| Co-Investigator(Kenkyū-buntansha) |
SEKIGUCHI Tsutomu Chuo Univ.Fac.Sci.&Eng.Prof., 理工学部, 教授 (70055234)
YAMAMOTO Makoto Chuo Univ.Fac.Sci.&Eng.Prof., 理工学部, 教授 (10158305)
MATSUYAMA Yoshio Chuo Univ.Fac.Sci.&Eng.Prof., 理工学部, 教授 (70112753)
SUGIYAMA Takakazu Chuo Univ.Fac.Sci.&Eng.Prof., 理工学部, 教授 (70090371)
MURAMATU Tosinobu Chuo Univ.Fac.Sci.&Eng.Prof., 理工学部, 教授 (60027365)
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| Research Abstract |
We discuss the problem of analytical simplification of equations (A) and (B) :
(A) x^2dy/dx=lambday + f(x, y, z), x dz/dx=betaz^2 + g(x, y, z), lambdabeta*0.
(B) x^2dy/dx=(l+alphax)y + f(x, y, z), x^2dz/dx=(-l+betax)z + g(x, y, z), alpha+beta> 0.
f, g are holomorphic functions of (x, y, z) at the origin of C^3 and their Taylor series expansions in powers of y and z involve higher order terms only. The simplified equations are of the form
(A') x^2du/dx=u(lambda+alpha_0(v)+xalpha_1(v)), x^2dv/dx=v^2beta+beta'v),
(B') x^2deta/dx = eta(1+alphax+alpha'etazeta+(gamma/delta)xalpha(x)etazeta+alpha(x)(etazeta)^2),
x^2dzeta/dx = zeta(-1+betax+beta'etazeta+(gamma/delta)xalpha(x)etazeta+beta(x)(etazeta)^2).
gamma=alpha+beta-1, delta=alpha'+beta'
Here alpha(x) is holomorphic at v=0, alpha1(x) is linear in v, beta' is a constant. In (B'), alpha(x) and beta(x) are expressed by power series in x. But, we can't give any analytical meaning to them without applying Borel-Ritt Theorem. To obtain (B') the assumption delta*0 is essential.
We construct stable domains for (A') and(B') in the case of gamma* 0. Based on the stable domains we can construct analytical expressions of transformations which change (A) to (A') and (B) to (B') respectively.
Stable domains of (B') in the case of gamma0 are narrower than they are expected. So, we need another investigation.
By the way, the origin is an irreegular type singular point of rank 1 for (A) and (B). The coefficients in the leading terms of (A) are (lambda, 0) and those for (B) are (1, -1). So, both equations d'not satisfy Poincare condition at the singular point.
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