Temat: sine mapping - Prolib Integro

Skocz do pozycji: 1.

Tytuł:: Non-Leibniz algebras with logarithms do not have the trigonometric identity
Autorzy:: Przeworska-Rolewicz, D.
Tematy:: sine mapping
algebra with unit
algebraic analysis
logarithmic mapping
Leibniz condition
cosine mapping; Pokaż więcej
Wydawca:: Polska Akademia Nauk. Instytut Matematyczny PAN
Powiązania:: https://bibliotekanauki.pl/articles/1207627.pdf Link otwiera się w nowym oknie
Opis:: Let X be a Leibniz algebra with unit e, i.e. an algebra with a right invertible linear operator D satisfying the Leibniz condition: D(xy) = xDy + (Dx)y for x,y belonging to the domain of D. If logarithmic mappings exist in X, then cosine and sine elements C(x) and S(x) defined by means of antilogarithmic mappings satisfy the Trigonometric Identity, i.e. $[C(x)]^2 + [S(x)]^2 = e$ whenever x belongs to the domain of these mappings. The following question arises: Do there exist non-Leibniz algebras with logarithms such that the Trigonometric Identity is satisfied? We shall show that in non-Leibniz algebras with logarithms the Trigonometric Identity does not exist. This means that the above question has a negative answer, i.e. the Leibniz condition in algebras with logarithms is a necessary and sufficient condition for the Trigonometric Identity to hold.
Dostawca treści:: Biblioteka Nauki

Artykuł

Skocz do pozycji: 2.

Tytuł:: Fourier-like methods for equations with separable variables
Autorzy:: Przeworska-Rolewicz, Danuta
Tematy:: algebraic analysis
commutative algebra with unit
Leibniz condition
logarithmic mapping
antilogarithmic mapping
right invertible operator
sine mapping
cosine mapping
initial value problem
boundary value problem
Fourier method; Pokaż więcej
Wydawca:: Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Powiązania:: https://bibliotekanauki.pl/articles/729392.pdf Link otwiera się w nowym oknie
Opis:: 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]).
Dostawca treści:: Biblioteka Nauki

Artykuł

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