Danuta Przeworska-Rolewicz

Postmodern logarithmo-technia

Abstract In order to solve ordinary differential equations, we use an equation with the so-called logarithmic derivative . Similar equations with a linear operator permit us to define logarithmic and antilogarithmic mappings and obtain some results unknown before for the classical derivation operator. This method and its applications are exposed in detail in the authors book [1] and may be treated as an introduction into the 21 st century logarithmo-technia (according to the original meaning of this word). In the first section, there are given basic notions and facts of algebraic analysis (without proofs). The second section consists of the most important definitions and theorems (without proofs) concerning logarithms and antilogarithms. The third section is concerned with properties of multiplicative true shifts. In the last section is given a generalization of the binomial theorem for harmonic logarithms.

True shifts revisited

True shifts introduced in PR[2] has been examined in several papers in various aspects (cf. for instance, PR[3]-PR[7]). Here we would like to give a survey of some of these results in order to recall the most important properties of true shifts. In particular, there is shown that true shifts are hypercyclic and that a necessary and sufficient condition for true shifts to be multiplicative is that the generating operator D satisfies the Leibniz condition. A consequence of this fact is that D is uniquely determined by an isomorphism acting on ^ .

Property (c) and interpolation formulae induced by right invertible operators

Dedicated to the memory of Professor Edward Otto In the present paper we shall solve the following problem: To find a D-polynomial, i.e. a polynomial induced by a right invertible operator D in a linear space X which admits the given values for given initial operators. When D = we have a classical interpolation problem. An unified approach to various interpolation problems in Hilbert spaces by applications of properties of right invertible operators has been given by M. Tasche 14J.

Generalized Bernoulli Operator and Euler-Maclaurin Formula

The purpose of the present paper is to construct an analogue of the classical Bernoulli operator for a right invertible operator in a complete linear metric locally convex space and to derive the corresponding Euler-Maclaurin Formula. We shall present some examples of applications of results obtained.

SYLVESTER INERTIA LAW IN COMMUTATIVE LEIBNIZ ALGEBRAS WITH LOGARITHMS

1. Quadratic forms in finite dimensional spaces We recall some notions and properties which will be used in the sequel. Let X be the Euclidian space E n over complexes with the inner product ( , ). It is well-known that every quadratic form, (i.e. quadratic functional) is of the form:

Shifts and periodicity in algebraic analysis

AbstractShifts and periodicity for functional-differential equations and their generalizations have been studied by the author in various aspects (cf. for instance, [17]–[20] and following papers). Here we would like to give a comprehensive survey of some of these results (without proofs) in order to recall the most important properties of the considered shifts. In particular, it is shown that the so-called true shifts in complete linear metric spaces are hypercyclic and that a necessary and sufficient condition for true shifts in commutative algebras to be multiplicative is that the generating operator D satisfies the Leibniz condition. A consequence of this fact is that in commutative Leibniz algebras with logarithms the operator D is uniquely determined by an isomorphism acting on

Logarithms and Antilogarithms of Higher Order

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Linear Equations in Leibniz Algebras

. There are also studied generalized periodic and exponential-periodic solutions of linear and some nonlinear equations with shifts and generalizations of the classical Birkhoff theorem and Floquet theorem. These results are obtained by means of tools given by Algebraic Analysis (cf. the author [17]). A generalization of binomial formula of Umbral Calculus is shown in Section 7 (cf. Roman and Rota [36]). Section 11 contains a perturbation theorem for linear differential-difference equations with non-commensurable deviations and some its consequences.