• 대한수학회논문집
• /
• 제13권4호
• /
• pp.759-764
• /
• 1998

We define the rational lens algebra (equation omitted)(n) as the crossed product by an action of Z on C( $S^{2n＋l}$). Assume the fibres are $M_{ k}$/(C). We prove that (equation omitted)(n) $M_{p}$ (C) is not isomorphic to C(Prim((equation omitted)(n))) $M_{kp}$ /(C) if k > 1, and that (equation omitted)(n) $M_{p{\infty}}$ is isomorphic to C(Prim((equation omitted)(n))) $M_{k}$ /(C) $M_{p{\infty}}$ if and only if the set of prime factors of k is a subset of the set of prime factors of p. It is moreover shown that if k > 1 then (equation omitted)(n) is not stably isomorphic to C(Prim(equation omitted)(n))) $M_{k}$ (c).

• 대한수학회보
• /
• 제59권5호
• /
• pp.1305-1315
• /
• 2022

Let u be a function on a connected finite graph G = (V, E). We consider the mean field equation (1) $-{\Delta}u={\rho}\({\frac{he^u}{\int_Vhe^ud{\mu}}}-{\frac{1}{{\mid}V{\mid}}}\),$ where ∆ is 𝜇-Laplacian on the graph, 𝜌 ∈ ℝ\{0}, h : V → ℝ⁺ is a function satisfying min_x∈V h(x) > 0. Following Sun and Wang [15], we use the method of Brouwer degree to prove the existence of solutions to the mean field equation (1). Firstly, we prove the compactness result and conclude that every solution to the equation (1) is uniformly bounded. Then the Brouwer degree can be well defined. Secondly, we calculate the Brouwer degree for the equation (1), say $$d_{{\rho},h}=\{{-1,\;{\rho}>0, \atop 1,\;{\rho}<0.}$$ Consequently, the equation (1) has at least one solution due to the Brouwer degree d_𝜌,h ≠ 0.

• 대한수학회보
• /
• 제60권5호
• /
• pp.1265-1280
• /
• 2023

The aim of this paper is to investigate Betchov-Da Rios equation by using null Cartan, pseudo null and partially null curve in Minkowski spacetime. Time derivative formulas of frame of s parameter null Cartan, pseudo null and partially null curve are examined, respectively. By using the obtained derivative formulas, new results are given about the solution of Betchov-Da Rios equation. The differential geometric properties of these solutions are obtained with respect to Lorentzian causal character of s parameter curve. For a solution of Betchov-Da Rios equation, it is seen that null Cartan s parameter curves are space curves in three-dimensional Minkowski space. Then all points of the soliton surface are flat points of the surface for null Cartan and partially null curve. Thus, it is seen from the results obtained that there is no surface corresponding to the solution of Betchov-Da Rios equation by using the pseudo null s parameter curve.

• Journal of the Korean Society for Industrial and Applied Mathematics
• /
• 제27권4호
• /
• pp.272-280
• /
• 2023

We study the numerical analysis for the Cahn-Hilliard (CH) equation using the decoupled projection (DP) method. The CH equation is a fourth order nonlinear partial differential equation that is hard to solve. Therefore, various of numerical schemes have been proposed to solve the CH equation. To verify the relation of each existing scheme for the CH equation, we consider the DP method for linear convex splitting schemes. We present the numerical experiments to demonstrate our analysis. Throughout this study, it is expected to construct a novel numerical scheme using the relation with existing numerical schemes.

• 한국신재생에너지학회:학술대회논문집
• /
• 한국신재생에너지학회 2010년도 추계학술대회 초록집
• /
• pp.65.1-65.1
• /
• 2010

Thermal performance tests of solar thermal collectors include determination of coefficient parameters in an efficiency equation. The parameters can be estimated using regression method to minimize an objective function as sum of differences between measured efficiency data and regressed efficiency equation. However, this conventional approach doesn't consider measurement uncertainties. In this presentation, a method to determine regression parameters in the efficiency equation and uncertainties of the parameters is described with mainly mathematical expressions based on literature reviews. In the method, parameters in the equation for collector efficiency can be determined using regression analysis with a weighting factor in the objective function. The weighting factor can be uncertainties of the differences between measured and fitted efficiencies. To evaluate the approach, performance estimation of a solar collector using the efficiency equation with uncertainties is compared to the result using the conventional efficiency equation by a simulated way for a case in one of previous studies.

