By Takashi Aoki, Hideyuki Majima, Yoshitsugu Takei, Nobuyuki Tose (eds.)
This quantity includes 23 articles on algebraic research of differential equations and comparable subject matters, such a lot of that have been provided 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 robust instruments which have been utilized to linear and non-linear differential equations in addition to many similar fields reminiscent of actual and complicated research, critical transforms, spectral conception, inverse difficulties, integrable structures, and mathematical physics. The articles contained the following current many new effects and concepts.
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 social gathering 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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Additional resources for Algebraic Analysis of Differential Equations: from Microlocal Analysis to Exponential Asymptotics Festschrift in Honor of Takahiro Kawai
It is clear that the sequence F0 , . . , F2m is a regular sequence in C[u0 , . . , u2m ] for every ﬁxed t. Hence (N Y )02m has a ﬁnite number of solutions for every ﬁxed t. Under suitable generic condition, the number is 22m and the number of ramiﬁcation points of solutions over t is 2m22m . These will be proved in our forthcoming paper. Remark 2. Of course we can consider (N Y )02m+1 for (N Y )2m+1 . The sequence of polynomials corresponding to (N Y )02m+1 is, however, not a regular sequence in C[u0 , .
Fl is a regular sequence at x0 . 2. For each k = 0, 1, . . , l, the dimension of V (x0 , f0 , . . , fk ) is equal to n − k − 1. 3. The dimension of V (x0 , f0 , . . , fl ) is n − l − 1. Thus, at least locally, the notion of regular sequences does not depend on the ordering of fj ’s. Deﬁnition 2. Let f0 , f1 , . . , fl be elements in Ox0 . 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 , .
Langman and A. B. Olde Daalhuis: On the higher-order Stokes phenomenon, Proc. R. Soc. Lond. A, 460 (2004), 2285-2303. T. Kawai, T. Koike, Y. Nishikawa and Y. Takei: On the Stokes geometry of higher order Painlev´e equations, Ast´erisque, No. 297, 2004, pp. 117166. T. Kawai and Y. Takei: Algebraic Analysis of Singular Perturbation Theory, Iwanami, Tokyo, 1998. (In Japanese; English translation, AMS, 2005) (1) M. Yamada: Higher order Painlev´e Equations of type Al , Funkcial. , 48 (1998), 483-503.
Algebraic Analysis of Differential Equations: from Microlocal Analysis to Exponential Asymptotics Festschrift in Honor of Takahiro Kawai by Takashi Aoki, Hideyuki Majima, Yoshitsugu Takei, Nobuyuki Tose (eds.)