scholarly journals A discrete version of Liouville’s theorem on conformal maps

2021 ◽  
Author(s):  
Ulrich Pinkall ◽  
Boris Springborn
Keyword(s):  
Simplicial Complexes ◽  
Discrete Version ◽  
Conformal Maps ◽  
Analogous Statement ◽  
Möbius Transformations ◽  
Factors Associated ◽  
Scale Factors ◽  
Liouville’S Theorem ◽  
Mobius Transformations

AbstractLiouville’s theorem says that in dimension greater than two, all conformal maps are Möbius transformations. We prove an analogous statement about simplicial complexes, where two simplicial complexes are considered discretely conformally equivalent if they are combinatorially equivalent and the lengths of corresponding edges are related by scale factors associated with the vertices.

scholarly journals Moduli of curve families and (quasi-)conformality of power-law entropies

2016 ◽  
Vol 13 (05) ◽  
pp. 1650063 ◽  
Author(s):  
Nikos Kalogeropoulos
Keyword(s):  
Power Law ◽  
Measure Space ◽  
Functional Form ◽  
Tsallis Entropy ◽  
Metric Measure Space ◽  
Conformal Maps ◽  
Möbius Transformations ◽  
Curve Family ◽  
Mobius Transformations

We present aspects of the moduli of curve families on a metric measure space which may prove useful in calculating, or in providing bounds to, non-additive entropies having a power-law functional form. We use as paradigmatic cases the calculations of the moduli of curve families for a cylinder and for an annulus in [Formula: see text]. The underlying motivation for these studies is that the definitions and some properties of the modulus of a curve family resembles those of the Tsallis entropy, when the latter is seen from a micro-canonical viewpoint. We comment on the origin of the conjectured invariance of the Tsallis entropy under Möbius transformations of the non-extensive (entropic) parameter. Needing techniques applicable to both locally Euclidean and fractal classes of spaces, we examine the behavior of the Tsallis functional, via the modulus, under quasi-conformal maps. We comment on properties of such maps and their possible significance for the dynamical foundations of power-law entropies.

scholarly journals Möbius geometry for hypersurfaces in S4

1995 ◽  
Vol 139 ◽  
pp. 1-20 ◽  
Author(s):  
Changping Wang
Keyword(s):  
Invariant System ◽  
Principal Curvatures ◽  
Möbius Transformations ◽  
Möbius Geometry ◽  
Mobius Transformations

Our purpose in this paper is to study Möbius geometry for those hypersurfaces in S4 which have different principal curvatures at each point. We will give a complete local Möbius invariant system for such hypersurface in S4 which determines the hypersurface up to Möbius transformations. And we will classify the so-called Möbius homogeneous hypersurfaces in S4.

MÖBIUS TRANSFORMATIONS IN QUANTUM AMPLITUDE AMPLIFICATION WITH GENERALIZED PHASES

2010 ◽  
Vol 08 (06) ◽  
pp. 923-935 ◽  
Author(s):  
CÉSAR BAUTISTA-RAMOS ◽  
NORA CASTILLO-TÉPOX
Keyword(s):  
Error Probability ◽  
Quantum Algorithm ◽  
Linear Transformations ◽  
Möbius Transformations ◽  
Good State ◽  
Quantum Amplitude ◽  
Initial States ◽  
Phase Angles ◽  
Amplitude Amplification ◽  
Mobius Transformations

The iteration of the operators employed in quantum amplitude amplification with generalized phases is analyzed by using elementary properties (geometric and algebraic) of the Möbius transformations (fractional linear transformations). It is shown that, for a given quantum algorithm without measurement, which produces a good state with probability a of success, if the phase angles φ and ϕ which mark the good and initial states respectively satisfy φ = ϕ with a small enough, then, for a number n of iterations with [Formula: see text] we get an error probability that is at most O(aϕ2).

scholarly journals Ellipses, near ellipses, and harmonic Möbius transformations

2005 ◽  
Vol 133 (9) ◽  
pp. 2705-2710 ◽  
Author(s):  
Martin Chuaqui ◽  
Peter Duren ◽  
Brad Osgood
Keyword(s):  
Möbius Transformations ◽  
Mobius Transformations
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