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

ISBN-10: 443173239X

ISBN-13: 9784431732396

This quantity includes 23 articles on algebraic research of differential equations and comparable issues, so much of that 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 strong instruments which were utilized to linear and non-linear differential equations in addition to many similar fields akin to genuine and intricate research, essential transforms, spectral idea, inverse difficulties, integrable structures, and mathematical physics. The articles contained right here current many new effects and concepts, offering researchers and scholars with worthwhile feedback and instructive information 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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**Sample text**

6 (ii), where C designates the crossing point of three Stokes curves observed there. We note that (21) and (24) immediately entail (23) in this case. This interpretation of a virtual turning point plays an important role in grasping the behavior of napping virtual turning points ([Sa2]) in the framework of Honda ([Ho]). References [AKKSST] T. Aoki, T. Kawai, T. Koike, S. Sasaki, S. Shudo and Y. 53-63. 42 Takashi Aoki et al. [AKKT] [AKSST] [AKT1] [AKT2] [AKT3] [AKT4] [AKT5] [BW] [BNR] [CH] [DDP] [H] [Ho] [HLO] [KKNT] [KT] [NY] [P] [Sa1] [Sa2] T.

Roughly speaking, in addition to an ordinary steepest descent path, we consider additional steepest descent paths which are bifurcated from an ordinary one at its crossing points with Stokes curves of the Laplace transformed equation. ) Thus, from the practical point of view, the determination of relevant virtual turning points is now mostly possible with the aid of a computer. As an application of exact WKB analysis for higher order diﬀerential equations we also discussed the transition probability for level crossing problems of Landau-Zener type in [112].

Alk−1 ). Note that the notion of tame regular sequences is independent of the ordering of aj ’s and that the second condition in Deﬁnition 3 is not assumed. This notion can be stated in terms of the Koszul complex. Let us denote by K(a0 , a1 , . . , al ; R) the Koszul complex associated with the sequence a0 , a1 , . . , al with coeﬃcients in R. Theorem 6. The following two conditions are equivalent: 1. The sequence a0 , a1 , . . , al is a tame regular sequence. 2. For any k = 0, . . , l and for any (k + 1) choice al0 , al1 , .

### Algebraic analysis of differential equations: from microlocal analysis to exponential asymptotics; Festschrift in honor Prof. Takahiro Kawai [on the occasion of his sixtieth birthday] by T. Aoki, H. Majima, Y. Takei, N. Tose

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