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

This quantity includes 23 articles on algebraic research of differential equations and comparable themes, 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 strong instruments which have been utilized to linear and non-linear differential equations in addition to many similar fields comparable to genuine and complicated research, critical transforms, spectral concept, inverse difficulties, integrable platforms, and mathematical physics. The articles contained right here current many new effects and ideas, delivering researchers and scholars with necessary 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 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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**Additional resources 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]**

**Example text**

Bender and T. T. Wu: Anharmonic oscillator, Phys. , 184 (1969), 1231-1260. H. L. Berk, W. M. Nevins and K. V. Roberts: New Stokes’ line in WKB theory, J. Math. , 23 (1982), 988-1002. R. Courant and D. Hilbert: Methods of Mathematical Physics, II, Interscience, 1962. E. 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.

Fl }, the dimension of V (x0 , fl0 , . . , flk ) is equal to n − k − 1 or V (x0 , fl0 , . . , flk ) is an empty set. Now let us study the relation between the notion of tame regular sequences and the Koszul complex. We denote by L· = K(f0 , f1 , . . , fl ; O) the Koszul complex associated with the sequence f0 , f1 , . . , fl with coeﬃcients in O. 4). If f0 , f1 , . . , fl is a tame regular sequence at x0 , then we have H k (L·x0 ) = 0 for every k ≤ l. If we look at the l-th order part of the Koszul complex, we have the following: Theorem 4.

L. Berk, W. M. Nevins and K. V. Roberts: New Stokes’ line in WKB theory, J. Math. , 23 (1982), 988-1002. R. Courant and D. Hilbert: Methods of Mathematical Physics, II, Interscience, 1962. E. 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.