• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제18권1호
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• pp.27-41
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• 2014

In this paper, we present an efficient numerical method for multiphase image segmentation using a multiphase-field model. The method combines the vector-valued Allen-Cahn phase-field equation with initial data fitting terms containing prescribed interface width and fidelity constants. An efficient numerical solution is achieved using the recently developed hybrid operator splitting method for the vector-valued Allen-Cahn phase-field equation. We split the modified vector-valued Allen-Cahn equation into a nonlinear equation and a linear diffusion equation with a source term. The linear diffusion equation is discretized using an implicit scheme and the resulting implicit discrete system of equations is solved by a multigrid method. The nonlinear equation is solved semi-analytically using a closed-form solution. And by treating the source term of the linear diffusion equation explicitly, we solve the modified vector-valued Allen-Cahn equation in a decoupled way. By decoupling the governing equation, we can speed up the segmentation process with multiple phases. We perform some characteristic numerical experiments for multiphase image segmentation.

• 대한수학회보
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• 제59권5호
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• pp.1305-1315
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• 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.

• 한국전기전자재료학회논문지
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• 제18권1호
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• pp.89-95
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• 2005

We derived a formula which can calculate the space distribution of refractive index variation by an applied electric field about Bi$_{12}$ GeO$_{20}$ single crystal. Stereographic projection maps of refractive index variation by an applied electric field were made out using numerical value to be calculated by this formula. By the calculated results, since an electric field had applied to [(equation omitted) 1 1] direction and [1 (equation omitted) 1] direction of Bi$_{12}$ GeO$_{20}$ crystal, positive variation of the refractive index of [(equation omitted) 1 1] direction and [1 (equation omitted) 1] direction was the largest. The incremented refractive index per unit electric field was +3.2410${\times}$10$^{-11}$ V$^{-1}$ for the wavelength of 6328 $\AA$. Since an electric field had applied to [1 1 1] direction and [(equation omitted) 1] direction, negative variation of the refractive index of [1 1 1] direction and [(equation omitted) 1] direction was the largest. The decremented refractive index per unit electric field was -3.2410${\times}$10$^{-11}$ V$^{-1}$ for the wavelength of 6328 $\AA$.

• 대한수학회지
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• 제35권4호
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• pp.903-931
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• 1998

Since the modular curves X(N) = $\Gamma$(N)\(equation omitted)＊ (N =1,2,3) have genus 0, we have field isomorphisms K(X(l))(equation omitted)C(J), K(X(2))(equation omitted)(λ) and K(X(3))(equation omitted)( $j_3$) where J, λ are the classical modular functions of level 1 and 2, and $j_3$ can be represented as the quotient of reduced Eisenstein series. When N = 4, we see from the genus formula that the curve X(4) is of genus 0 too. Thus the field K(X(4)) is a rational function field over C. We find such a field generator $j_4$(z) = x(z)/y(z) (x(z) = $\theta$$_3$((equation omitted)), y(z) = $\theta$$_4$((equation omitted)) Jacobi theta functions). We also investigate the structures of the spaces $M_{k}$($\Gamma$(4)), $S_{k}$($\Gamma$(4)), M(equation omitted)((equation omitted)(4)) and S(equation omitted)((equation omitted)(4)) in terms of x(z) and y(z). As its application, we apply the above results to quadratic forms.rms.

• 한국소음진동공학회:학술대회논문집
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• 한국소음진동공학회 2007년도 춘계학술대회논문집
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• pp.1062-1073
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• 2007

Acoustic Target Strength (TS) is a major parameter of the active sonar equation, which indicates the ratio of the radiated intensity from the source to the re-radiated intensity by a target. In developing a TS equation, it is assumed that the radiated pressure is known and the re-radiated intensity is unknown. This research provides a TS equation for polygonal plates, which is applicable to near field acoustics. In this research, Helmholtz-Kirchhoff formula is used as the primary equation for solving the re-radiated pressure field; the primary equation contains a surface (double) integral representation. The double integral representation can be reduced to a closed form, which involves only a line (single) integral representation of the boundary of the surface area by applying Stoke's theorem. Use of such line integral representations can reduce the cost of numerical calculation. Also Kirchhoff approximation is used to solve the surface values such as pressure and particle velocity. Finally, a generalized definition of Sonar Cross Section (SCS) that is applicable to near field is suggested. The TS equation for polygonal plates in near field is developed using the three prescribed statements; the redection to line integral representation, Kirchhoff approximation and a generalized definition of SCS. The equation developed in this research is applicable to near field, and therefore, no approximations are allowed except the Kirchhoff approximation. However, examinations with various types of models for reliability show that the equation has good performance in its applications. To analyze a general shape of model, a submarine type model was selected and successfully analyzed.

