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On the Schröder equation

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CONTENTS PART I Introduction............................................................................................... 3 1. General solution.................................................................................. 4 2. Preliminaries and notation................................................................ 5 3. $C^p$ solutions in $\mathfrak{C}$*................................................ 7 4. Change of variables........................................................................... 13 5. Continuous solutions in $\mathfrak{C}$....................................... 15 6. $C^p$ solutions in a neighbourhood of the origin....................... 18 7. Final comments.................................................................................. 26 PART II 8. Introductory remarks.......................................................................... 28 9. $C^1$ solutions.................................................................................. 30 10. Monotonic solutions......................................................................... 35 11. Monotonic solutions of the equation ϱ[f(x)] — ϱ(x) = h(x)........... 36 12. Convex solutions............................................................................... 38 13. Local $C^p$ transformations of the real line............................... 43 14. Iterations and commuting functions.............................................. 45 REFERENCES
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Remark on a difference equation

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Basic concepts of the theory of geometric objects

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Contents Introduction............................................................................................................................................................................... 3 § 1. Historical development of the concept of a geometric object..................................................................................4 § 2. Manifold, coordinate system, transformations of the coordinate system............................................................. 7 § 3. Group, pseudogroup, groupoid.................................................................................................................................... 9 § 4. Wundheiler's definition of a geometric object........................................................................................................... 15 § 5. Special geometric objects............................................................................................................................................. 18 § 6. Abstract geometric object. The Haantjes-Laman definition of a geometric object............................................. 24 § 7. Classification of geometric objects.............................................................................................................................. 28 § 8. Equivalence of geometric objects................................................................................................................................ 39 § 9. Fibres of equivalent geometric objects....................................................................................................................... 42 § 10. Concomitants................................................................................................................................................................. 47 § 11. Algebra of geometric objects...................................................................................................................................... 54 § 12. Differential concomitants............................................................................................................................................. 57 References............................................................................................................................................................................... 63
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Some models of plane geometries and a functional equation

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