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

This quantity comprises 23 articles on algebraic research of differential equations and similar themes, such a lot of which have been offered as papers on the foreign convention ''Algebraic research of Differential Equations вЂ“ 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 were utilized to linear and non-linear differential equations in addition to many comparable fields resembling actual and complicated research, imperative transforms, spectral idea, inverse difficulties, integrable platforms, and mathematical physics. The articles contained the following current many new effects and concepts, offering researchers and scholars with precious 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 party 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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**Additional resources for Algebraic Analysis of Differential Equations: from Microlocal Analysis to Exponential Asymptotics**

**Example text**

Then we have x0 ι ι (ξ1 − ξ3 )dx = (ξ2 − ξ1 )dx + τ1 ι (ξ3 − ξ2 )dx. (21) τ2 Hence the comparison of (16) and (21) entails that x0 is coincident with a virtual turning point x∗ . Actually all the virtual turning points studied by Sasaki ([Sa1], [Sa2]) and Honda ([Ho]) have been detected by this method. Furthermore virtual turning points thus detected play important roles in describing the phase function φ(t) used in the instanton expansion of a solution of the Noumi-Yamada system: for example the function φ(t) is given by v1 (t) (ξ1 − ξ3 )dx (22) d(t) in the situation of Fig.

Fig. 8. Here, and in what follows, a wiggly line designates a cut to ﬁx the branch of a characteristic root, and the symbol j > k attached to a Stokes curve indicates the dominance relation along the Stokes curve. ) By letting the parameter t sit on the curve we then obtain the following Fig. 9 through the limiting procedure. (Cf. [AKSST, Fig. 2]) Fig. 9. 1: virtual turning points v1 , v2 and v3 should have been taken into account in Fig. 6 (ii). The degeneration of the Stokes geometry observed in Fig.

1 1 1 0 ⎜ −1 0 1 . . 1 1 1 0 ⎟ ⎟ ⎜ ⎜ −1 −1 0 . . 1 1 1 0 ⎟ ⎟ ⎜ ⎜ .. .. ⎟ . (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 ...