By Takashi Aoki, Hideyuki Majima, Yoshitsugu Takei, Nobuyuki Tose (eds.)
This quantity includes 23 articles on algebraic research of differential equations and similar issues, so much of that have been awarded 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 comparable fields equivalent to actual and complicated research, critical transforms, spectral thought, inverse difficulties, integrable platforms, and mathematical physics. The articles contained right here current many new effects and ideas.
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 $64000 contributions he has made to the sphere. Introductory notes at the clinical works of Professor Kawai also are included.
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Extra resources for Algebraic Analysis of Differential Equations: from Microlocal Analysis to Exponential Asymptotics Festschrift in Honor of Takahiro Kawai
Delabaere, H. Dillinger and F. Pham: Exact semi-classical expansions for one dimensional quantum oscillators, J. Math. , 38 (1997), 61266184. L. , 127 (1971), 79183. N. Honda: Toward the complete description of the Stokes geometry, in prep. C. J. Howls, P. J. 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.
Proof. Let mj denote the degree of fj (j = 0, 1, . . , k) and set μ = max deg(Rj fj ). j We prove the lemma by induction on μ. The lemma trivially holds if μ < 0. Suppose that the lemma is proved for μ − 1. Taking σμ of the relation k Rj fj = 0, (10) j=0 we have k σμ−mj (Rj )σ(fj ) = 0. j=0 Since the sequence σ(f0 ), σ(f1 ), . . , σ(fk ) is a tame regular sequence, there exist hij ∈ R satisfying deg(hij ) = μ − mi − mj , hij = −hji and σμ−mi (Ri ) = hij σ(fj ). j Now we set ˜ i = Ri − R hij fj .
Here we say “most likely” just because we have not conﬁrmed that x∗ and ι belong to the same connected component of the real one-dimensional curve deﬁned by (17). This point is, however, almost automatically checked in the computer-assisted study. This reasoning can be converted to ﬁnd out a virtual turning point relevant to ι: We ﬁrst consider a curve γ deﬁned by x (ξ1 − ξ3 )dx = 0. Im (18) ι Deﬁning a function ρ(x) by x (ξ1 − ξ3 )dx, Re ι we seek for a point x0 in γ at which the following relation holds: (19) Virtual turning points ι (ξ2 − ξ1 )dx + ρ(x0 ) = 41 ι τ1 (ξ3 − ξ2 )dx.
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.)