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1
Content available remote

Some highlights in the development of algebraic analysis

100%
Synowiec J. A.
Banach Center Publications
|
2000
|
tom 53
|
nr 1
11-46
2
Content available remote

Reduction of differential equations

80%
Skórnik K., Wloka J.
Banach Center Publications
|
2000
|
tom 53
|
nr 1
199-204
EN
Let (F,D) be a differential field with the subfield of constants C (c ∈ C iff Dc=0). We consider linear differential equations (1) $Ly = D^{n}y + a_{n-1}D^{n-1}y+...+ a_{0}y = 0$, where $a_0,... ,a_n ∈ F$, and the solution y is in F or in some extension E of F (E ⊇ F). There always exists a (minimal, unique) extension E of F, where Ly=0 has a full system $y_1,... ,y_n$ of linearly independent (over C) solutions; it is called the Picard-Vessiot extension of F E = PV(F,Ly=0). The Galois group G(E|F) of an extension field E ⊇ F consists of all differential automorphisms of E leaving the elements of F fixed. If E = PV(F,Ly=0) is a Picard-Vessiot extension, then the elements g ∈ G(E|F) are n × n matrices, n= ord L, with entries from C, the field of constants. Is it possible to solve an equation (1) by means of linear differential equations of lower order ≤ n-1? We answer this question by giving neccessary and sufficient conditions concerning the Galois group G(E|F) and its Lie algebra A(E|F).
3
Content available remote

Two centuries of the term "algebraic analysis"

80%
Przeworska-Rolewicz D.
Banach Center Publications
|
2000
|
tom 53
|
nr 1
47-70
EN
The term "Algebraic Analysis" in the last two decades is used in two completely different senses. It seems that at least one is far away from its historical roots. Thus, in order to explain this misunderstanding, the history of this term from its origins is recalled.
4
Content available remote

Modulating element method in the identification of a generalized dynamical system

61%
Wysocki H., Zellma M.
Applicationes Mathematicae
|
1993-1995
|
tom 22
|
nr 4
447-467
EN
In this paper the identification of generalized linear dynamical differential systems by the method of modulating elements is presented. The dynamical system is described in the Bittner operational calculus by an abstract linear differential equation with constant coefficients. The presented general method can be used in the identification of stationary continuous dynamical systems with compensating parameters and for certain nonstationary compensating or distributed parameter systems.
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