Radius Of Convergence Research Articles

Böttcher coordinates at wild superattracting fixed points

AbstractLet be a prime number, let with , and let be the Böttcher coordinate satisfying . Salerno and Silverman conjectured that the radius of convergence of in is . In this article, we confirm that this conjecture is true by showing that it is a special case of our more general result.

Olsson.wl & ROC2.wl: Mathematica packages for transformations of multivariable hypergeometric functions & regions of convergence for their series representations in the two variables case

We present the Olsson.wlMathematica package which aims to find transformations for some classes of multivariable hypergeometric functions. It is based on a well-known method developed by P. O. M. Olsson [1] (1964) to derive the analytic continuations of the Appell F1 double hypergeometric series using the linear transformations of the Gauss F12 hypergeometric function. We provide a brief description of the method of Olsson and demonstrate the use of the commands of the Olsson.wl package using some examples that are presented in the text and in some ancillary Mathematica notebooks. In particular, we reproduce various results of the literature on multivariable hypergeometric functions and show practical applications of this package in the derivation of novel formulas. In the context of high energy physics, we also demonstrate how it can be used to disentangle some known results about the analytic continuation of some series representations of the one-loop pentagon in multi-Regge kinematics and D=6−2ϵ. We also provide a companion package, called ROC2.wl, which is dedicated to the derivation of the regions of convergence of double hypergeometric series. This package can be used independently of Olsson.wl. Program summaryProgram Title:Olsson.wlCPC Library link to program files:https://doi.org/10.17632/gc63xzwzz5.1Licensing provisions: GNU General Public License v3.0.Programming language: Wolfram Mathematica version 11.3 and beyond.Nature of problem: To find the transformation formulas of multivariable hypergeometric series appearing in Feynman integral calculus.Solution method: Mathematica implementation of the method of Olsson [1]. The method uses the transformation theory of lower variable hypergeometric functions to find the transformation formulas of higher variable hypergeometric functions.Companion package:ROC2.wl, which is a Mathematica package used inside Olsson.wl and can also be used as a standalone package. The package is used to find the region of convergence of double hypergeometric functions, using Horn's theorem.

Norm and Numerical Radius Inequalities for Sums of Power Series of Operators in Hilbert Spaces

The main focus of this paper is on establishing inequalities for the norm and numerical radius of various operators applied to a power series with the complex coefficients h(λ)=∑k=0∞akλk and its modified version ha(λ)=∑k=0∞|ak|λk. The convergence of h(λ) is assumed on the open disk D(0,R), where R is the radius of convergence. Additionally, we explore some operator inequalities related to these concepts. The findings contribute to our understanding of operator behavior in bounded operator spaces and offer insights into norm and numerical radius inequalities.

Taylor Polynomials in a High Arithmetic Precision as Universal Approximators

Function approximation is a fundamental process in a variety of problems in computational mechanics, structural engineering, as well as other domains that require the precise approximation of a phenomenon with an analytic function. This work demonstrates a unified approach to these techniques, utilizing partial sums of the Taylor series in a high arithmetic precision. In particular, the proposed approach is capable of interpolation, extrapolation, numerical differentiation, numerical integration, solution of ordinary and partial differential equations, and system identification. The method employs Taylor polynomials and hundreds of digits in the computations to obtain precise results. Interestingly, some well-known problems are found to arise in the calculation accuracy and not methodological inefficiencies, as would be expected. In particular, the approximation errors are precisely predictable, the Runge phenomenon is eliminated, and the extrapolation extent may a priory be anticipated. The attained polynomials offer a precise representation of the unknown system as well as its radius of convergence, which provides a rigorous estimation of the prediction ability. The approximation errors are comprehensively analyzed for a variety of calculation digits and test problems and can be reproduced by the provided computer code.

Hölder-Type Inequalities for Power Series of Operators in Hilbert Spaces

Consider the power series with complex coefficients h(z)=∑k=0∞akzk and its modified version ha(z)=∑k=0∞|ak|zk. In this article, we explore the application of certain Hölder-type inequalities for deriving various inequalities for operators acting on the aforementioned power series. We establish these inequalities under the assumption of the convergence of h(z) on the open disk D(0,ρ), where ρ denotes the radius of convergence. Additionally, we investigate the norm and numerical radius inequalities associated with these concepts.

