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Spaces of D-paraanalytic elements

100%

Przeworska-Rolewicz D.

Instytut Matematyczny Polskiej Akademii Nauk

EN

CONTENTS Introduction..............................................................................5 1. Preliminaries........................................................................5 2. Spaces of smooth elements.................................................8 3. Spaces of D-analytic elements...........................................21 4. Spaces of D-paraanalytic elements....................................32 5. Characterization of .D-paraanalytic elements by means of property (c) and canonical mappings.............39 6. D-R paraanalytic functions .................................................47 7. Smooth elements in Leibniz D-algebras..............................59 8. Weak E-solutions................................................................63 9. Linear systems with scalar coefficients...............................79 10. Linear boundary value problems .....................................82 References.............................................................................97

2

Algebraic analysis in structures with the Kaplansky-Jacobson property

94%

Przeworska-Rolewicz D.

Studia Mathematica

|

2005

|

tom 168

|

nr 2

165-186

EN

In 1950 N. Jacobson proved that if u is an element of a ring with unit such that u has more than one right inverse, then it has infinitely many right inverses. He also mentioned that I. Kaplansky proved this in another way. Recently, K. P. Shum and Y. Q. Gao gave a new (non-constructive) proof of the Kaplansky-Jacobson theorem for monoids admitting a ring structure. We generalize that theorem to monoids without any ring structure and we show the consequences of the generalized Kaplansky-Jacobson theorem for the theory of linear operators, and even for the classical Calculus. In order to do that, we recall some multiplicative systems, called pseudocategories, very useful in the algebraic theory of perturbations of linear operators. In the second part of the paper, basing on the Kaplansky-Jacobson theorem, we show how to use the above mentioned structures for building Algebraic Analysis of linear operators over a class of linear spaces. We also define (non-linear) logarithmic and antilogarithmic mappings on these structures.

3

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

94%

Przeworska-Rolewicz D.

Banach Center Publications

|

2000

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tom 53

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nr 1

177-189

EN

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.

4

Nonlinear separable equations in linear spaces and commutative Leibniz algebras

94%

Przeworska-Rolewicz D.

Annales Polonici Mathematici

|

2010

|

tom 97

|

nr 3

219-241

EN

We consider nonlinear equations in linear spaces and algebras which can be solved by a "separation of variables" obtained due to Algebraic Analysis. It is shown that the structures of linear spaces and commutative algebras (even if they are Leibniz algebras) are not rich enough for our purposes. Therefore, in order to generalize the method used for separable ordinary differential equations, we have to assume that in algebras under consideration there exist logarithmic mappings. Section 1 contains some basic notions and results of Algebraic Analysis. In Section 2 we consider equations in linear spaces. Section 3 contains results for commutative Leibniz algebras. In Section 4 basic notions and facts concerning logarithmic and antilogarithmic mappings are collected. Section 5 is devoted to separable nonlinear equations in commutative Leibniz algebras with logarithms.

5

Some boundary value problems with transformed argument

94%

Przeworska-Rolewicz D.

Commentationes Mathematicae

|

1974

|

tom 17

|

nr 2

EN

The article contains no abstract

6

Sur les équations intégrales singulières dans l'espace résolubles par la méthode des approximations successives

82%

Przeworska-Rolewicz D.

Annales Polonici Mathematici

|

1962-1963

|

tom 12

|

nr 3

289-299

7

On equations with several involutions of different orders and its applications to partial differential-difference equations

71%

Przeworska-Rolewicz D.

Studia Mathematica

|

1969

|

tom 32

|

nr 2

101-113

8

Sur l'application de la méthode des approximations successives à une équation intégrale à forte singularité

71%

Przeworska-Rolewicz D.

Annales Polonici Mathematici

|

1959

|

tom 6

|

nr 2

161-170

9

On linear differential equations with transformed argument solvable by means of right invertible operators

71%

Przeworska-Rolewicz D.

Annales Polonici Mathematici

|

1974-1975

|

tom 29

|

nr 2

141-148

10

Remarks on $\phi$-operators in linear topological spaces

60%

Przeworska-Rolewicz D., Rolewicz S.

Commentationes Mathematicae

|

1965

|

tom 9

|

nr 1

EN

The article contains no abstract

11

The only continuous Volterra right inverses in $C_c[0,1]$ of the operator $d/dt$ are $ʃ^t_a$

60%

Przeworska-Rolewicz D., Rolewicz S.

Colloquium Mathematicum

|

1987

|

tom 51

|

nr 1

281-285

12

Admissible initial operators for superpositions of right invertible operators

59%

Przeworska-Rolewicz D.

Annales Polonici Mathematici

|

1976-1977

|

tom 33

|

nr 2

311-318

13

On d-characteristic and $d_Ξ$-characteristic of linear operators

48%

Przeworska-Rolewicz D., Rolewicz S.

Annales Polonici Mathematici

|

1967

|

tom 19

|

nr 2

117-121

14

On integrals of functions with values in a complete linear metric spaces

48%

Przeworska-Rolewicz D., Rolewicz S.

Studia Mathematica

|

1965-1966

|

tom 26

|

nr 2

121-131

15

Sur l'intégrale de Cauchy pour un are fermé à l'infini

47%

Przeworska-Rolewicz D.

Annales Polonici Mathematici

|

1960

|

tom 8

|

nr 2

155-171

16

On periodic solutions of linear differential-difference equations with constant coefficients

47%

Przeworska-Rolewicz D.

Studia Mathematica

|

1968

|

tom 31

|

nr 5

507-511

17

Sur les systèmes d'équations intégrales singulières pour des lignes fermées

47%

Przeworska-Rolewicz D.

Studia Mathematica

|

1959

|

tom 18

|

nr 3

247-268

18

Green's formula for right invertible operators

47%

Przeworska-Rolewicz D.

Annales Polonici Mathematici

|

1983

|

tom 42

|

nr 1

283-295

19

Problème non linéaire d'Hilbert pour un système infini de fonctions

47%

Przeworska-Rolewicz D.

Annales Polonici Mathematici

|

1958-1959

|

tom 5

|

nr 3

293-301

20

A characterization of commutators with Hilbert transforms

42%

Przeworska-Rolewicz D.

Studia Mathematica

|

1972

|

tom 44

|

nr 1

27-30

Ograniczanie wyników

Spaces of D-paraanalytic elements

Algebraic analysis in structures with the Kaplansky-Jacobson property

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

Nonlinear separable equations in linear spaces and commutative Leibniz algebras

Some boundary value problems with transformed argument

Sur les équations intégrales singulières dans l'espace résolubles par la méthode des approximations successives

On equations with several involutions of different orders and its applications to partial differential-difference equations

Sur l'application de la méthode des approximations successives à une équation intégrale à forte singularité

On linear differential equations with transformed argument solvable by means of right invertible operators

Remarks on \(\phi\)-operators in linear topological spaces

The only continuous Volterra right inverses in $C_c[0,1]$ of the operator $d/dt$ are $ʃ^t_a$

Admissible initial operators for superpositions of right invertible operators

On d-characteristic and $d_Ξ$-characteristic of linear operators

On integrals of functions with values in a complete linear metric spaces

Sur l'intégrale de Cauchy pour un are fermé à l'infini

On periodic solutions of linear differential-difference equations with constant coefficients

Sur les systèmes d'équations intégrales singulières pour des lignes fermées

Green's formula for right invertible operators

Problème non linéaire d'Hilbert pour un système infini de fonctions

A characterization of commutators with Hilbert transforms