Wyniki wyszukiwania

Wyszukiwano:
w słowach kluczowych: cosine mapping

Sortuj według:

Ogranicz wyniki do:

Non-Leibniz algebras with logarithms do not have the trigonometric identity

100%

Przeworska-Rolewicz D.

Banach Center Publications

2000

tom 53

nr 1

177-189

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.

Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik

Fourier-like methods for equations with separable variables

67%

Przeworska-Rolewicz D.

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

2009

tom 29

nr 1

19-42

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]).

Ograniczanie wyników

1 Banach Center Publications

1 Discussiones Mathematicae, Differential Inclusions, Control and Optimization

2 Przeworska-Rolewicz D.

1 2009

1 2000

Wyniki wyszukiwania

Non-Leibniz algebras with logarithms do not have the trigonometric identity

Fourier-like methods for equations with separable variables