• Korean Journal of Mathematics
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• 제18권1호
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• pp.21-28
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• 2010

In the unified field theory(UFT), many works on the solutions of Einstein's equation have been published. The main goal in the present paper is to obtain some geometric results on a particular solution of Einstein's equation under some condition in even-dimensional UFT $X_n$.

• 대한전기학회논문지:전기물성ㆍ응용부문C
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• 제52권4호
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• pp.185-191
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• 2003

In this Paper, we propose an accurate and stable solution of the transient electromagnetic response from three-dimensional arbitrarily shaped conducting objects by using a time domain magnetic field integral equation. This method does not utilize the conventional marching-on in time (MOT) solution. Instead we solve the time domain integral equation by expressing the transient behavior of the induced current in terms of temporal expansion functions with decaying exponential functions and Laguerre·polynomials. Since these temporal expansion functions converge to zero as time progresses, the transient response of the induced current does not have a late time oscillation and converges to zero unconditionally. To show the validity of the proposed method, we solve a time domain magnetic field integral equation for three closed conducting objects and compare the results of Mie solution and the inverse discrete Fourier transform (IDFT) of the solution obtained in the frequency domain.

• KIEE International Transactions on Electrophysics and Applications
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• 제4C권2호
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• pp.45-50
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• 2004

The field utilization factor (equation omitted) (the mean electric field / the maximum electric field) of standard sphere gaps was calculated by the charge simulation method, taking into account the ground plane and shanks. n changes mainly with g/r and slightly with 1$_1$, 1$_2$ and 1, where D=2r is the sphere diameter, g is the gap length, 1$_1$ and 1$_2$, respectively, are the lengths of the upper and lower shank, and t is the shank diameter. Generally, (equation omitted) increases as 1$_1$,1$_2$ and t each becomes larger. IEC standard 60052(2002) limits t$\leq$0.2D 1$_1$$\geq$1D and prescribes A=1$_2$＋D＋g where A is the height of the spark point on the upper sphere. Therefore, (equation omitted) is the largest when A=9D and the smallest when A=3D. The simple equation of a straight line, (equation omitted)=1- (g/3r), can generally be used as a representative value of (equation omitted) for a wide variety of sphere diameters that are permitted by the IEC standard. The maximum electric field E$_{m}$ at sparkover of standard air gaps has also been calculated by the relation E$_{m}$=V/(equation omitted)g). E$_{m}$ describes a U-curve for g/r, up to the sphere diameter of 1 m. Moreover, for 1.5-m and 2-m diameters and especially .for negative polarity, sparkover voltages have been calculated by integration of the ionization index.index.

• 한국초전도저온공학회:학술대회논문집
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• 한국초전도저온공학회 2002년도 학술대회 논문집
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• pp.315-317
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• 2002

The value of I$_{c}$(critical current) in HTS (High Temperature Superconducting) tape has a great influence on B(equation omitted) (magnetic field amplitude applied perpendicular to the tape surface). Therefore, I$_{c}$ of HTS magnet is determined by not only operating temperature but also the B(equation omitted). In shape design of field coil for the HTS motor, a method to reduce the B(equation omitted) and to determine operating current should be considered in order to optimal design. On the basis of the magnetic field analysis, this paper deals with various field coil shape to obtain operating current of HTS motor by using analytical method. And also this paper discusses the operating current of 100hp class HTS motor by using I$_{c}$-B(equation omitted) curve.curve.

• 한국진공학회지
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• 제17권5호
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• pp.400-407
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• 2008

본 연구에서는 축전 결합형 플라즈마원에 대한 전자기장의 2차원 공간 의존성을 계산하였다. 1차원 유체 방정식을 기반으로 축전 전기장과 전도 전류 밀도의 axial 방향 공간 의존성을 계산한 후, radial 방향으로는 맥스웰 방정식의 해를 ${\omega}r/c$에 대한 power series로 전개하여 전자기장의 2차원 공간 의존성을 계산하였다.