Schwarzian Research Articles

Rational functions with positive Schwarzian derivative

This paper examines the sign of the Schwarzian derivative of real valued rational functions of degree 2 and higher. We prove that rational functions whose two polynomials have real, simple, and interlaced zeros have positive Schwarzian. We show the converse holds for rational functions of degree 2 and 3 but fails for higher degrees, as they may have non-real roots. Nonetheless, we prove the real zeros are still interlaced and simple, and we show that the two polynomials have the same number of non-real roots. These results settle two conjectures in the literature.

Differential subordination for bounded turning functions using pre-Schwarzian and the Schwarzian derivatives

Differential subordination for bounded turning functions using pre-Schwarzian and the Schwarzian derivatives

KdV conformal symmetry breaking in nearly AdS2

We study the gauge theory formulation of Jackiw-Teitelboim gravity and propose Korteweg-de Vries asymptotic conditions that generalize the asymptotic dynamics of the theory. They permit to construct an enlarged set of boundary actions formed by higher order generalizations of the Schwarzian derivative that contain the Schwarzian as lower term in a tower of SL(2, ℝ) invariants. They are extracted from the KdV Hamiltonians and can be obtained recursively. As a result, the conformal symmetry breaking observed in nearly AdS2 is characterized by a much larger set of dynamical modes associated to the stationary KdV hierarchy. We study quantum perturbation theory for the generalized Schwarzian action including the symplectic measure and compute the one-loop correction to the partition function. We find that despite the non-linear nature of the higher-Schwarzian contribution, it acquires a manageable expression that renders a curious leading temperature dependence on the entropy S = #Ta for a an odd integer.

A Characterization of Concave Mappings Using the Carathéodory Class and Schwarzian Derivative

A Characterization of Concave Mappings Using the Carathéodory Class and Schwarzian Derivative

The Schwarzian Derivative and the Degree of a Classical Minimal Surface

Using the Schwarzian derivative we construct a sequence ( P l ) l ⩾ 2 of meromorphic differentials on every non-flat oriented minimal surface in Euclidean 3-space. The differentials ( P l ) l ⩾ 2 are invariant under all deformations of the surface arising via the Weierstrass representation and depend on the induced metric and its derivatives only. A minimal surface is said to have degree n if its n-th differential is a polynomial expression in the differentials of lower degree. We observe that several well-known minimal surfaces have small degree, including Enneper’s surface, the helicoid/catenoid and the Scherk—as well as the Schwarz family. Furthermore, it is shown that locally and away from umbilic points every minimal surface can be approximated by a sequence of minimal surfaces of increasing degree.

Dynamical system analysis of quintessence dark energy model

Our work deals with the dynamical system analysis of quintessence dark energy scalar field model with exponential potential. A dynamical system analysis has been applied at the background level. Using suitable transformation of variables, the evolution equations are reduced to an autonomous system for exponential form of the scalar potential. The critical points are analyzed with center manifold theory and stability has been discussed by using Schwarzian derivative. Finally, cosmological implications of the critical points are discussed and it is found that the stability of the late-time attractor changes for quintessence dark energy model.

Certain sharp estimates of Ozaki close-to-convex functions

The goal of this paper is to establish the sharp estimates on coefficient functionals like Hermitian–Toeplitz determinant of second-order involving logarithmic coefficients, initial logarithmic inverse coefficients as well as sharp bounds on initial order Schwarzian derivatives of the Ozaki close-to-convex functions.

The Schwarzian Approach in Sturm–Liouville Problems

A novel method for finding the eigenvalues of a Sturm–Liouville problem is developed. Following the minimalist approach, the problem is transformed to a single first-order differential equation with appropriate boundary conditions. Although the resulting equation is nonlinear, its form allows us to find the general solution by adding a second part to a particular solution. This splitting of the general solution into two parts involves the Schwarzian derivative: hence, the name of the approach. The eigenvalues that correspond to acceptable solutions can be found by requiring the second part to correct the asymptotically diverging behavior of the particular solution. The method can be applied to many different areas of physics, such as the Schrödinger equation in quantum mechanics and stability problems in fluid dynamics. Examples are presented.

