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

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Przeworska-Rolewicz D.

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.

On a sum form functional equation related to entropies and some moments of a discrete random variable

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Nath P. , Singh D.

tom Vol. 42, nr 1

83-96

The general solutions of a sum form functional equation have been obtained. The importance of its solutions in relation to the entropies and some moments of a discrete random variable has been discussed.