By T. Aoki, H. Majima, Y. Takei, N. Tose
This quantity includes 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 hooked up and supply strong instruments which have been utilized to linear and non-linear differential equations in addition to many similar fields comparable to actual and intricate research, imperative transforms, spectral idea, inverse difficulties, integrable structures, and mathematical physics. The articles contained the following current many new effects and concepts, offering researchers and scholars with worthy feedback and instructive assistance 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 real 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]
See [Sh] for such equations. For example, in the case of the operator P given by (3), the associated bicharacteristic strip passing through (x, y; ξ, η) = (1, 0; −i, 1) is given by ⎧ x(t) = −4(t + 1/2)(t2 + t − 1/2) ⎪ ⎪ ⎪ ⎨ y(t) = −6it2 (t + 1)2 ⎪ ξ(t) = −2it − i ⎪ ⎪ ⎩ η(t) = 1. (4) Hence its projection to the base space forms a (unique) self-intersection point at (x, y) = (0, −3i/2). The situation is schematically illustrated in Fig. 3 with an appropriate labelling of solutions of the following characteristic equation of P : ξ 3 + 3η 2 ξ + 2ixη 3 = 0 with η = 1.
A natural question then arises: What is a new Stokes curve? Bringing in the microlocal analysis to the study of singularities of Borel transformed WKB solutions, Kawai gave the following intriguing answer to the above question in : Let P (x, ∂x , ∂y ) be the Borel transform of the operator P (x, ∂x , η) in question and take a self-intersection point of the bicharacteristic curve of P (x, ∂x , ∂y ). , the projection of a self-intersection point to the x-space) a “new turning point”, then a new Stokes curve is a Stokes curve (in the ordinary sense) that emanates from a new turning point —.
Hereafter let R denote the ring C[u0 , . . , un−1 ] of polynomials of u0 , u1 , . . , un−1 with complex coeﬃcients and let f0 , f1 , . . , fl be elements of R. Theorem 7. The following two conditions are equivalent: 1. The sequence f0 , f1 , . . , fl is a tame regular sequence in R. 2. For any point x0 in Cn , the sequence f0 , f1 , . . , fl is a tame regular sequence at x0 . Regular sequences associated with the Noumi-Yamada equations 49 Proof. We denote respectively by L· and by L· the Koszul complexes associated with the sequence f0 , f1 , .
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