On a multiplier operator induced by the Schwarzian derivative of univalent functions

AbstractIn this paper, we study a multiplier operator which is induced by the Schwarzian derivative of univalent functions with a quasiconformal extension to the extended complex plane. As applications, we show that the Brennan conjecture is satisfied for a large class of quasidisks. We also establish a new characterization of asymptotically conformal curves and of the Weil–Petersson curves in terms of the multiplier operator.

The Loewner–Kufarev energy and foliations by Weil–Petersson quasicircles

AbstractWe study foliations by chord–arc Jordan curves of the twice punctured Riemann sphere using the Loewner–Kufarev equation. We associate to such a foliation a function on the plane that describes the “local winding” along each leaf. Our main theorem is that this function has finite Dirichlet energy if and only if the Loewner driving measure has finite Loewner–Kufarev energy, defined by whenever is of the form , and set to otherwise. Moreover, if either of these two energies is finite, they are equal up to a constant factor, and in this case, the foliation leaves are Weil–Petersson quasicircles. This duality between energies has several consequences. The first is that the Loewner–Kufarev energy is reversible, that is, invariant under inversion and time reversal of the foliation. Furthermore, the Loewner energy of a Jordan curve can be expressed using the minimal Loewner–Kufarev energy of those measures that generate the curve as a leaf. This provides a new and quantitative characterization of Weil–Petersson quasicircles. Finally, we consider conformal distortion of the foliation and show that the Loewner–Kufarev energy satisfies an exact transformation law involving the Schwarzian derivative. The proof of our main theorem uses an isometry between the Dirichlet energy space on the unit disc and that we construct using Hadamard's variational formula expressed by means of the Loewner–Kufarev equation. Our results are related to ‐parameter duality and large deviations of Schramm–Loewner evolutions coupled with Gaussian random fields.

On special solutions to the Ermakov–Painlevé XXV equation

In this paper, we study a nonlinear second-order ordinary differential equation which we call the Ermakov–Painlevé XXV equation since under certain restrictions on its coefficients it can be reduced either to the Ermakov or the Painlevé XXV equation. The Ermakov–Painlevé XXV equation arises from a generalized Riccati equation and the related third-order linear differential equation via the Schwarzian derivative. The generalized Riccati equation has two families of Riccati solutions and we study the corresponding solutions to the Ermakov–Painlevé XXV equation. We show that one of these families appears only in the Ermakov case. We give examples of the Ermakov–Painlevé XXV equations and show how to construct their solutions expressed in terms of elementary or in terms of the classical special functions.

ON RESIDUES AND CONJUGACIES FOR GERMS OF 1-D PARABOLIC DIFFEOMORPHISMS IN FINITE REGULARITY

AbstractWe study conjugacy classes of germs of nonflat diffeomorphisms of the real line fixing the origin. Based on the work of Takens and Yoccoz, we establish results that are sharp in terms of differentiability classes and order of tangency to the identity. The core of all of this lies in the invariance of residues under low-regular conjugacies. This may be seen as an extension of the fact (also proved in this article) that the value of the Schwarzian derivative at the origin for germs of $C^3$ parabolic diffeomorphisms is invariant under $C^2$ parabolic conjugacy, though it may vary arbitrarily under parabolic $C^1$ conjugacy.

Nowhere coexpanding functions.

We define a family of C1 functions, which we call "nowhere coexpanding functions," that is closed under composition and includes all C3 functions with non-positive Schwarzian derivatives. We establish results on the number and nature of the fixed points of these functions, including a generalization of a classic result of Singer.

