Schwarzian Research Articles

Investigation of breaking dynamics for Riemann waves in shallow water

Abstract Breaking dynamics of Riemann waves, has a profound impact on studying the dynamics of shock-breaking systems. Calogero– Bogoyavlenskii– Schiff (CBS) equation describes the interaction between Riemann propagating wave along y-axis with long propagating wave along x-axis. Two extensions of the CBS (2+1)-dimensional equation are investigated. Optimization of the commutative product for the nonlocal symmetry is constructed and two successive symmetry reductions reduced the equations to ordinary differential equations (ODEs). Hidden symmetries are used in the second reduction step. The singular manifold method (SMM) is then applied to the ordinary differential equations. This results in a Schwarzian derivative whose solution leads to Cuspon and Kink waves.

The properties of certain linear and nonlinear differential equations of the fourth order arising in beam models

The purpose of this paper is to present several new results concerning relations between linear differential equations of the fourth order and nonlinear differential equations of the fourth order. These equations are involved in the description of models of building structures, where there are beams with small deflections or curved axes. We consider linear differential equations of the second, the third and the fourth order and nonlinear fourth order differential equations related via the Schwarzian derivative. As a result, we obtain new relations between the solutions of these linear and nonlinear equations. For example, assuming that we know a solution of linear differential equation of the fourth order and a solution of the third order linear differential equation, then the Schwarzian derivative of their ratio solves certain nonlinear differential equation of the fourth order. We also present some conditions on the coefficients when this statement holds. Two more similar statements are presented. To illustrate theorems and our constructive approach we give two examples. The given method may be generalized to differential equations of higher orders.

Nonlinearly determined wavefronts of the Nicholson's diffusive equation: when small delays are not harmless

By proving the existence of non-monotone and non-oscillating wavefronts for the Nicholson's blowflies diffusive equation (the NDE), we answer an open question from [16]. Surprisingly, wavefronts of such a kind can be observed even for arbitrarily small delays. Similarly to the pushed fronts, obtained waves are not linearly determined. In contrast, a broader family of eventually monotone wavefronts for the NDE is indeed determined by properties of the spectra of the linearized equations. Our proofs use essentially several specific characteristics of the blowflies birth function (its unimodal form and the negativity of its Schwarz derivative, among others). One of the key auxiliary results of the paper shows that the Mallet-Paret–Cao–Arino theory of super-exponential solutions for scalar equations can be extended for some classes of second order delay differential equations. For the new type of non-monotone waves to the NDE, our numerical simulations also confirm their stability properties established by Mei et al.

Stability for one-dimensional discrete dynamical systems revisited

We present a new method to study the stability of one-dimensional discrete-time models, which is based on studying the graph of a certain family of functions. The method is closely related to exponent analysis, which the authors introduced to study the global stability of certain intricate convex combinations of maps. We show that the new strategy presented here complements and extends some existing conditions for the global stability. In particular, we provide a global stability condition improving the condition of negative Schwarzian derivative. Besides, we study the relation between this new method and the enveloping technique.

Sharp coefficient bounds for starlike functions associated with the Bell numbers

Abstract Let $\begin{array}{} \mathcal{S}^*_B \end{array}$ be the class of normalized starlike functions associated with a function related to the Bell numbers. By establishing bounds on some coefficient functionals for the family of functions with positive real part, we derive for functions in the class $\begin{array}{} \mathcal{S}^*_B \end{array}$ several sharp coefficient bounds on the first six coefficients and also further sharp bounds on the corresponding Hankel determinants. Bounds on the first three consecutive higher-order Schwarzian derivatives for functions in the class $\begin{array}{} \mathcal{S}^*_B \end{array}$ are investigated.

Exact solutions of nonlinear fractional order partial differential equations via singular manifold method

Singular manifold method is applied to solve nonlinear fractional partial differential equations (NLFPDEs). Fractional complex transformation is firstly used to reduce the equations to ordinary differential equations (ODEs). Using Bäcklund transformation the Schwarzian derivatives for the Eigen functions are obtained leading to a new variety of exact solutions. The method is applied to nonlinear time fractional Klein-Gordon, Cahn-Hilliard, Burger and Cahn-Allen equations. Some of the resulting solutions are graphically illustrated showing periodic kink, multi soliton and kink solutions.

