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Logarithmic and antilogarithmic mappings

Euler in his paper De la controverse entre Mrs. Leibniz and Bernoulli sur les logarithmes des nombres négatifs and imaginairesg (Mémoires de l'Académie des Sciences de Berlin 5 (1749), 139-171, in: Opera, (1) 17, 195-232; cf. C. G. Fraser [1]) considered the rule d(log x) = dx/x. He rejected an earlier suggestion of Leibniz that this rule is only valid for positive real values of x with the following observation:"(...) Car, comme ce calcul roule sur les quantités variables, c. à d. sur des quantités...

Two centuries of the term "algebraic analysis"

Danuta Przeworska-Rolewicz — 2000

Banach Center Publications

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.

Fourier-like methods for equations with separable variables

Danuta Przeworska-Rolewicz — 2009

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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...

Powers and Logarithms

Przeworska-Rolewicz, Danuta — 2004

Fractional Calculus and Applied Analysis

There are applied power mappings in algebras with logarithms induced by a given linear operator D in order to study particular properties of powers of logarithms. Main results of this paper will be concerned with the case when an algebra under consideration is commutative and has a unit and the operator D satisfies the Leibniz condition, i.e. D(xy) = xDy + yDx for x, y ∈ dom D. Note that in the Number Theory there are well-known several formulae expressed by means of some combinations of powers...

Equations in linear spaces

CONTENTSPreface...................... 5Acknowledgment...................... 7PART A. LINEAR OPERATORS IN LINEAR SPACESCHAPTER I. Operators with a finite and semifinite dimensional characteristic........ 25CHAPTER II. Algebraic and almost algebraic operators........ 65CHAPTER III. Φ_Ξ-operators........ 90CHAPTER IV. Determinant theory of Φ_Ξ-operators........ 102PART B. LINEAR OPERATORS IN LINEAR TOPOLOGICAL SPACESCHAPTER I. Linear topological and linear metric space........ 115CHAPTER II. Continuous...

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