By T. Aoki, H. Majima, Y. Takei, N. Tose

This quantity includes 23 articles on algebraic research of differential equations and comparable subject matters, such a lot of which have been offered as papers on the overseas convention "Algebraic research of Differential Equations – from Microlocal research to Exponential Asymptotics" at Kyoto college in 2005. Microlocal research and exponential asymptotics are in detail hooked up and supply robust instruments which have been utilized to linear and non-linear differential equations in addition to many comparable fields comparable to genuine and complicated research, vital transforms, spectral idea, inverse difficulties, integrable platforms, and mathematical physics. The articles contained right here current many new effects and concepts, delivering researchers and scholars with beneficial feedback and instructive tips for his or her paintings. This quantity is devoted to Professor Takahiro Kawai, who's one of many creators of microlocal research and who brought the means of microlocal research into exponential asymptotics. This commitment is made at the get together of Professor Kawai's sixtieth birthday as a token of deep appreciation of the real contributions he has made to the sector. Introductory notes at the clinical works of Professor Kawai also are included.

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F2m is a tame regular sequence in C[u0 , . . , u2m , t]. Moreover, we can see that (N Y )02m has solutions as follows. For a fixed t, (N Y )02m has solutions in P2m+1 ([Ha]. 2). 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.

J. Silverstone: JWKB connection-formula problem revisited via Borel summation, Phys. Rev. , 55 (1985), 2523-2526. Y. Takei: Exact WKB analysis, and exact steepest descent method, – A sequel to “Algebraic analysis of singular perturbations”, Sˆ ugaku, 55 (2003), 350-367. (In Japanese. ) A. Voros: The return of the quartic oscillator. The complex WKB method, Ann. Inst. Henri Poincar´e, 39 (1983), 211-338. J. Zinn-Justin: Instantons in quantum mechanics: Numerical evidence for a conjecture, J. Math.

Fk ) is a tame regular sequence, there exist hij ∈ R satisfying deg(hij ) = μ − mi − mj , hij = −hji and σμ−mi (Ri ) = hij σ(fj ). j Now we set ˜ i = Ri − R hij fj . (11) j Eliminating Ri ’s by using (11) and (10), we have ˜ i fi = R hij fj fi + i j i ˜ i fi = 0. R i Since ˜ i fi ) ≤ μ − 1, max deg(R i ˜ i ’s and this proves the lemma for we can use the assumption of induction to R μ. Regular sequences associated with the Noumi-Yamada equations 51 3 Regular sequences associated with (N Y )02m Let us consider (N Y )02m in the ring C[u0 , u1 , .

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