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

This quantity includes 23 articles on algebraic research of differential equations and similar issues, so much of which have been provided as papers on the overseas convention ''Algebraic research of Differential Equations – from Microlocal research to Exponential Asymptotics'' at Kyoto collage in 2005. Microlocal research and exponential asymptotics are in detail attached and supply robust instruments which have been utilized to linear and non-linear differential equations in addition to many similar fields resembling genuine and complicated research, essential transforms, spectral concept, inverse difficulties, integrable platforms, and mathematical physics. The articles contained right here current many new effects and ideas, delivering researchers and scholars with necessary feedback and instructive advice 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 celebration 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 medical works of Professor Kawai also are included.

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Parenthetically we note that the importance of the Borel resummation in WKB analysis is also shown by Bender-Wu ([BW]), Voros ([V]), Zinn-Justin ([Z]), Pham ([P]) and others from several points of view and that the exact WKB analysis means the WKB analysis based on the Borel resummation. , a point a satisfying ξj (a) = ξk (a), (2) where ξj (x) and ξk (x) are characteristic roots of the operator P . , points satisfying (2)) does not suffice to describe the Stokes region as Berk-Nevins-Roberts ([BNR]) first pointed out; we need a “new Stokes curve” that does not emanate from a traditional turning point if we want to find correct Stokes regions.

K) and set μ = max deg(Rj fj ). j We prove the lemma by induction on μ. The lemma trivially holds if μ < 0. Suppose that the lemma is proved for μ − 1. Taking σμ of the relation k Rj fj = 0, (10) j=0 we have k σμ−mj (Rj )σ(fj ) = 0. j=0 Since the sequence σ(f0 ), σ(f1 ), . . , σ(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.

The sequence f0 , f1 , . . , fl is said to be a tame regular sequence at x0 if for any integer k so that 0 ≤ k ≤ l and for any (k + 1) choice fl0 , fl1 , . . , flk of elements in {f0 , f1 , . . , fl }, the element flk is not a zero divisor on Ox0 /(fl0 , . . , flk−1 ). Note that we do not assume x0 to be a common zero of f0 , f1 , . . , fl . As in the case of regular sequences, we have Theorem 2. Let f0 , f1 , . . , fl be elements in Ox0 . Then the following two conditions are equivalent: 1.

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