On the complexity of extending the convergence domain of Newton’s method under the weak majorant condition

Abstract The local analysis of convergence for Newton’s method has been extensively studied by numerous researchers under a plethora of sufficient conditions. However, the complexity of extending the convergence domain requires very general conditions such as the ones depending on the majorant principle in order to include as large classes of operators as possible. In the present article, such an analysis is developed under the weak majorant condition. The new results extend earlier ones using similar information. Finally, the numerical examples complement the theory.

Sliding mode control for Markovian jump systems under a switched scheduling protocol

SummaryThis article is concerned with the sliding mode control for Markovian jump systems under transmission constraints. To deal with node congestions, a switched channel scheduling scheme integrating round‐robin (RR) protocol and weight try‐once‐discard (WTOD) protocol is proposed to orchestrate the priority of multiple sensor nodes. At each transmission instant, a system performance‐based detector is adopted to select one scheduling protocol between RR and WTOD in a certain event‐triggering manner, by which only one sensor node has the priority to access the network. Compared with conventional protocols, the proposed scheduling scheme establishes a dynamic coupling mechanism between channel resource and scheduling rules, aiming to strike a balance between improving system performance and reducing communication resource requirements. Then, a token‐dependent sliding mode controller is constructed, and a genetic algorithm‐based optimization problem is given to achieve smaller convergent domain. Finally, numerical examples are shown to verify the present control strategy.

A FIXED-POINT TYPE RESULT FOR SOME NON-DIFFERENTIABLE FREDHOLM INTEGRAL EQUATIONS

In this paper, we present a new fixed-point result to draw conclusions about the existence and uniqueness of the solution for a nonlinear Fredholm integral equation of the second kind with non-differentiable Nemytskii operator. To do this, we will transform the problem of locating a fixed point for an integral operator into the problem of locating a solution of an integral equation. Thus, assuming conditions on the Nemytskii operator, we will obtain a global convergence domain for the solution of the considered integral equation, taking for this a uniparametric family of derivativefree iterative processes with quadratic convergence. This result provides us a new fixed-point result for the integral operator considered.

Robust Stabilization of Uncertain Jerk Chaotic Control Systems with Mixed Uncertainties

In this paper, the problem of robust stabilizability of jerk chaotic control systems with mixed uncertainties is investigated. Combining robust control theory and differential-integral inequalities, a nonlinear controller will be derived and guaranteed to achieve the goal of practical stabilization. Besides, both the convergence radius and the exponential convergence rate can be specified in advance. Finally, some numerical simulation results are supplemented to demonstrate the correctness and effectiveness of the main result

Hybrid-scheduling-based sliding mode control for Markovian jump systems under transmission constraints

This paper presents a new hybrid-scheduling-based design of sliding mode control for Markovian jump systems against transmission constraints.The hybrid scheduling scheme integrates stochastic communication protocol and event-triggering mechanism. At each transmission instant, a discrete-time Markovian chain is employed to orchestrate the stochastic transmission of multiple sensor nodes, by which only one sensor node has the priority to be evaluated whether it meets the triggering condition to access the network. This hybrid scheduling scheme aims to generate a sparser sampling pattern than the existing one while improving channel resource utilization. Based on which a feasible compensator and token-dependent sliding mode controller is constructed. Moreover, a genetic algorithm-based optimization problem is given to achieve a smaller convergent domain. Finally, a numerical example is shown to verify the present control strategy.

On a New Two-Point Taylor Expansion With Applications

A new two-point Taylor series expansion is proposed. The expansion is slightly different than the classical definition. The coefficients are calculated as recursive relations in a general form. The two-point Taylor expansion is applied to several functions which are odd, even, neither odd nor even. Functions having finite interval of convergence or infinite interval of convergence are investigated. The conditions for convergence are derived and the results are compared with the results of single-point Taylor expansions as well as two-point Taylor expansions reported in the literature. It is found that for a finite radius of convergence, two-point Taylor expansions can have a single convergence interval as well as two separate convergence intervals. Generally speaking, two-point Taylor expansions better represent the real function when the series is truncated. The new two-point expansion and the classical two-point expansion produced identical results for all the problems treated. Based on the results of this analysis, the asymmetric two-point Taylor expansion presented here does not have an advantage compared to the classical symmetric expansion. An application of the series to solution of a variable coefficient differential equation is also treated.

