Coefficients Of Formal Power Series Research Articles

On initials and the fundamental theorem of tropical partial differential algebraic geometry

Tropical Differential Algebraic Geometry considers difficult or even intractable problems in Differential Equations and tries to extract information on their solutions from a restricted structure of the input. The fundamental theorem of Tropical Differential Algebraic Geometry and its extensions state that the support of power series solutions of systems of ordinary differential equations (with formal power series coefficients over an uncountable algebraically closed field of characteristic zero) can be obtained either, by solving a so-called tropicalized differential system, or by testing monomial-freeness of the associated initial ideals. Tropicalized differential equations work on a completely different algebraic structure which may help in theoretical and computational questions, particularly on the existence of solutions.We show here that both of these methods can be generalized to the case of systems of partial differential equations, this is, one can go either with the solution of tropicalized systems, or test monomial-freeness of the ideal generated by the initials when looking for supports of power series solutions of systems of differential equations, regardless the (finite) number of derivatives. The key are the vertex sets of Newton polytopes, upon which relies the definition of both tropical vanishing condition and the initial of a differential polynomial.

On Mahler's Classification of Formal Power Series Over a Finite Field

Abstract Let K be a finite field, K(x) be the field of rational functions in x over K and K be the field of formal power series over K. We show that under certain conditions integral combinations with algebraic formal power series coefficients of a U 1-number in K are Um -numbers in K , where m is the degree of the algebraic extension of K(x), determined by these algebraic formal power series coefficients.

On full rank differential systems with power series coefficients

We consider the following problem: given a linear ordinary differential system of arbitrary order with formal power series coefficients, decide whether the system has non-zero Laurent series solutions, and find all such solutions if they exist (in a truncated form preserving the space dimension). If the series coefficients of the original systems are represented algorithmically then these problems are algorithmically undecidable. However, it turns out that they are decidable in the case when we know in advance that a given system is of full rank.We define the width of a given full rank system S with formal power series coefficients as the smallest non-negative integer w such that any l-truncation of S with l⩾w is a full rank system. We prove that the value w exists for any full rank system and can be found algorithmically.We propose corresponding algorithms and their Maple implementation, and report some experiments.

High Frequency Wave Packet Concentrated near the moving Line on the Liquid Surface

Wave wall type solutions for waves on the surface of an ocean with slowly varying depth are constructed. It is a generalization of the solution of quasi-photon type. Wave wall is a packet of short waves concentrated near the moving line which forms the wave front. The approach is based on the representation of the solution in the form of asymptotic space-time rays (STR) series each term of which is a formal power series. The appropriate form of the ansatz is suggested. Chains of equations for coefficients of formal power series for phase and amplitude terms are derived. These equations are investigated and the solvability of the problem at each step is proved. As a result, asymptotic space-time series for the wave wall moving with group velocity on the surface of a heavy liquid is obtained.

On the Moser- and super-reduction algorithms of systems of linear differential equations and their complexity

The notion of irreducible forms of systems of linear differential equations with formal power series coefficients as defined by Moser [Moser, J., 1960. The order of a singularity in Fuchs’ theory. Math. Z. 379–398] and its generalisation, the super-irreducible forms introduced in Hilali and Wazner [Hilali, A., Wazner, A., 1987. Formes super-irréductibles des systèmes différentiels linéaires. Numer. Math. 50, 429–449], are important concepts in the context of the symbolic resolution of systems of linear differential equations [Barkatou, M., 1997. An algorithm to compute the exponential part of a formal fundamental matrix solution of a linear differential system. Journal of App. Alg. in Eng. Comm. and Comp. 8 (1), 1–23; Pflügel, E., 1998. Résolution symbolique des systèmes différentiels linéaires. Ph.D. Thesis, LMC-IMAG; Pflügel, E., 2000. Effective formal reduction of linear differential systems. Appl. Alg. Eng. Comm. Comp., 10 (2) 153–187]. In this paper, we reduce the task of computing a super-irreducible form to that of computing one or several Moser-irreducible forms, using a block-reduction algorithm. This algorithm works on the system directly without converting it to more general types of systems as needed in our previous paper [Barkatou, M., Pflügel, E., 2007. Computing super-irreducible forms of systems of linear differential equations via Moser-reduction: A new approach. In: Proceedings of ISSAC’07. ACM Press, Waterloo, Canada, pp. 1–8]. We perform a cost analysis of our algorithm in order to give the complexity of the super-reduction in terms of the dimension and the Poincaré-rank of the input system. We compare our method with previous algorithms and show that, for systems of big size, the direct block-reduction method is more efficient.

