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1 Acta Universitatis Lodziensis. Folia Oeconomica

1 1996

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Quasihomographies in the theory of Teichmüller spaces

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Zając J.

Instytut Matematyczny Polskiej Akademii Nauk

CONTENTS Introduction............................................................................................................................5 I. Special functions of quasiconformal theory.....................................................................10 1. Introduction.................................................................................................................10 2. The distortion function $Φ_K$.....................................................................................11 3. Quasisymmetric functions............................................................................................19 4. Functional identities for special functions....................................................................27 5. Applications..................................................................................................................38 II. Quasihomographies of a circle.......................................................................................42 1. Introduction..................................................................................................................42 2. Introduction to quasihomographies..............................................................................42 3. Quasihomographies and quasisymmetric functions on the real line.............................45 4. Quasihomographies and quasisymmetric functions on the unit circle...........................48 5. Quasisymmetric functions as quasihomographies.........................................................51 III. Distortion theorems for quasihomographies....................................................................57 1. Introduction...................................................................................................................57 2. Similarities.....................................................................................................................57 3. Distortion theorems.......................................................................................................60 4. Normal and compact families of quasihomographies.....................................................67 5. Topological characterization of quasihomographies.......................................................69 IV. Quasihomographies of a Jordan curve ...........................................................................72 1. Introduction...................................................................................................................72 2. Harmonic cross-ratio.....................................................................................................72 3. One-dimensional quasiconformal mappings..................................................................76 4. Complete boundary transformations.............................................................................78 5. Quasicircles...................................................................................................................80 V. The universal Teichmüller space.......................................................................................84 1. Introduction....................................................................................................................84 2. The universal Teichmüller space of a circle...................................................................85 3. The universal Teichmüller space of an oriented Jordan curve........................................87 4. The space of normalized quasihomographies................................................................91 5. A linearization formula....................................................................................................94 Acknowledgements...................................................................................................................97 References...............................................................................................................................98

Analiza statystyczna parametrów ekonomicznych z ustaloną strukturą algebraiczną

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Fałda B. , Zając J. , Państwowa Wyższa Szkoła Zawodowa w Chełmie; Instytut Matematyki i Informatyki , Katolicki Uniwersytet Lubelski Jana Pawła II; Wydział Nauk Społecznych; Instytut Ekonomii i Zarządzania

Substantiated study of economic processes by examining the behavior of economic parame- ters requires taking into account their algebraic structures. It is known that these parameters have their own, natural algebraic structure which establishment is the first step to correct the accounting of analysis and reasoning. In the article, the authors present an algebraic algorithm that can be applied to the classification of basic economic parameters. Its consequence will be an indication of the mathematical and probabilistic structures, within which the calculation of values characterizing studied processes can be done. As the examples mentioned issues we will present several algebraic structures and their corresponding types of nonlinear probability space.