The Higgs-gluon form factor at three loops in QCD with three mass scales

We report on the complete three-loop corrections to the Higgs-gluon form factor in QCD. While previous calculations are based on QCD with a single heavy quark of arbitrary mass, we extend the study to QCD involving two different massive quark flavors. Thereby, the full set of possible Feynman diagrams at three-loop order is taken into account. Employing differential equations for the relevant master integrals, we determine the form factor in terms of analytic expansions. Outside the radii of convergence, we compute high-precision numerical samples over the two-dimensional physical parameter space. Our new findings will enter as virtual corrections the computation of the top-bottom interference in hadronic Higgs-boson production at next-to-next-to-leading order (NNLO) in QCD.

A Convex Combination–Variable-Step-Size Least Mean p-Norm Algorithm

Underwater acoustic channels often have to face the interference of impulsive noise, which is usually modeled by α-stable distribution in simulation experiments. To solve the problem of underwater acoustic channel estimation under impulsive noise, this paper proposes a convex combination–variable-step-size least mean p-norm algorithm. The algorithm incorporates a convex combination into the variable-step-size least mean p-norm algorithm and uses the convex combination of different convergence domains provided by changing the parameters of the Gaussian function to further improve the effect after convergence. The simulation results of channel estimation show that the convex combination–variable-step-size least mean p-norm algorithm provides a more stable, robust, and universal solution than the variable-step-size least mean p-norm algorithm.

Practical Aspects for Applying Picard Iterations to the SIR Model Using Actual Data

The updated version of the Picard method for solving systems of differential equations is employed to solve the SIR system. A local performance of the Picard iteration algorithm combined with the Gauss-Seidel approach is applied to the SIR model. The integral form of the SIR model, in addition to the use of Gauss-Seidel philosophy (using the most recent calculated values), achieved more accuracy in the computational work than those obtained using the differential forms. Documented data regarding the spread of corona virus 19 in the Kingdom of Saudi Arabia region from April to the end of December 2020 were used to calculate the corresponding actual values for the parameters and the initial conditions. Due to efficient management and to obtain representable behaviours, we restricted the size of the study to only 1% of the population. The global characteristics of the integral formulation have affected the calculations accurately. The initial conditions and the model’s parameters are established depending on the documented data. The results illustrate the superiority of the updated Picard formulation over the classical Picard within their domain of convergence. The results of this study illuminate and validate the importance of mathematical modeling. These findings can provide valuable insights into mathematical modeling for those involved in environmental health research, especially those responsible for devising strategic plans.

On the Analytic Extension of Lauricella–Saran’s Hypergeometric Function FK to Symmetric Domains

In this paper, we consider the representation and extension of the analytic functions of three variables by special families of functions, namely branched continued fractions. In particular, we establish new symmetric domains of the analytical continuation of Lauricella–Saran’s hypergeometric function FK with certain conditions on real and complex parameters using their branched continued fraction representations. We use a technique that extends the convergence, which is already known for a small domain, to a larger domain to obtain domains of convergence of branched continued fractions and the PC method to prove that they are also domains of analytical continuation. In addition, we discuss some applicable special cases and vital remarks.

A cleavage-based conceptualisation of politicised global integration

ABSTRACT How can we conceptualise politicised integration at the global level? Functionalism and neo-functionalism are de-politicised, technocratic theories of integration dispensing with ideology and contestation, while post-functionalist approaches address politicisation but not supra-national integration. Drawing on comparative macro-historical theories, this article suggests a conceptualisation of politicised integration that extends exit–voice and political structuring models used at national and European levels to global integration. The replacement of territorial divisions with cleavages cutting across world regions – core–periphery and North–South divisions, civilisations and ‘multiple modernities’ – creates the conditions for cross-territorial dimensions of contestation, along which cross-regional membership groups link, coordinate and form bonds of solidarity. The de-territorialisation of global cleavages and the regional convergence it brings about, provide the social bases for mass politics rather than elitist policy integration. The conceptualisation of integration is political rather than technocratic, and one in which collective decision making is a matter of pluralist contestation rather than problem-solving expertise.