Jordan totient quotients

The Jordan totient Jk(n) can be defined by Jk(n)=nk∏p|n(1−p−k). In this paper, we study the average behavior of fractions P/Q of two products P and Q of Jordan totients, which we call Jordan totient quotients. To this end, we describe two general and ready-to-use methods that allow one to deal with a larger class of totient functions. The first one is elementary and the second one uses an advanced method due to Balakrishnan and Pétermann. As an application, we determine the average behavior of the Jordan totient quotient, the kth normalized derivative of the nth cyclotomic polynomial Φn(z) at z=1, the second normalized derivative of the nth cyclotomic polynomial Φn(z) at z=−1, and the average order of the Schwarzian derivative of Φn(z) at z=1.

On the Slice Regular Schwarz Derivative

The aim of this paper is to introduce, in the context of slice regular functions, three notions of the quaternionic Schwarz derivative with their the basic properties: the characterization of zeros, the conformal covariant property and the conformal invariant property. These concepts mimic the structure of the complex Schwarz derivative which is a differential operator with several properties and applications. Particularly, one of these operators is curiously related to the generalization of the schwarzian derivative over vector spaces given by Ryan (Ann Pol Math 57:29–44, 1992).

Hausdorff Dimension of Escaping Sets of Nevanlinna Functions

Abstract We determine the exact values of Hausdorff dimensions of escaping sets of meromorphic functions with polynomial Schwarzian derivatives. This will follow from the relation between these functions and the second-order differential equations in the complex plane.

Schwarzian mechanics via nonlinear realizations

The method of nonlinear realizations is used to clarify some conceptual and technical issues related to the Schwarzian mechanics. It is shown that the Schwarzian derivative arises naturally, if one applies the method to SL(2,R)×R group and decides to keep the number of the independent Goldstone fields to a minimum. The Schwarzian derivative is linked to the invariant Maurer–Cartan one–forms, which make its SL(2,R)–invariance manifest. A Lagrangian formulation for a variant of the Schwarzian mechanics studied recently in A. Galajinsky (2018) [5] is built and its geometric description in terms of 4d metric of the ultrahyperbolic signature is given.

Maximal chaos from strings, branes and Schwarzian action

In this article, we explicitly demonstrate that, for a sufficiently generic class of examples, an open string embedded in an AdS-background yields an effective Schwarzian action and, in the semi-classical description, the fluctuation modes of the open string couple to this Schwarzian sector. This leads to a maximal chaos, observed in the open string degrees of freedom, irrespective of the gravitational background. This corresponds to the dynamics of quark-like degrees of freedom in a strongly coupled large Nc gauge theory. We also demonstrate a maximal chaos, resulting from an inherent D-brane horizon, by computing the four-point out-of-time-ordered correlator of spin-one operators. We also offer some observations and comments regarding a class of effective theories that is described by a generic functional of a Schwarzian derivative.

Who Should Exert More Effort? Risk Aversion, Downside Risk Aversion and Optimal Prevention

I provide new results on how risk preferences affect optimal prevention. I identify a comparative risk aversion and a comparative downside risk aversion effect and emphasize those cases where both effects are aligned. Alignment depends on a probability threshold, which, in turn, only depends on the preferences of a benchmark agent. This allows to define an entire class of decision-makers who all share the same comparative static prediction relative to the reference agent. I interpret my results in terms of Arrow-Pratt risk aversion and the Schwarzian derivative, and also study several classes of parametric preferences changes.

On Möbius‐invariant and symmetry‐integrable evolution equations and the Schwarzian derivative

AbstractWe consider symmetry‐integrable evolution equations in 1 + 1 dimensions of order 3 and order 5. We show that there exist only three equations in this class that are invariant under the Möbius transformation, and we name those Schwarzian equations. We report an interesting relation between the recursion operators of the Schwarzian equations and the corresponding adjoint operators that generate hierarchies of Schwarzian systems in terms of the Schwarzian derivative. This indicates a deep relation between the Schwarzian equations and the Schwarzian derivative. A classification of the fully nonlinear third‐order Schwarzian equations is also reported.

