• Kyungpook Mathematical Journal
• /
• v.47 no.1
• /
• pp.69-76
• /
• 2007

Let $f$ denote a mapping from an orthogonality space ($\mathcal{X}$, ${\bot}$) into a real Banach space $\mathcal{Y}$. In this paper, we prove the Hyers-Ulam-Rassias stability of the orthogonally cubic functional equations $f(2x+y)+f(2x-y)=2f(x+y)+2f(x-y)+12f(x)$ and $f(x+y+2z)+f(x+y-2z)+f(2x)+f(2y)=2f(x+y)+4f(x+z)+4f(x-z)+4f(y+z)+4f(y-z)$, where $x{\bot}y$, $y{\bot}z$, $x{\bot}z$.

• Journal of the Chungcheong Mathematical Society
• /
• v.19 no.3
• /
• pp.275-281
• /
• 2006

In this article, we establish the stability of the orthogonally cubic type functional equation 2f(x + 2y) + 2f(x - 2y) + 2f(2x)+7[f(x)+f(-x)] = 4f(x)+8[f(x+y)+f(x-y)], $x{\bot}y$ in which ${\bot}$ is the orthogonality in the sense in the R$\ddot{a}$tz.

• Transactions of the Korean Society of Automotive Engineers
• /
• v.2 no.4
• /
• pp.80-90
• /
• 1994

Basic constraint equations derived from orthogonality conditions between a pair of body-fixed vectors and a body-fixed vector or a vector between two bodies are reformulated by using relative coordinate kinematics between two adjacent reference frames. Arithmetic numbers of operations required to compute derivatives of the constraint equations are drastically reduced. A mixed formulation of relative and cartesian coordinates is developed to further simplify derivatives of the constraints. Advantages and disadvantages of the new formulation are discussed. Possible singularity problem of para llelism constraints is resolved by introducing an extra generalized coordinate. Kinematic analysis of a McPherson strut suspension system are carried out to illustrate use and efficiency of the new formulation.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
• /
• 1998.04a
• /
• pp.536-541
• /
• 1998

Cylindrical chamber silencers with an extended inlet and outlet are widely used to reduce the propagated noise in ducts in many application fields. In this study, an acoustic analysis is carried out for concentric extended pipes inserted into a simple expansion chambers An algebraic equation is derived by using the eigenfunction expansion and orthogonality principle. Using this analytical method, transmission losses are predicted for several configurations of the concentric extended systems.

• Bulletin of the Korean Mathematical Society
• /
• v.41 no.3
• /
• pp.493-505
• /
• 2004

We introduce certain local Thomsen condition in a 3-web and prove that it is equivalent to the equation a－(a－b)＝b in its corresponding loop, where we denote the loop operation additively for convenience and simplicity, even though the loop is neither associative nor commutative. Also we interpret such local Thomsen condition using orthogonality of chains in a web.

• Transactions of the Korean Society of Mechanical Engineers A
• /
• v.21 no.12
• /
• pp.2038-2046
• /
• 1997

In this paper the instability of the system which has a disk and a mass-spring-damper system interacting through a medium having stiffness and damping is analyzed. To solve the equations of motion of this systme, it is assumed that the solution consists of the eigenfunctions which are the products of the Bessel functions and sine or cosine functions. The former represents the radial characteristics of the disk and the latter represents the circumferential characteristics. Using this assumed solution and the orthogonality of the eigenfunctions, the equations of motion can be transformed into a set of equations of motion with variables dependent only on the time. After this set is changed to the state equation, the eigenvalue problem can be made. Once the eigenvalues are calculated according to the angular velocity of the disk, the dynamic characteristics ofthis system is obtained. Because the thickness of the disk and the element characteristics of the mass-spring-damper system have important effects on the stability of the system, it will be understood how these factors affect the system and then a method to ameliorate the stability of the system with a disk will be presented.

