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

This quantity includes 23 articles on algebraic research of differential equations and similar issues, so much of which have been provided as papers on the overseas convention ''Algebraic research of Differential Equations вЂ“ from Microlocal research to Exponential Asymptotics'' at Kyoto collage in 2005. Microlocal research and exponential asymptotics are in detail attached and supply robust instruments which have been utilized to linear and non-linear differential equations in addition to many similar fields resembling genuine and complicated research, essential 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 advice 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 celebration 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.

**Read or Download Algebraic Analysis of Differential Equations: from Microlocal Analysis to Exponential Asymptotics PDF**

**Similar mathematics books**

**Trigonometric Delights (Princeton Science Library)**

Trigonometry has constantly been an underappreciated department of arithmetic. It has a name as a dry and tough topic, a glorified kind of geometry advanced by means of tedious computation. during this ebook, Eli Maor attracts on his amazing abilities as a advisor to the area of numbers to dispel that view. Rejecting the standard arid descriptions of sine, cosine, and their trigonometric family, he brings the topic to lifestyles in a compelling mixture of historical past, biography, and arithmetic.

**Mathematical Olympiad Challenges**

Mathematical Olympiad demanding situations is a wealthy selection of difficulties prepare by way of skilled and recognized professors and coaches of the U. S. overseas Mathematical Olympiad staff. hundreds of thousands of demanding and instructive difficulties from algebra, geometry, trigonometry, combinatorics, and quantity concept have been chosen from quite a few mathematical competitions and journals.

**Introduction to Mathematical Philosophy**

Advent to Mathematical Philosophy is a ebook that was once written by means of Bertrand Russell and released in 1919. the point of interest of the booklet is at the idea of description and it provides the information present in Principia Mathematica in a better approach to comprehend. Bertrand Russell was once a British thinker, philosopher, and mathematician.

- Equivalence Transformations for Classes of Differential Equations
- Stochastic differential equations theory and applications
- Select Exercises for Young Proficients in the Mathematicks.
- After Gödel: Platonism and Rationalism in Mathematics and Logic
- Quo Vadis, Graph Theory?A Source Book for Challenges and Directions
- Manifolds in image science and visualization

**Extra info for Algebraic Analysis of Differential Equations: from Microlocal Analysis to Exponential Asymptotics**

**Sample text**

Parenthetically we note that the importance of the Borel resummation in WKB analysis is also shown by Bender-Wu ([BW]), Voros ([V]), Zinn-Justin ([Z]), Pham ([P]) and others from several points of view and that the exact WKB analysis means the WKB analysis based on the Borel resummation. , a point a satisfying ξj (a) = ξk (a), (2) where ξj (x) and ξk (x) are characteristic roots of the operator P . , points satisfying (2)) does not suﬃce to describe the Stokes region as Berk-Nevins-Roberts ([BNR]) ﬁrst pointed out; we need a “new Stokes curve” that does not emanate from a traditional turning point if we want to ﬁnd correct Stokes regions.

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 . (11) j Eliminating Ri ’s by using (11) and (10), we have ˜ i fi = R hij fj fi + i j i ˜ i fi = 0.

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 , . . , flk of elements in {f0 , f1 , . . , fl }, the element flk is not a zero divisor on Ox0 /(fl0 , . . , flk−1 ). Note that we do not assume x0 to be a common zero of f0 , f1 , . . , fl . As in the case of regular sequences, we have Theorem 2. Let f0 , f1 , . . , fl be elements in Ox0 . Then the following two conditions are equivalent: 1.