Computer Algebra in Scientific Computing: 12th International by Sergey Abrahamyan (auth.), Vladimir P. Gerdt, Wolfram Koepf,

By Sergey Abrahamyan (auth.), Vladimir P. Gerdt, Wolfram Koepf, Ernst W. Mayr, Evgenii V. Vorozhtsov (eds.)

The CASC Workshops are typically held in flip within the Commonwealth of IndependentStates(CIS)andoutsideCIS(Germanyinparticular,but,attimes, additionally different nations with full of life CA communities). the former CASC Wo- store was once held in Japan, and the twelfth workshop was once held for the ?rst time in Armenia, that's one of many CIS republics. it may be famous that greater than 35 institutes and scienti?c facilities functionality in the nationwide Academy of S- ences of Armenia (further information about the constitution of the academy may be foundhttp://www. sci. am). those associations are involved, particularly, with difficulties in such branches of common technological know-how as arithmetic, informatics, physics, astronomy, biochemistry, and so on. It follows from the talks offered on the prior CASC workshops that the tools and structures of computing device algebra could be utilized effectively in the entire above-listed branches of typical sciences. as a result, the organizers of the twelfth CASC Workshop wish that the current workshop may also help the Armenian scientists to turn into much more acquainted with the functions of complicated machine algebra equipment and platforms and to get involved with experts in computing device algebra from different nations. The eleven past CASC meetings, CASC 1998, CASC 1999, CASC 2000, CASC 2001, CASC 2002, CASC 2003, CASC 2004, CASC 2005, CASC 2006, CASC 2007, and CASC 2009 have been held, respectively, in St. Petersburg (R- sia), Munich (Germany), Samarkand (Uzbekistan), Konstanz (Germany), Yalta (Ukraine), Passau (Germany), St.

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Additional info for Computer Algebra in Scientific Computing: 12th International Workshop, CASC 2010, Tsakhkadzor, Armenia, September 6-12, 2010. Proceedings

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Then we write for nonlinearities of the third order ∞ 3 M≤3 ◦ X = M11 X+ m=2 k=1 Pk1 m X[k(m−1)+1] + k! ∞ ∞ + l=1 k=1 1 kl l (k+1) [2l+k+1] P P X . l! 2 3 Repeating described procedure we can evaluate M≤k ◦X for any k with necessary calculation accuracy. But, as M≤k is a truncated form of the map M (we use information about generating function up to the kth order only) then we have to evaluate up to terms of the kth order for obtaining desired result for M≤k . So we can write M≤2 ◦ X ≈ M11 X + M12 X[2] , M12 = M11 P11 2 , 1 21 [2] M≤3 ◦ X ≈ M11 X + P11 + P11 X[3] 2 X 3 + P2 2 = = M11 X + M12 X[2] + M13 X[3] , 1 21 P , 2!

2 p= Decomposition Algorithms Our version of the decomposition algorithm in each round treats one system, potentially splitting it into several subsystems. For this purpose, one polynomial is chosen from a list of polynomials to be processed. This polynomial is pseudoreduced modulo the system and afterwards combined with the polynomial in the Thomas Decomposition of Algebraic and Differential Systems 35 system having the same leader. To ensure that all polynomials are square-free and their initials do not vanish, the system may be split into several ones by initials of polynomials or subresultants.

With the Janet decomposition being defined for sets of differential variables, we will assign admissible derivations to differential polynomials according to their leaders. In particular, we extend the definitions of ΔW (w) for finite W ⊂ F {U } and w ∈ W by defining ΔW (w) := Δld(W ) (ld(w)). A differential polynomial q ∈ F {U } is called reducible with respect to p ∈ F {U }, if there exists i ∈ Zn≥0 such that ∂1i1 · . . · ∂nin ld(p) = ld(∂1i1 · . . · ∂nin p) = ld(q) and rank(∂1i1 · . . · ∂nin p) ≤ rank(q).

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