Quadratic Rational Function Research Articles

Arboreal Galois groups for quadratic rational functions with colliding critical points

Arboreal Galois groups for quadratic rational functions with colliding critical points

Cyclotomic and abelian points in backward orbits of rational functions

We prove several results on backward orbits of rational functions over number fields. First, we show that if K is a number field, ϕ∈K(x) and α∈K then the extension of K generated by the abelian points (i.e. points that generate an abelian extension of K) in the backward orbit of α is ramified only at finitely many primes. This has the immediate strong consequence that if all points in the backward orbit of α are abelian then ϕ is post-critically finite. We use this result to prove two facts: on the one hand, if ϕ∈Q(x) is a quadratic rational function not conjugate over Qab to a power or a Chebyshev map and all preimages of α are abelian, we show that ϕ is Q-conjugate to one of two specific quadratic functions, in the spirit of a recent conjecture of Andrews and Petsche. On the other hand we provide conditions on a quadratic rational function in K(x) for the backward orbit of a point α to only contain finitely many cyclotomic preimages, extending previous results of the second author. Finally, we give necessary and sufficient conditions for a triple (ϕ,K,α), where ϕ is a K-Lattès map over a number field K and α∈K, for the whole backward orbit of α to only contain abelian points.

Study on Characteristics of Compression Deformation and Post-Peak Stress Rebound for Solid Waste Cemented Body

Most of the previous studies focused on the mechanical characteristics before the stress peak of solid waste cemented backfill, but in the compression process of a solid waste cemented body, the phenomenon of post-peak stress rebound often occurs. Through the uniaxial compression experiment of a solid waste cemented body composed of coal gangue, fly ash, desulfurization gypsum, gasification slag, and furnace bottom slag, this paper analyzed the compression deformation characteristics of a solid waste cemented body with different mix proportions before and after the stress peak, established the stress–strain curve model of rebound stress in the rising and descending section after the stress peak, and revealed the reasons for the rebound stress and secondary unloading of the cemented body after the stress peak. The results showed that the maximum rebound stress accounts for 40%–80% of the compressive strength, and the changes in the two are positively correlated. The stress–strain curve model is a cubic function in the post-peak stress rising section and a quadratic rational function in the descending section. With the increase in the maximum compressive strength of the cemented body, its maximum rebound stress also increases, but its corresponding compressive strain generally shows a downward trend. There is a positive correlation between the rebound stress increment and strain increment of the cemented body. The change in the supporting structure and the evolution of the failure form of the cemented body before and after the maximum rebound stress indicate that the compression failure of the residual supporting structure caused by the main crack is the main reason for the rebound of the stress after the peak value of the cemented body to the complete unloading.

Parabolic bifurcation loci in the spaces of rational functions

We give a geometric description of the parabolic bifurcation locus in the space Rat d of all rational functions on of degree d > 1, generalising the study by Morton and Vivaldi in the case of monic polynomials. The results are new even for quadratic rational functions.

Quadratic Rational Maps with a 2-Cycle of Siegel Disks

For the family of quadratic rational functions having a $2$-cycle of bounded type Siegel disks, we prove that each of the boundaries of these Siegel disks contains at most one critical point. In the parameter plane, we prove that the locus for which the boundaries of the $2$-cycle of Siegel disks contain two critical points is a Jordan curve.

On Deformation Spaces of Quadratic Rational Functions

AbstractWe study the group of self-equivalences of a partially postcritically finite branched cover and show that certain deformation spaces of rational maps are not contractible.