Revisiting the Seasonal Cycle of Rainfall over Central Africa

Abstract The intertropical convergence zone (ITCZ), with its twice-annual passage over central Africa, is considered as the main driver of the rainfall seasonality. In this ITCZ paradigm, high rainfall occurs over regions of large low-level convergence. But recently, this paradigm was challenged over central Africa. Here, we show that a shallow meridional overturning circulation—driven by surface conditions—plays a thermodynamical control on the rainfall seasonality over central Africa. Indeed, due to the local evaporative cooling effect, the foot of the ascending branch of Hadley cells occurs where the temperature is the warmest, indicating a thermal low. This distorts the southern Hadley cell by developing its bottom-heavy structure. As result, both shallow and deep Hadley cells coexist over central Africa year-round. The deep mode is associated with the poleward transport of atmospheric energy at upper levels. The shallow mode is characterized by a shallow meridional circulation, with its moisture transport vanishing and converging in the midtroposphere rather than at lower troposphere. This midtropospheric moisture convergence is also the dominant component that shapes the vertically integrated moisture flux convergence, with little contribution of African easterly jets. This convergence zone thus controls the precipitating convection. Its meridional migration highlights the interhemispheric rainfall contrast over central Africa and outlines the unimodal seasonality. On the other hand, forced by the Congo basin cell, the precipitable water regulates the deep convection from the vegetated surface of Congo basin, acting as a continental sea. This nonlinear mechanism separates the rainfall into three distinct regimes: the moisture-convergence-controlled regime, with convective rainfall exclusively occurring in the rainy season; the local evaporation-controlled regime with drizzle in the dry season; and the precipitable-water-controlled regime, with exponential rainfall increase in the dry season.

Larger convergence regions for an efficient two-step iterative method

Larger convergence regions for an efficient two-step iterative method

Composition-driven phase coexistence and functional properties of the (1-x)BZT-(x)BCT ceramics near the phase convergence region

The concept of composition-induced phase transformation in Lead Zirconate Titanate (PZT) at the Morphotropic Phase Boundary (MPB) has been employed to improve functional properties of the (1-x)BZT-(x)BCT ceramic. However, it was observed that the phase diagram of the (1-x)BZT-(x)BCT ceramic is different to the PZT. As a result, the nature of the superior functional properties found in (1-x)BZT-(x)BCT ceramic is unlike PZT and still unclear so far. In this work, functional properties; dielectric, ferroelectric, energy storage, and piezoelectric properties, of the (1-x)BZT-(x)BCT ceramics where x = 0.3 mol% to 0.6 mol% were evaluated at room temperature in comparison to the identification of phase coexistence using synchrotron x-ray powder diffraction (SXPD). This work found that changes of BCT content had a strong impact on the observed coexisting phases and functional properties. Moreover, the composition that showed the highest piezoelectric properties did not present the largest of saturation polarization. This implies that the functional properties of the (1-x)BZT-(x)BCT ceramics are not dependent on the presence of polarizations under the application of electric field. The contribution of non-180° domain switching also plays a vital role, especially in the piezoelectric properties. These findings would help to extend our knowledge of the nature of the (1-x)BZT-(x)BCT ceramic.

On a New Two Point Taylor Expansion With Applications

A new two point Taylor series expansion is proposed. The expansion is slightly different than the classical definition. The coefficients are calculated as recursive relations in a general form. The two point Taylor expansion is applied to expressing two different functions, one of which has a finite interval of convergence and the other infinite interval of convergence. The conditions for convergence are derived and the results are compared with the results of single point Taylor expansions as well as two point Taylor expansions reported in the literature. It is found that for a finite radius of convergence, two point Taylor expansions can have a single convergence interval as well as two separate convergence intervals. Results of the new expansion are compared with the single point Taylor expansions as well as the classical two point Taylor expansion. Generally speaking two point Taylor expansions better represent the real function when the series is truncated. The new two point expansion and the classical two point expansion produced identical results for the problems treated. An application of the series to solution of a variable coefficient differential equation is also treated.