Newton's method and Baker domains

We show that there exists an entire function f without zeros for which the associated Newton function N(z)=z-f(z)/f'(z) is a transcendental meromorphic functions without Baker domains. We also show that there exists an entire function f with exactly one zero for which the complement of the immediate attracting basin has at least two components and contains no invariant Baker domains of N. The second result answers a question of J. Rueckert and D. Schleicher while the first one gives a partial answer to a question of X. Buff.Comment: 6 page

Hyperbolic entire functions and the Eremenko–Lyubich class: Class B or not class B?

Hyperbolicity plays an important role in the study of dynamical systems, and is a key concept in the iteration of rational functions of one complex variable. Hyperbolic systems have also been considered in the study of transcendental entire functions. There does not appear to be an agreed definition of the concept in this context, due to complications arising from the non-compactness of the phase space. In this article, we consider a natural definition of hyperbolicity that requires expanding properties on the preimage of a punctured neighbourhood of the isolated singularity. We show that this definition is equivalent to another commonly used one: a transcendental entire function is hyperbolic if and only if its postsingular set is a compact subset of the Fatou set. This leads us to propose that this notion should be used as the general definition of hyperbolicity in the context of entire functions, and, in particular, that speaking about hyperbolicity makes sense only within the Eremenko–Lyubich classB of transcendental entire functions with a bounded set of singular values. We also considerably strengthen a recent characterisation of the class B, by showing that functions outside of this class cannot be expanding with respect to a metric whose density decays at most polynomially. In particular, this implies that no transcendental entire function can be expanding with respect to the spherical metric. Finally we give a characterisation of an analogous class of functions analytic in a hyperbolic domain

Normal families and fixed points of iterates

Let F be a family of holomorphic functions and let K be a constant less than 4. Suppose that for all f in F the second iterate of f does not have fixed points for which the modulus of the multiplier is greater than K. We show that then F is normal. This is deduced from a result about the multipliers of iterated polynomials.Comment: 5 page

Slow escaping points of quasiregular mappings

This article concerns the iteration of quasiregular mappings on Rd and entire functions on C. It is shown that there are always points at which the iterates of a quasiregular map tend to infinity at a controlled rate. Moreover, an asymptotic rate of escape result is proved that is new even for transcendental entire functions. Let f:Rd→Rd be quasiregular of transcendental type. Using novel methods of proof, we generalise results of Rippon and Stallard in complex dynamics to show that the Julia set of f contains points at which the iterates fn tend to infinity arbitrarily slowly. We also prove that, for any large R, there is a point x with modulus approximately R such that the growth of |fn(x)| is asymptotic to the iterated maximum modulus Mn(R,f)

Proof of a conjecture of Polya on the zeros of successive derivatives of real entire functions

We prove Polya's conjecture of 1943: For a real entire function of order greater than 2, with finitely many non-real zeros, the number of non-real zeros of the n-th derivative tends to infinity with n. We use the saddle point method and potential theory, combined with the theory of analytic functions with positive imaginary part in the upper half-plane.Comment: 26 page

Synchronization of organ pipes: experimental observations and modeling

We report measurements on the synchronization properties of organ pipes. First, we investigate influence of an external acoustical signal from a loudspeaker on the sound of an organ pipe. Second, the mutual influence of two pipes with different pitch is analyzed. In analogy to the externally driven, or mutually coupled self-sustained oscillators, one observes a frequency locking, which can be explained by synchronization theory. Further, we measure the dependence of the frequency of the signals emitted by two mutually detuned pipes with varying distance between the pipes. The spectrum shows a broad ``hump'' structure, not found for coupled oscillators. This indicates a complex coupling of the two organ pipes leading to nonlinear beat phenomena.Comment: 24 pages, 10 Figures, fully revised, 4 big figures separate in jpeg format. accepted for Journal of the Acoustical Society of Americ

Synchronization of Sound Sources

Sound generation and -interaction is highly complex, nonlinear and self-organized. Already 150 years ago Lord Rayleigh raised the following problem: Two nearby organ pipes of different fundamental frequencies sound together almost inaudibly with identical pitch. This effect is now understood qualitatively by modern synchronization theory (M. Abel et al., J. Acoust. Soc. Am., 119(4), 2006). For a detailed, quantitative investigation, we substituted one pipe by an electric speaker. We observe that even minute driving signals force the pipe to synchronization, thus yielding three decades of synchronization -- the largest range ever measured to our knowledge. Furthermore, a mutual silencing of the pipe is found, which can be explained by self-organized oscillations, of use for novel methods of noise abatement. Finally, we develop a specific nonlinear reconstruction method which yields a perfect quantitative match of experiment and theory.Comment: 5 pages, 4 figure

Rigidity of escaping dynamics for transcendental entire functions

We prove an analog of Boettcher's theorem for transcendental entire functions in the Eremenko-Lyubich class B. More precisely, let f and g be entire functions with bounded sets of singular values and suppose that f and g belong to the same parameter space (i.e., are *quasiconformally equivalent* in the sense of Eremenko and Lyubich). Then f and g are conjugate when restricted to the set of points which remain in some sufficiently small neighborhood of infinity under iteration. Furthermore, this conjugacy extends to a quasiconformal self-map of the plane. We also prove that this conjugacy is essentially unique. In particular, we show that an Eremenko-Lyubich class function f has no invariant line fields on its escaping set. Finally, we show that any two hyperbolic Eremenko-Lyubich class functions f and g which belong to the same parameter space are conjugate on their sets of escaping points.Comment: 28 pages; 2 figures. Final version (October 2008). Various modificiations were made, including the introduction of Proposition 3.6, which was not formally stated previously, and the inclusion of a new figure. No major changes otherwis

On permutable meromorphic functions

We study the class ${\mathcal{M}}$ of functions meromorphic outside a countable closed set of essential singularities. We show that if a function in ${\mathcal{M}}$, with at least one essential singularity, permutes with a non-constant rational map $g$, then $g$ is a Möbius map that is not conjugate to an irrational rotation. For a given function ${f \in \mathcal{M}}$ which is not a Möbius map, we show that the set of functions in ${\mathcal{M}}$ that permute with ƒ is countably infinite. Finally, we show that there exist transcendental meromorphic functions ${f : \mathbb{C} \to \mathbb{C}}$ such that, among functions meromorphic in the plane, ƒ permutes only with itself and with the identity map

A digital shadow cloud-based application to enhance quality control in manufacturing

In Industry 4.0 era, rapid changes to the global landscape of manufacturing are transforming industrial plants in increasingly more complex digital systems. One of the most impactful innovations generated in this context is the "Digital Twin", a digital copy of a physical asset, which is used to perform simulations, health predictions and life cycle management through the use of a synchronized data flow in the manufacturing plant. In this paper, an innovative approach is proposed in order to contribute to the current collection of applications of Digital Twin in manufacturing: a Digital Shadow cloud-based application to enhance quality control in the manufacturing process. In particular, the proposal comprises a Digital Shadow updated on high performance computing cloud infrastructure in order to recompute the performance prediction adopting a variation of the computer-aided engineering model shaped like the actual manufactured part. Thus, this methodology could make possible the qualification of even not compliant parts, and so shift the focus from the compliance to tolerance requirements to the compliance to usage requirements. The process is demonstrated adopting two examples: the structural assessment of the geometry of a shaft and the one of a simplified turbine blade. Moreover, the paper presents a discussion about the implications of the use of such a technology in the manufacturing context in terms of real-time implementation in a manufacturing line and lifecycle management. Copyright (C) 2020 The Authors