• Title, Summary, Keyword: Jacobi equation

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VOLUME PROBLEMS ON LORENTZIAN MANIFOLDS

  • Kim, Seon-Bu
    • Communications of the Korean Mathematical Society
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    • v.10 no.1
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    • pp.163-173
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    • 1995
  • Inspired in [2,9,10,17], pp. E. Ehrlich and S. B. Kim in [4] cultivated the Riccati equation related to the Raychaudhuri equation of General Relativity for the stable Jacobi tensor along the geodesics to extend the Hawking-Penrose conjugacy theorem to $$ f(t) = Ric(c(t)',c'(t)) + tr(\sigma(A)^2) $$ where $\sigma(A)$ is the shear tensor of A for the stable Jacobi tensor A with $A(t_0) = Id$ along the complete Riemannian or complete nonspacelike geodesics c.

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NEW EXACT TRAVELLING WAVE SOLUTIONS FOR SOME NONLINEAR EVOLUTION EQUATIONS

  • Lee, Youho;An, Jaeyoung;Lee, Mihye
    • Journal of the Chungcheong Mathematical Society
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    • v.24 no.2
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    • pp.359-370
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    • 2011
  • In this work, we obtain new solitary wave solutions for some nonlinear partial differential equations. The Jacobi elliptic function rational expansion method is used to establish new solitary wave solutions for the combined KdV-mKdV and Klein-Gordon equations. The results reveal that Jacobi elliptic function rational expansion method is very effective and powerful tool for solving nonlinear evolution equations arising in mathematical physics.

ON A MOVING GRID NUMBERICAL SCHEME FOR HAMILTON-JACOBI EQUATIONS

  • Hong, Bum-Il
    • Journal of the Korean Mathematical Society
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    • v.33 no.2
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    • pp.249-258
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    • 1996
  • Analysis by the method of characteristics shows that if f and $u_0$ are smooth and $u_0$ has compact support, then the Hamilton-Jacobi equation $$ (H-J) ^{u_t + f(u_x) = 0, x \in R, t > 0, } _{u(x, 0) = u_0(x), x \in R, } $$ has a unique $C^1$ solution u on some maximal time interval $0 \leq t < T$ for which $lim_{t \to T}u(x, t) exists uniformly; but this limiting function is not continuously differentiable.

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ON THE DIOPHANTINE EQUATION (5pn2 - 1)x + (p(p - 5)n2 + 1)y = (pn)z

  • Kizildere, Elif;Soydan, Gokhan
    • Honam Mathematical Journal
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    • v.42 no.1
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    • pp.139-150
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    • 2020
  • Let p be a prime number with p > 3, p ≡ 3 (mod 4) and let n be a positive integer. In this paper, we prove that the Diophantine equation (5pn2 - 1)x + (p(p - 5)n2 + 1)y = (pn)z has only the positive integer solution (x, y, z) = (1, 1, 2) where pn ≡ ±1 (mod 5). As an another result, we show that the Diophantine equation (35n2 - 1)x + (14n2 + 1)y = (7n)z has only the positive integer solution (x, y, z) = (1, 1, 2) where n ≡ ±3 (mod 5) or 5 | n. On the proofs, we use the properties of Jacobi symbol and Baker's method.

ON THE MODULAR FUNCTION $j_4$ OF LEVEL 4

  • Kim, Chang-Heon;Koo, Ja-Kyung
    • Journal of the Korean Mathematical Society
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    • v.35 no.4
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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.

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Nonlinear $H_{\infty}$ control to semi-active suspension

  • Sampei, Mitsuji;Kubota, Kenta;Hosokawa, Atsukuni;Laosuwan, Patpong
    • 제어로봇시스템학회:학술대회논문집
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    • pp.287-290
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    • 1995
  • Recently H$_{\infty}$ control theory for nonlinear systems based on the Hamilton-Jacobi inequality has been developed. In this paper, we apply the state feedback controller solved via Riccati equation to a semi-active suspension model, two degree of freedom vehicle model, and show that it is effective for vibration control..

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Nonlinear H$\infty$ Control for Linear Systems using Nonlinear Weight

  • Kubota, K.;Samei, M.;Shimizu, E;Koga, M.
    • 제어로봇시스템학회:학술대회논문집
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    • pp.60-63
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    • 1996
  • This study deals with the nonlinear H$_{\infty}$ control problem of linear system using nonlinear weight. Generally the solvable condition of nonlinear H$_{\infty}$ control problem is given by the Hamilton Jacobi equality or inequality, but it is very difficult to solve. In this study, some constraints of nonlinear weight reduce the solvable condition to linear Riccati equation. Some examples of the control system design using nonlinear weight are shown.n.

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Computational Solution of a H-J-B equation arising from Stochastic Optimal Control Problem

  • Park, Wan-Sik
    • 제어로봇시스템학회:학술대회논문집
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    • pp.440-444
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    • 1998
  • In this paper, we consider numerical solution of a H-J-B (Hamilton-Jacobi-Bellman) equation of elliptic type arising from the stochastic control problem. For the numerical solution of the equation, we take an approach involving contraction mapping and finite difference approximation. We choose the It(equation omitted) type stochastic differential equation as the dynamic system concerned. The numerical method of solution is validated computationally by using the constructed test case. Map of optimal controls is obtained through the numerical solution process of the equation. We also show how the method applies by taking a simple example of nonlinear spacecraft control.

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