Schwarzian derivative in higher-order Riccati equations

Schwarzian derivative in higher-order Riccati equations

BC-system, absolute cyclotomy and the quantized calculus

We give a short survey on several developments on the BC-system, the adele class space of the rationals, and on the understanding of the “zeta sector” of the latter space as the Scaling Site. The new result that we present concerns the description of the BC-system as the universal Witt ring (i.e., K -theory of endomorphisms) of the “algebraic closure” of the absolute base \mathbb{S} . In this way we attain a conceptual meaning of the BC dynamical system at the most basic algebraic level. Furthermore, we define an invariant of Schwartz kernels in one dimension and relate the Fourier transform (in one dimension) to its role over the algebraic closure of \mathbb{S} . We implement this invariant to prove that, when applied to the quantized differential of a function, it provides its Schwarzian derivative. Finally, we survey the roles of the quantized calculus in relation to Weil's positivity, and that of spectral triples in relation to the zeros of the Riemann zeta function.

A Study of the Higher-Order Schwarzian Derivatives of Hirotaka Tamanoi

ABSTRACT In this paper, we study the higher-order Schwarzian derivative Sn (f) proposed by H. Tamanoi [Higher Schwarzian operators and combinatorics of the Schwarzian derivative, Math. Ann. 305 (1996), 127–151]. For the strongly starlike functions of order α and strongly convex functions of order α, the sharp bound of |S 3(f)(0)| is obtained. When n ∈ [2, 7], we prove that the higher Bers maps induced by Sn (f) on Weil-Petersson Teichmüller space and BMO-Teichmüller space are holomorphic.

Some Transformation Properties with Omitted Value

The purpose of this paper is to show that for a holomorphic and univalent function in class S, an omitted –value transformation yields a class of starlike functions as a rotation transformation of the Koebe function, allowing both the image and rotation of the function to be connected. Furthermore, these functions have several properties that are not far from a convexity properties. We also show that Pre- Schwarzian derivative is not invariant since the convexity property of the function is so weak.

Remarks on higher Schwarzians

The Schwarzian derivative has recently received renewed attention in connection with the study of the Sachdev–Ye–Kitaev model. In mathematics literature, various higher order generalizations of the Schwarzian derivative are known due to Aharonov, Bertilsson, and Schippers. Physical applications of the higher Schwarzian derivatives have not yet been discussed in any detail. In this work, we link Bertilsson's variant to the ℓ–conformal Galilei group, as well as discuss some of its interesting peculiarities. These include a recurrence relation, which allows one to construct the higher Schwarzians iteratively, a composition law, and symmetry transformations.

Noisy prediction-based control leading to stability switch

Applying Prediction-Based Control (PBC) xn+1=(1−αn)f(xn)+αnxn with stochastically perturbed control coefficient αn=α+ℓξn+1, n∈N, where ξ are bounded identically distributed independent random variables, we globally stabilize the unique equilibrium K of the equation xn+1=f(xn) in a certain domain. In our results, the noisy control α+ℓξ provides both local and global stability, while the mean value α of the control does not guarantee global stability, for example, the deterministic controlled system can have a stable two-cycle, and non-controlled map be chaotic. In the case of unimodal f with a negative Schwarzian derivative, we get sharp stability results generalizing Singer’s famous statement ‘local stability implies global’ to the case of the stochastic control. New global stability results are also obtained in the deterministic settings for variable αn and, generally, continuous but not differentiable at K map f.

Schwarzian derivative, Painlevé XXV–Ermakov equation, and Bäcklund transformations

AbstractThe role of Schwarzian derivative in the study of nonlinear ordinary differential equations is revisited. Solutions and invariances admitted by Painlevé XXV–Ermakov equation, Ermakov equation, and third‐order linear equation in a normal form are shown to be based on solutions of the Schwarzian equation. Starting from the Riccati equation and the second‐order element of the Riccati chain as the simplest examples of linearizable equations, by introducing a suitable change of variables, it is shown how the Schwarzian derivative represents a key tool in the construction of solutions. Two families of Bäcklund transformations, which link the linear and nonlinear equations under investigation, are obtained. Some analytical examples are given and discussed.