A Differential–Algebraic Projective Framework for the Deformable Single-View Geometry of the 1D Perspective Camera

Single-View Geometry (SVG) studies the world-to-image mapping or warp, which is the relationship that exists between a body’s model and its image. For a rigid body observed by a projective camera, the warp is described by the usual camera matrix and its properties. However, it is clear that for a body whose deformation state changes between the body’s model and its image, the ‘simple,’ globally parameterized warp described solely by the camera matrix, breaks down. Existing work has exploited deformation to reconstruct the deformed body from its image, but did not establish the properties of the deformable warp. Studying these properties is part of deformable SVG and forms a recent research topic. Because deformations may take place anywhere in the object’s body, and because they may be uncorrelated, the warp is local in nature. Using a differential framework is thus an obvious choice. We propose a differential–algebraic projective framework based on modeling the body’s surface by a locally rational projective embedding and on the 1D projective camera. We show that this leads, via the study of univariate rational functions, to differential invariants that the warp must satisfy. It may seem surprising, given the generic hypothesis made on the observed body, hardly stronger than mere local smoothness, that constraints can still be found. Our framework generalizes the Schwarzian derivative, the first-order projective differential invariant, which holds under the assumption that the body’s shape is locally linear. Our invariants may be used to construct regularizers to be used in warp estimation. We report experimental results of two types on simulated and real data. The first type shows that the proposed invariants hold well for an independently estimated warp. The second type shows that the proposed regularizers improve warp estimation from point correspondences compared to the classical derivative-penalizing regularizers.

A Shimorin-Type Estimate for Higher-Order Schwarzian Derivatives

Using Grunsky inequalities, an estimate on the norm of the multiplication by higher-order Schwarzian derivatives in weighted Bergman spaces is given.

Dynamics analysis and cryptographic application of fractional logistic map

Based on Lyapunov exponent, Schwarzian derivative, Shannon entropy and Kolmogorov entropy, we will firstly study chaos and bifurcation of fractional (semi-) logistic map (FLM). It is derived from fractional integration (not fractional derivative) of the classical logistic map. Then, this paper put forward a new accumulated coupled fractional (semi-) logistic map lattice (ACFLML) whose lattice function is the FLM. Local stability, pattern information, high-dimensional chaos and bifurcation of the ACFLML are analyzed based on stability theory, pattern simulation, 0–1 test for chaos, Lyapunov exponent spectrum and Kolmogorov entropy. Finally, the chaotic ACFLML is successfully applied to encryption and decryption of digital image. To evaluate security, histogram analysis, correlation analysis, differential attack, key space, key sensitivity, encryption time, computational complexity and chosen/known-plaintext attacks, analysis is performed. Simulation analysis shows that this encryption scheme is effective and has good statistical effect.

A New Formula to Get Sharp Global Stability Criteria for One-Dimensional Discrete-Time Models

We present a new formula that makes it possible to get sharp global stability results for one-dimensional discrete-time models in an easy way. In particular, it allows to show that the local asymptotic stability of a positive equilibrium implies its global asymptotic stability for a new family of difference equations that finds many applications in population dynamics, economic models, and also in physiological processes governed by delay differential equations. The main ingredients to prove our results are the Schwarzian derivative and some dominance arguments.

Higher Schwarzian derivative and Dirichlet Morrey space

We treat the logarithmic derivative model and Schwarzian derivative model of the Dirichlet-Morrey Teichm?ller space. It is shown that the higher Bers maps, induced by the higher Schwarzian differential operators, are holomorphic in Dirichlet-Morrey Teichm?ller space. It is also shown that the logarithmic derivative model of this Teichm?ller space is connected.

Conformal Mapping onto a Doubly Connected Circular Arc Polygonal Domain

This paper presents a construction principle for the Schwarzian derivative of conformal mappings from an annulus onto doubly connected domains bounded by polygons of circular arcs.

Bounded type Siegel disks of finite type maps with few singular values

Let $U$ be an open subset of the Riemann sphere $\S$.We give sufficient conditions for which a finite type map $f:U\to~\S$ with at most three singular values has a Siegel disk compactly contained in $U$ and whose boundary is a quasicircle containing a unique critical point. The main tool is quasiconformal surgery ala Douady-Ghys-Herman-§w. We also give sufficient conditions for which, instead, $\Delta$ has not compact closure in $U$. The main tool is the Schwarzian derivative and area inequalities ala Graczyk-§w.