Prediction of gas-liquid two-phase choke flow using Gaussian process regression

In the process of shale gas production, it is of great significance to select an appropriate mathematical method to accurately predict the gas flow rate of wellhead choke for the rational formulation of shale gas well production plan. Gas-liquid two-phase flow occurs in most of the time from the flowback to the production period in shale gas wells. Wellhead chokes play key roles in regulating the flowing rates of both the flowback fluid and shale gas. Therefore, it is important to study the law of two-phase choke flow clearly so as to accurately predict gas flow rate through wellhead chokes. Up to now, previous studies have proposed a variety of applicable empirical methods, including Gilbert-type correlation (GC), artificial neural network (ANN) and support vector machine (SVM). The analysis of training data and the establishment of accurate prediction models determine the accuracy of prediction. In this study, Gaussian process regression (GPR) was adopted to learn and predict the behavior of gas-liquid two-phase flow through wellhead chokes, and huge amounts of data collected from Chuannan Shale gas wells were used to verify the effectiveness of the GPR method. The prediction accuracy of the GPR method was compared with those of other methods like GC, ANN and SVM. In addition, we also compared the prediction accuracy of different kernel functions to select the best kernel function for GPR. The kernel functions considered are exponential function, squared exponential function, rational quadratic function and Matérn function. The results showed that GPR method is accurate and applicable for analyzing the behavior of gas-liquid two-phase flow through wellhead chokes, and GPR method with exponential kernel function could achieve greater prediction accuracy than other kernel functions.

Quadratic rational functions with a rational periodic critical point of period 3

Quadratic rational functions with a rational periodic critical point of period 3

An Extended Newmark-FDTD Method for Complex Dispersive Media

Based on polarizability in the form of a complex quadratic rational function, a novel finite-difference time-domain (FDTD) approach combined with the Newmark algorithm is presented for dealing with a complex dispersive medium. In this paper, the time-stepping equation of the polarization vector is derived by applying simultaneously the Newmark algorithm to the two sides of a second-order time-domain differential equation obtained from the relation between the polarization vector and electric field intensity in the frequency domain by the inverse Fourier transform. Then, its accuracy and stability are discussed from the two aspects of theoretical analysis and numerical computation. It is observed that this method possesses the advantages of high accuracy, high stability, and a wide application scope and can thus be applied to the treatment of many complex dispersion models, including the complex conjugate pole residue model, critical point model, modified Lorentz model, and complex quadratic rational function.

Modeling carbon stock dynamics under fallow and cocoa agroforest systems in the shifting agricultural landscape of Central Cameroon

With increasing concerns raised by climate change, understanding biological processes within cocoa (Theobroma cacao L.) agroforest (CAF) and fallow systems is a prerequisite for developing actions related to emission reduction in the shifting agricultural landscape of Cameroon. Carbon (C) stocks and accretion were assessed and modeled in various C components (large trees, small trees, dead wood, litter, roots, soil, and total C) of fallow and CAF systems along a 50-year chronosequence. Several functions were empirically fitted to a time series of C stocks. Large tree, soil, and total C stocks were best described by a logistic growth function while that for small trees by a rational quadratic function. The best-fitted functions explained 72–96 % of C stock accumulation over time. Two metrics describing C stock accretion were derived from these functions: the point of maximum C growth and the C growth coefficient (GC). The rate of maximum growth of total C stock was reached after 12–13 years in both fallow and CAF, with maximum GCs of 6.9 and 6.3 Mg C ha−1 year−1, respectively. Over the 50-year period, the GCs of total C stocks varied between 0.2 and 6.9 Mg C ha−1 year−1, with quick accumulation within the first decade that then slowed until it levelled off after 45 years. Over a period of about 30 years, both systems sequestered a total of ~200 Mg C ha−1. This indicates that cocoa agroforests, a main source of income for local populations, can also provide significant climate change mitigation services.

ON THE DYNAMICS OF A FAMILY OF RATIONAL FUNCTIONS WHICH CONTAINS KOEBE FUNCTION

We study the dynamics of a parameterized family of quadratic rational functions z 7→�z/(1−z) 2 , � ∈ (0,∞). We have investigated the dynamical and geometric properties of Julia sets and the nature of the variance of these properties according to the parameter. We have shown the Julia sets are connected when � > 1 and are dynamically defined cantor sets when � < 1. We also have shown the continuity of geometric properties specially in terms of Hausdorff dimensions of these Julia sets.

On the Numerical Accuracy of Finite-Difference Time-Domain Dispersive Modeling Based on a Complex Quadratic Rational Function

Recently, based on a quadratic complex rational function, an attractive finite-difference time-domain algorithm was suggested for dispersive modeling of complex media because it is accurate and easy to implement. To fully utilize the quadratic complex rational function finite-difference time-domain, it is essential to investigate its numerical errors based on an exact mathematical approach. Toward this purpose, the exact expression of the numerical permittivity is first derived. From this numerical permittivity, the numerical dispersion, numerical dissipation, and numerical anisotropy inherent to the quadratic complex rational function finite-difference time-domain are examined. Numerical examples illustrate that the numerical errors of the quadratic complex rational function finite-difference time-domain is almost same as those of the nondispersive finite-difference time-domain.