• Journal of the Korean Society for Industrial and Applied Mathematics
• /
• v.18 no.4
• /
• pp.305-316
• /
• 2014

The Hodge-Helmholtz decomposition splits a vector field into the unique sum of a divergence-free vector field (solenoidal part) and a gradient field (irrotational part). In a bounded domain, a boundary condition needs to be supplied to the decomposition. The decomposition with the non-penetration boundary condition is equivalent to solving the Poisson equation with the Neumann boundary condition. The Gibou-Min method is an application of the Poisson solver by Purvis and Burkhalter to the decomposition. Using the $L^2$-orthogonality between the error vector and the consistency, the convergence for approximating the divergence-free vector field was recently proved to be $O(h^{1.5})$ with step size h. In this work, we analyze the convergence of the irrotattional in the decomposition. To the end, we introduce a discrete version of the Poincare inequality, which leads to a proof of the O(h) convergence for the scalar variable of the gradient field in a domain with general intersection property.

• Honam Mathematical Journal
• /
• v.41 no.2
• /
• pp.403-420
• /
• 2019

The Helmholtz equation represents acoustic or electromagnetic scattering phenomena. The Method of Lines are known to have many advantages in simulation of forward and inverse scattering problems due to the usage of angle rays and Bessel functions. However, the method does not account for the jump phenomena on obstacle boundary and the approximation includes many high order Bessel functions. The high order Bessel functions have extreme blow-up or die-out features in resonance region obstacle boundary. Therefore, in particular, when we consider shape reconstruction problems, the method is suffered from severe instabilities due to the logical confliction and the severe singularities of high order Bessel functions. In this paper, two approximation formulas for the Helmholtz equation are introduced. The formulas are new and powerful. The derivation is based on Method of Lines, Huygen's principle, boundary jump relations, Addition Formula, and the orthogonality of the trigonometric functions. The formulas reduce the approximation dimension significantly so that only lower order Bessel functions are required. They overcome the severe instability near the obstacle boundary and reduce the computational time significantly. The convergence is exponential. The formulas adopt the scattering jump phenomena on the boundary, and separate the boundary information from the measured scattered fields. Thus, the sensitivities of the scattered fields caused by the boundary changes can be analyzed easily. Several numerical experiments are performed. The results show the superiority of the proposed formulas in accuracy, efficiency, and stability.

• Journal of the Korean Society for Precision Engineering
• /
• v.23 no.8 s.185
• /
• pp.72-79
• /
• 2006

We measured grating pitch standards using optical diffractometry and analyzed measurement uncertainty. Grating pitch standards have been used widely as a magnification standard for a scanning electron microscope (SEM) and a scanning probe microscope (SPM). Thus, to establish the meter-traceability in nano-metrology using SPM and SEM, it is important to certify grating pitch standards accurately. The optical diffractometer consists of two laser sources, argon ion laser (488 nm) and He-Cd laser (325 nm), optics to make an incident beam, a precision rotary table and a quadrant photo-diode to detect the position of diffraction beam. The precision rotary table incorporates a calibrated angle encoder, enabling the precise and accurate measurement of diffraction angle. Applying the measured diffraction angle to the grating equation, the mean pitch of grating specimen can be obtained very accurately. The pitch and orthogonality of two-dimensional grating pitch standards were measured, and the measurement uncertainty was analyzed according to the Guide to the Expression of Uncertainty in Measurement. The expanded uncertainties (k = 2) in pitch measurement were less than 0.015 nm and 0.03 nm for the specimen with the nominal pitch of 300 nm and 1000 nm. In the case of orthogonality measurement, the expanded uncertainties were less than $0.006^{\circ}$. In the pitch measurement, the main uncertainty source was the variation of measured pitch values according to the diffraction order. The measurement results show that the optical diffractometry can be used as an effective calibration tool for grating pitch standards.

• Journal of the Society of Naval Architects of Korea
• /
• v.36 no.1
• /
• pp.70-81
• /
• 1999

To predict the viscous boundary layers and wakes around a ship, the CFD techniques are commonly used. For the efficient application of CFD tools to practical hull farms, a 3-D field grid generating system is developed. The present grid generating system utilizes the solution of Poisson equation. Sorenson's method developed for 2-D is extended into 3-D to provide the forcing functions controling grid interval and orthogonality on hull surface, etc. The weighting function scheme is used for the discretization of the Poisson equation and the linear equations are solved by using MSIP salver. The trans-finite interpolation is also employed to assure the smooth transition into boundary surface grids. To rove the applicability, the field grid systems around a container ship and a VLCC with bow and stem bulb are illustrated, and the procedures for generating 3-D field grid system are explained.