200
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• 2002

In this paper, roll damping coefficients for a non-conventional cross section, which is herein named as 'step' model, are investigated numerically and experimentally. Experiments are extensively carried out to estimate the roll damping coefficients. Numerical estimations are also made with the help of numerical codes. For convenience, the roll damping is divided into wave-making component and viscous component. The wave-making component is determined using a potential code and the viscous component using a viscous flow code, in which the fluid domain is taken as unbounded. In order to validate the present approach, a typical cross section with bilge is considered and our results are compared with published data. The comparison shows a good agreement qualitatively. For the step model, numerical results are compared well with experimental data besides some quantitative discrepancies at a certain range of frequency. It is thought that the discrepancy might be caused by the ignorance of the free surface in viscous computations. It is found in the case of the step model that not only the viscous component but also the wave component increases considerably compared to the section with bilge.

• Coupled systems mechanics
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• v.12 no.2
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• pp.127-149
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• 2023

A theoretical method is developed to analyze the free vibration of an elastic annular plate in contact with an ideal liquid. The displacement potential functions of the contained liquid are expressed as a combination of the Bessel functions that satisfy the Laplace equation and the liquid boundary conditions. The compatibility condition along the interface between the annular plate and the contained liquid is taken into account to consider the fluid-structure coupling. The dynamic displacement of the wet annular plate is assumed to be a combination of dry eigenfunctions, allowing for prediction of the natural frequencies using the Rayleigh-Ritz method. The study investigates the effect of radial liquid boundary conditions on the natural frequencies of the wet annular plate, considering four types of liquid bounding: outer container bounded, outer and inner bounded, inner bounded, and radially unbounded. The proposed theoretical method is validated by comparing the predicted wet natural frequencies with those obtained from finite element analysis, showing excellent accuracy. The results indicate that the radial liquid bounding effect on the natural frequencies is negligible for the axisymmetric vibrational mode, but relatively significant for the mode with one nodal diameter (n =1) and no nodal circle (m' = 0). Furthermore, the study reveals that the wet natural frequencies are the largest for the plate with an inner bounded cylinder among the radial liquid boundary cases, regardless of the vibration mode.