Galois theory of quadratic rational functions

For a number field K with absolute Galois group G_K , we consider the action of G_K on the infinite tree of preimages of \alpha \in K under a degree-two rational function \phi \in K(x) , with particular attention to the case when \phi commutes with a non-trivial Möbius transformation. In a sense this is a dynamical systems analogue to the \ell -adic Galois representation attached to an elliptic curve, with particular attention to the CM case. Using a result about the discriminants of numerators of iterates of \phi , we give a criterion for the image of the action to be as large as possible. This criterion is in terms of the arithmetic of the forward orbits of the two critical points of \phi . In the case where \phi commutes with a non-trivial Möbius transformation, there is in effect only one critical orbit, and we give a modified version of our maximality criterion. We prove a Serre-type finite-index result in many cases of this latter setting.

Qualitative theory of differential equations and dynamics of quadratic rational functions

We study the qualitative theory of first order di fferential equations consisting of the iteration of complex quadratic rational functions and we focus on the configuration, namely location and stability, of simple equilibrium points which correspond to periodic points of the quadratic rational functions. Our main tools are properties of Julia sets of the quadratic rational functions and the Euler-Jacobi formula.

Copulas with Given Rational Function Sections

We introduce a construction method with rational function horizontal or vertical sections based on the relationship between copula function and its generator. The copulas formula with fraction linear horizontal (vertical) sections is given. In addition, nonexistence about Archimedean copulas with quadratic rational function horizontal or vertical sections has been proved.

Minimal models for rational functions in a dynamical setting

AbstractWe present a practical algorithm to compute models of rational functions with minimal resultant under conjugation by fractional linear transformations. We also report on a search for rational functions of degrees 2 and 3 with rational coefficients that have many integers in a single orbit. We find several minimal quadratic rational functions with eight integers in an orbit and several minimal cubic rational functions with ten integers in an orbit. We also make some elementary observations on possibilities of an analogue of Szpiro’s conjecture in a dynamical setting and on the structure of the set of minimal models for a given rational function.

Captures, matings and regluings

In parameter slices of quadratic rational functions, we identify arcs represented by matings of quadratic polynomials. These arcs are on the boundaries of hyperbolic components.

Some cubic Blaschke products and quadratic rational functions with Siegel disks

We show that for any given complex number „ with modulus at most one and any given real number fi, there exists a cubic Blaschke product such that the point at infinity is its fixed point with multiplier „ and its restriction on the unit circle is a critical circle map with rotation number fi. Moreover if the given real number fi is irrational of bounded type, then a modified Blaschke product is quasiconformally conjugate to some quadratic rational function with a Siegel disk whose boundary is a quasicircle containing its critical point and the point at infinity is its fixed point with multiplier „.

Non-monotone bifurcations for quadratic rational functions mf( x) m

Non-monotone bifurcations for quadratic rational functions mf( x) m

A kinetic analysis of the actions of l-noradrenaline and its related agonists and antagonists on in vitro lipolysis in rat adipose tissue

(1) Iterative non-linearising optimisation techniques have been used to fit three alternative models to relationships between β-adrenergic effector concentrations and the lipolytic response which they elicit in dispersed adipocytes derived from rat epididymal fat pads. (2) The models, which consist of a simple hyperbolic relationship, a Hill-type function and a rational quadratic formulation, were fitted to data obtained with agonists, ‘partial’ agonists and antagonists both alone and in combination. (3) Whereas the hyperbolic relationship was inadequate in all circumstances, the Hill-type function accommodated dose-response curves which exhibit no ‘auto-inhibitory’ hook or bell-shaped feature. However, the rational quadratic function could be satisfactorily fitted to the data whether or not the auto-inhibitory phase was apparent. (4) The mechanisms that govern the steepness of the dose-response relationships and their bell-shaped feature are discussed. Evidence is presented that the latter originates at the level of adenylate cyclase.