On the Check of Accuracy of the Coefficients of Formal Power Series

Let M(y) be a matrix whose entries are polynomial in y, λ(y) and v(y) be a set of eigenvalue and eigenvector of M(y). Then, λ(y) and v(y) are algebraic functions of y, and λ(y) and v(y) have their power series expansions λ(y) = β0 + β1y + ₀ + βkyk + ₀ (βj e C),            (1) v(y) = γ0 + γ1y + ₀ + γkyk + ₀ (γj e Cn),            (1) provided that y = 0 is not a singular point of λ(y) or v(y). Several algorithms are already proposed to compute the above power series expansions using Newton's method (the algorithm in [4]) or the Hensel construction (the algorithm in [5],[12]). The algorithms proposed so far compute high degree coefficients βk and γk, using lower degree coefficients βj and γj (j = 0,1,₀,k-1). Thus with floating point arithmetic, the numerical errors in the coefficients can accumulate as index k increases. This can cause serious deterioration of the numerical accuracy of high degree coefficients βk and γk, and we need to check the accuracy. In this paper, we assume that given matrix M(y) does not have multiple eigenvalues at y = 0 (this implies that y = 0 is not singular point of λ(y) or v(y)), and presents an algorithm to estimate the accuracy of the computed power series βi,γj in (1) and (2). The estimation process employs the idea in [9] which computes a coefficient of a power series with Cauchy's integral formula and numerical integrations. We present an efficient implementation of the algorithm that utilizes Newton's method. We also present a modification of Newton's method to speed up the procedure, introducing tuning parameter p. Numerical experiments of the paper indicates that we can enhance the performance of the algorithm by 12 ~ 16%, choosing the optimal tuning parameter p.

Formal Expansion of Harish-Chandra Homomorphism

Let 𝔤 be a complex semisimple Lie algebra with adjoint group G and let 𝔥 be a Cartan subalgebra of 𝔤. Let Â(𝔤) and Â(𝔥) denote the algebra of differential operators with formal power series coefficients on 𝔤 and 𝔥 respectively. We construct a subalgebra A 𝔤 of Â(𝔤) containing all the pull-backs of the differential operators in G attached to any element x in 𝔤. We also consider the projection P: A 𝔤 → Â 𝔥. Then, we calculate explicity the pull-back of the differential operator in G attached to an element h in 𝔥 modulo Ker P.

Sur l'inverse de l'application de Dixmier pour une algèbre de Lie résoluble

Let g be a completely solvable Lie algebra over a field of characteristic zero. Let Â( g ) be the algebra of differential operators with formal power series coefficients. The subalgebra generated by the Weyl algebra A( g ) on g and the formal power series with rational coefficients in the weights of the adjoint representation is denoted by P̂( g ). There is a canonical imbedding L g of the enveloping algebra U( g ) in P̂( g ). For every derivation x of g , the adjoint vector field on g defined by − x is denoted by W g ( x ). Let n a be the biggest nilpotent ideal in the algebraic hull of the image of g by the adjoint representation. For every prime ideal I in U( g ), let M g ( I ) be the quotient of P̂( g ) by the left ideal generated by L g ( I ) and W g ( n a ). The main result connects β g ( I ) and M g ( I ) where β g ( I ) is the inverse image of the Dixmier map for g . To state this result, certain categories Mod J [P̂( g )] are introduced and β g ( I ) is described in terms of the set of all J such that M g ( I ) is in Mod J [P̂( g )].

Formal Solutions and Factorization of Differential Operators with Power Series Coefficients

The topic of this paper is formal solutions of linear differential equations with formal power series coefficients. The method proposed for computing these solutions is based on factorization of differential operators. The notion of exponential parts is introduced to give a description of factorization properties and to characterize the formal solutions. The algorithms will be described and their implementation is available.

A formal power series approach to the construction of minimal fuzzy automata

A new approach to the problem of construction of minimal fuzzy automata using formal power-series representation is presented. The proceduie involves automated inference of regular fuzzy grammar from a set of sample strings. The technique yields a minimal fuzzy automaton that recognizes the positive samples of the language. The desired automata are inferred from a finite submatrix of the fuzzy Hankel matrix of the coefficients of formal power series representing the set of strings of the given language. An application in pattern recognition is shown for the inference of fuzzy grammar.

Bell polynomials

An introduction to Bell polynomials is given in order to see how they can be used to compute some of the classical counting functions of Combinatorics. They are also used to compute the Taylor coefficients of formal power series given as a composition of two power series along with computing the composition inverse of a power series.

Inference of fuzzy regular pattern grammar

This paper contains an approach to the inference of fuzzy pattern grammar from a set of sample strings. The grammar is inferred from a finite submatrix of the fuzzy Hankel matrix of the coefficients of formal power series representing the set of sample strings from each pattern class.