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1

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

Nath P., Singh D.

Demonstratio Mathematica

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2009

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Vol. 42, nr 1

83-96

EN

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.

2

Sylvester inertia law in commutative Leibniz algebras with logarithms

Przeworska-Rolewicz D.

Demonstratio Mathematica

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2007

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Vol. 40, nr 3

659-669

EN

In algebras with logarithms induced by a given right invertible operator D one can define quadratic forms by means of power mappings induced by logarithmic mapping. Main results of this paper will be concerned with the case when an algebra X 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 is an element of dom D. If X is an locally m-convex algebra then these forms have the similar properties as quadratic forms in the Euclidian spaces En, including the Sylvester inertia law.

3

Some summations formulae in commutative Leibniz algebras with logarithms

Przeworska-Rolewicz D.

Control and Cybernetics

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2007

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Vol. 36, no 3

841-857

EN

A survey of summation formulae in commutative Leibniz algebras with logarithms is given. New results concerning generic functions and related summation formulae, which generalize well-known properties of the Bessel functions, are demonstrated.

4

Multidimensional Riemann-Hilbert type problems in Leibniz algebras with logarithms

Przeworska-Rolewicz D.

Demonstratio Mathematica

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2002

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Vol. 35, nr 3

629-644

EN

Riemann-Hilbert type problems in Leibniz algebras with logarithms have been studied in PR[8] (cf. Chapter 14). These problems correspond to such classical problems when the Cauchy transformation is an involution. It was shown that this involution is not multiplicative. On the other hand, in the same book equations with multiplicative involutions were considered. These results can be applied to equations with an involutive transformation of argument, in particular, to equations with transformed argument by means of a function of Carleman type. Riemann-Hilbert type problems with an additional multiplicative involution in commutative Leibniz algebras with logarithms are examined in PR[13]. Results obtained there can be applied not only to problems with a transformation of argument but also to problems with the conjugation (in the complex sense). In the present paper there are considered similar problems in several variables with Riemann- Hilbert condition posed on each variable separately. For instance, these problems correspond in the classical case to problems for polyanalytic functions on polydiscs (cf. HD[1], Ms[1).

5

True shifts revisited

Przeworska-Rolewicz D.

Demonstratio Mathematica

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2001

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Vol. 34, nr 1

111-122

6

Linear combinations of right invertible operators in commutative algebras with logarithms

Przeworska-Rolewicz D.

Demonstratio Mathematica

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1998

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Vol. 31, nr 4

887-898

EN

It is well known that a power of a right invertible operators 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 results concerning the Fourier method are proved under weaker assumptions than those obtained in PR[l] (cf. also PR[2]). The extensive bibliography of the subject can be found in PR[2] and PR[4].