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

This quantity comprises 23 articles on algebraic research of differential equations and similar subject matters, so much of which have been offered as papers on the foreign 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 comparable fields equivalent to genuine and intricate research, crucial transforms, spectral idea, inverse difficulties, integrable platforms, and mathematical physics. The articles contained the following current many new effects and ideas, delivering researchers and scholars with important 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 $64000 contributions he has made to the sphere. Introductory notes at the medical works of Professor Kawai also are included.

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**Extra info for Algebraic analysis of differential equations: from microlocal analysis to exponential asymptotics; Festschrift in honor Prof. Takahiro Kawai [on the occasion of his sixtieth birthday]**

**Sample text**

9 (i) and (ii) are switched by the following Fig. 10, which is observed when another simple turning point s2 hits the Stokes curves in question and causes their bifurcation as we explained in Section 2. Comparison of Fig. 2. Resolution of the degeneration in Fig. 9 (ii) by a tiny change of t induces the change of topological conﬁgurations of Stokes curves of (12), as is observed in Fig. 8 (ii)+ and (ii)− ; it is exactly in the same manner as the result of the resolution of the degeneration observed in Fig.

I sincerely wish he may enjoy mathematics and stimulate us as ever even after he is over sixty years old. M. T. Wu: Anharmonic oscillator, Phys. , 184(1969), 1231–1260. L. M. V. Roberts: New Stokes’ line in WKB theory, J. Math. , 23(1982), 988–1002. [DDP] E. Delabaere, H. Dillinger et F. Pham: R´esurgence de Voros et p´eriodes des courbes hyperelliptiques, Ann. Inst. Fourier (Grenoble), 43(1993), 163–199. [E] J. Ecalle: Cinq applications des fonctions r´esurgentes, Pr´epublication d’Orsay 84T62, Univ.

Actually a traditional turning point of a Schr¨ odinger operator P = d2 /dx2 − η 2 Q(x) is of this character: the ϕ of the equation P ϕ = 0 has Borel transform ϕB (x, y) of a WKB solution x√ two singularities s± = {(x, y); y = ± a Qdx} with Q(a) = 0, and they coalesce at (x, y) = (a, 0). Let us now raise the following question: In what sense are s+ and s− cognate? To answer this question, we have to understand the structure of singularities of ϕB (x, y). , a is a simple zero of the potential Q(x), there exists a non-singular bicharacteristic strip of the Borel transform PB of P whose projection to the base manifold C2(x,y) is s+ ∪ s− near (x, y) = (a, 0).