Abstract
Generalized numerical systems. Different numerical systems: decimal, dyadic, positional, continuous fractions. A new kind of function—translators from one numerical system to another. Application to basic harmonic functions.
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Notes
- 1.
The notion of conformal mappings makes sense for every Riemannian manifold, but we prefer to keep the exposition on an elementary level.
- 2.
Recall that a matrix A is said to be orthogonal if it satisfies the equation A t = A − 1; the corresponding linear operator in \({\mathbb{R}}^{n}\) preserves the dot product of vectors.
- 3.
Here we use the terminology and elementary facts of group theory. The reader can find all necessary information in textbooks on abstract algebra, such as Artin, Michael (1991), Algebra, Prentice Hall, ISBN 978-0-89871-510-1.
- 4.
Until now, we have defined conformal mappings only for domains in extended Euclidean spaces. So strictly speaking, we can not call s a conformal mapping. But using s, we can identify S n with \({\overline{\mathbb{R}}}^{n}\), and conformal mappings of \({\overline{\mathbb{R}}}^{n}\) become transformations of S n. The fact is that they are exactly conformal transformations of S n as a Riemannian manifold.
- 5.
Do not confuse \(p \in {\mathbb{R}}^{1,n}\) with \(\vec{p} \in {\mathbb{R}}^{n}\), introduced below.
- 6.
These two groups coincide, because every matrix from \(\mathrm{GL}(2,\, \mathbb{C})\) is proportional to a matrix from \(\mathrm{SL}(2,\, \mathbb{C})\).
- 7.
The algebra of quaternions is a four-dimensional real noncommutative algebra. It can be realized as a subalgebra of \(\mathrm{Mat}_{2}(\mathbb{C})\) or of \(\mathrm{Mat}_{4}(\mathbb{R})\).
- 8.
Recall that the hyperbolic functions are defined as \(\cosh t = \frac{{\mathrm{e}}^{t}+{\mathrm{e}}^{-t}} {2}\), \(\sinh \,t = \frac{{\mathrm{e}}^{t}-{\mathrm{e}}^{-t}} {2}\).
- 9.
The reason why curvatures are better than radii will be explained later, when we develop a group-theoretic approach to the problem. We shall see that the group transformations act linearly in terms of curvatures but not in terms of radii.
- 10.
In fact, the solutions of the second kind also can be associated with tangent disks, but on the sphere with the opposite orientation.
- 11.
Compare with the properties of the usual orthogonal matrices: all rows (columns) have length 1 and are orthogonal to one another.
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Kirillov, A.A. (2013). Circles and Disks on Spheres. In: A Tale of Two Fractals. Birkhäuser, New York, NY. https://doi.org/10.1007/978-0-8176-8382-5_4
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DOI: https://doi.org/10.1007/978-0-8176-8382-5_4
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Publisher Name: Birkhäuser, New York, NY
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