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Pham: Resurgence, quantized canonical transformations, and multiinstanton expansions, Algebraic Analysis, vol. II, Acad. Press, 1988, pp. 699-726. S. Sasaki: On the role of virtual turning points in the deformation of higher order linear ordinary differential equations, RIMS Koukyuuroku, (ISSN 1880-2818), No. 1433, 2005, pp. 27-64. , pp. 65-109. ) Virtual turning points [SKK] [Sh] [S] [T] [V] [Z] 43 M. Sato, T. Kawai and M. Kashiwara: Microfunctions and pseudodifferential equations, Lect. , No. 287, Springer, 1973, pp.

14) ⎟ ⎜ . . . ⎟ ⎜ ⎜ −1 −1 −1 . . 0 1 1 0 ⎟ ⎟ ⎜ ⎝ −1 −1 −1 . . −1 0 1 0 ⎠ −1 1 −1 . . −1 1 −1 −1 Suppose that we choose uj = 0 for some j’s instead of gj = 0. That is, let I be a subset of {0, 1, 2, . . , 2m−1} and we consider the case where uj = 0 for j ∈ I and gj = 0 for j ∈ / I. Then the coefficient matrix of the corresponding system of linear equations with respect to unknowns (−1)j+1 uj (j ∈ / I) becomes ⎞ ⎛ 0 1 1 ... 1 1 1 0 ⎜ −1 0 1 . . 1 1 1 0 ⎟ ⎟ ⎜ ⎜ −1 −1 0 . . 1 1 1 0 ⎟ ⎟ ⎜ ⎜ ..

To see that there is no u solution at the infinity, we replace u by u/λ in (N Y )02m . Then we have ⎧ ⎨ uj (uj+1 − uj+2 + · · · − uj+2m ) + αj λ2 = 0 (0 ≤ j ≤ 2m − 1), (19) ⎩ u0 + u1 + · · · + u2m = tλ. Putting λ = 0 yields Regular sequences associated with the Noumi-Yamada equations 53 ⎧ ⎨ uj (uj+1 − uj+2 + · · · − uj+2m ) = 0 (0 ≤ j ≤ 2m − 1), ⎩ (20) u0 + u1 + · · · + u2m = 0. By the arguments in the proof of Theorem 9, we see that (20) has the unique . Hence we have solution u0 = u1 = · · · = u2m = 0 which does not lie in P2m+1 u Theorem 11.

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A New Equation of State for Fluids IV. An Equation Expressing the volume as an Explicit Function of the Pressure and Temperature by Beattie J. A.


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