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Two centuries of the term "algebraic analysis"

100%
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.
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Spaces of D-paraanalytic elements

89%
Przeworska-Rolewicz D.
Instytut Matematyczny Polskiej Akademii Nauk
EN
CONTENTS Introduction..............................................................................5 1. Preliminaries........................................................................5 2. Spaces of smooth elements.................................................8 3. Spaces of D-analytic elements...........................................21 4. Spaces of D-paraanalytic elements....................................32 5. Characterization of .D-paraanalytic elements by means of property (c) and canonical mappings.............39 6. D-R paraanalytic functions .................................................47 7. Smooth elements in Leibniz D-algebras..............................59 8. Weak E-solutions................................................................63 9. Linear systems with scalar coefficients...............................79 10. Linear boundary value problems .....................................82 References.............................................................................97
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Fourier-like methods for equations with separable variables

51%
Przeworska-Rolewicz D.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
2009
|
tom 29
|
nr 1
19-42
EN
It is well known that a power of a right invertible operator is again right invertible, as well as a polynomial in a right invertible operator under appropriate assumptions. However, a linear combination of right invertible operators (in particular, their sum and/or difference) in general is not right invertible. It will be shown how to solve equations with linear combinations of right invertible operators in commutative algebras using properties of logarithmic and antilogarithmic mappings. The used method is, in a sense, a kind of the variables separation method. We shall obtain also an analogue of the classical Fourier method for partial differential equations. Note that the results concerning the Fourier method are proved under weaker assumptions than these obtained in [6] (cf. also [7, 8, 11]).
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