• Journal of Advanced Marine Engineering and Technology
• /
• 제22권5호
• /
• pp.670-679
• /
• 1998

The solution of hyperbolic equation with wave nature has sharp discontinuties in the medium at the wave front. Difficulties encounted in the numrtical solution of such problem in clude among oth-ers numerical oscillation and the representation of sharp discontinuities with good resolution at the wave front. In this work inviscid Burgers equation and modified heat conduction equation is intro-duced as hyperboic equation. These equations are caculated by numerical methods(explicit method MacCormack method Total Variation Diminishing(TVD) method) along various Courant numbers and numerical solutions are compared with the exact analytic solution. For inviscid Burgers equa-tion TVD method remains stable and produces high resolution at sharp wave front but for modified heat Conduction equation MacCormack method is recommmanded as numerical technique.

• 한국철도학회:학술대회논문집
• /
• 한국철도학회 2008년도 춘계학술대회 논문집
• /
• pp.1676-1682
• /
• 2008

The vertical curve of railway is installed in changing point where the gradient is more than a certain degree to ensure safe operation of train and comfort of passengers. As a result of CAD work with 3D coordination by using the existing equation, it was found that the simple equation about vertical curve installation suggested in the surveying engineering or railway regulations did not reflect theoretical curvature of circular curve in a case of steep grade and large vertical radius. In this study, for derive precision equation and transition curve about vertical curve, gradient parameters, equation of circles and transition curve equation were used, and it was verified by CAD that the derived equation is exactly satisfied with parameters of geometrical circular and transition curve. This equation could be used to provide more smoothing operation of train and comfort of passengers on vertical curves, especially in case that railway requires precision vertical alignment of track such as super high-speed railway, LRT or Maglev system.

• 한국해양공학회지
• /
• 제18권2호
• /
• pp.18-24
• /
• 2004

In this study, the Mild slope equation is extended to both rapidly varying topography and nonlinear waves, using the Hamiltonian principle. It is shown that this equation is equivalent to the modified mild-slope equation (Kirby and Misra, 1998) for small amplitude wave, and it is the same form with the nonlinear mild-slope equation (Isobe, 1994) for slowly varying bottom topography. Comparing its numerical solutions with the results of some hydraulic experiments, there is good agreement between them.

• 한국수학교육학회지시리즈B:순수및응용수학
• /
• 제15권2호
• /
• pp.169-178
• /
• 2008

In this paper we generalize the superstability of the exponential functional equation proved by J. Baker et al. [2], that is, we solve an exponential type functional equation $$f(x+y)\;=\;a^{xy}f(x)f(y)$$ and obtain the superstability of this equation. Also we generalize the stability of the exponential type equation in the spirt of R. Ger[4] of the following setting $$|{\frac{f(x\;+\;y)}{{a^{xy}f(x)f(y)}}}\;-\;1|\;{\leq}\;{\delta}.$$

• Tribology and Lubricants
• /
• 제29권5호
• /
• pp.275-285
• /
• 2013

The Reynolds equation is commonly used to investigate the lubrication characteristics of a spool valve. However, the applicability of the Reynolds equation is questionable for analyzing a spool valve because cavitation often occurs in the grooves of the valve and the depth of a groove is much higher than the clearance in most cases. In this study, the validity of the Reynolds equation in the spool valve analysis is investigated by comparing the results obtained from the Reynolds equation and those obtained from the Navier-Stokes equation. The results are compared in terms of the lateral forces, friction forces, and volume flow rates (leakages). A significant difference of more than 20% is found in the lateral forces in cases where cavitation occurs and there are many grooves. Therefore, the Navier-Stokes equation should be used to investigate the lubrication characteristics of a spool valve when cavitation occurs and when the spool valve contains many grooves.