Remarks on nehari's theorem about the Schwarzian derivative and schlicht functions

1979 ◽  
Vol 36 (1) ◽  
pp. 184-190 ◽  
Author(s):  
Olli Lehto
Keyword(s):  
Schwarzian Derivative

Geometric estimates for the Schwarzian derivative

2017 ◽  
Vol 72 (3) ◽  
pp. 479-511 ◽  
Author(s):  
V N Dubinin
Keyword(s):  
Schwarzian Derivative ◽  
Geometric Estimates

UNIMODAL INTERVAL MAPS OBTAINED FROM THE MODIFIED CHUA EQUATIONS

1993 ◽  
Vol 03 (02) ◽  
pp. 323-332 ◽  
Author(s):  
MICHAŁ MISIUREWICZ
Keyword(s):  
Periodic Orbit ◽  
Critical Point ◽  
Real Line ◽  
Positive Measure ◽  
Schwarzian Derivative ◽  
Large Range ◽  
Unimodal Maps ◽  
Interval Maps ◽  
Central Piece ◽  
Piecewise Continuous

Following Brown [1992, 1993] we study maps of the real line into itself obtained from the modified Chua equations. We fix our attention on a one-parameter family of such maps, which seems to be typical. For a large range of parameters, invariant intervals exist. In such an invariant interval, the map is piecewise continuous, with most of pieces of continuity mapped in a monotone way onto the whole interval. However, on the central piece there is a critical point. This allows us to find sometimes a smaller invariant interval on which the map is unimodal. In such a way, we get one-parameter families of smooth unimodal maps, very similar to the well-known family of logistic maps x ↦ ax(1−x). We study more closely one of those and show that these maps have negative Schwarzian derivative. This implies the existence of at most one attracting periodic orbit. Moreover, there is a set of parameters of positive measure for which chaos occurs.

scholarly journals Dirichlet and VMOA domains via Schwarzian derivative

2009 ◽  
Vol 359 (2) ◽  
pp. 543-546 ◽  
Author(s):  
Fernando Pérez-González ◽  
Jouni Rättyä
Keyword(s):  
Schwarzian Derivative

scholarly journals Universe as Klein–Gordon eigenstates

2021 ◽  
Vol 81 (12) ◽  
Author(s):  
Marco Matone
Keyword(s):  
Linear Form ◽  
Schwarzian Derivative ◽  
Eigenvalue Problems ◽  
Measurement Problem ◽  
Linear Equations ◽  
Flat Space ◽  
Curvature Term ◽  
Klein Gordon ◽  
Linear Differential ◽  
Jacobi Equations

AbstractWe formulate Friedmann’s equations as second-order linear differential equations. This is done using techniques related to the Schwarzian derivative that selects the$$\beta $$ β -times $$t_\beta :=\int ^t a^{-2\beta }$$ t β : = ∫ t a - 2 β , where a is the scale factor. In particular, it turns out that Friedmann’s equations are equivalent to the eigenvalue problems $$\begin{aligned} O_{1/2} \Psi =\frac{\Lambda }{12}\Psi , \quad O_1 a =-\frac{\Lambda }{3} a , \end{aligned}$$ O 1 / 2 Ψ = Λ 12 Ψ , O 1 a = - Λ 3 a , which is suggestive of a measurement problem. $$O_{\beta }(\rho ,p)$$ O β ( ρ , p ) are space-independent Klein–Gordon operators, depending only on energy density and pressure, and related to the Klein–Gordon Hamilton–Jacobi equations. The $$O_\beta $$ O β ’s are also independent of the spatial curvature, labeled by k, and absorbed in $$\begin{aligned} \Psi =\sqrt{a} e^{\frac{i}{2}\sqrt{k}\eta } . \end{aligned}$$ Ψ = a e i 2 k η . The above pair of equations is the unique possible linear form of Friedmann’s equations unless $$k=0$$ k = 0 , in which case there are infinitely many pairs of linear equations. Such a uniqueness just selects the conformal time $$\eta \equiv t_{1/2}$$ η ≡ t 1 / 2 among the $$t_\beta $$ t β ’s, which is the key to absorb the curvature term. An immediate consequence of the linear form is that it reveals a new symmetry of Friedmann’s equations in flat space.

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