Roots Of Polynomial Research Articles

Approximating roots by quadratic iteration

We apply a coctel of elementary methods to the problem of finding the roots of an arbitrary polynomial. Specifically, we combine properties of the iteration z → z2 + c with rudimentary Galois theory in order to justify an algorithm to find the roots of a complex polynomial.

Efficient Inverse Fractional Neural Network-Based Simultaneous Schemes for Nonlinear Engineering Applications

Finding all the roots of a nonlinear equation is an important and difficult task that arises naturally in numerous scientific and engineering applications. Sequential iterative algorithms frequently use a deflating strategy to compute all the roots of the nonlinear equation, as rounding errors have the potential to produce inaccurate results. On the other hand, simultaneous iterative parallel techniques require an accurate initial estimation of the roots to converge effectively. In this paper, we propose a new class of global neural network-based root-finding algorithms for locating real and complex polynomial roots, which exploits the ability of machine learning techniques to learn from data and make accurate predictions. The approximations computed by the neural network are used to initialize two efficient fractional Caputo-inverse simultaneous algorithms of convergence orders ς+2 and 2ς+4, respectively. The results of our numerical experiments on selected engineering applications show that the new inverse parallel fractional schemes have the potential to outperform other state-of-the-art nonlinear root-finding methods in terms of both accuracy and elapsed solution time.

Fullerenes via their counting polynomials

For a graph G with orbits O1,…,Ok, the orbit polynomial is defined as OG(x)=∑i=1kx|Oi|. In the current work, we will characterize all fullerenes concerning their pentagonal orbit polynomials (namely, the orbit polynomials for orbits of pentagons). Besides, the orbit polynomial, the orbit-entropy, the symmetry index and the unique positive root of orbit polynomial are computed for several infinite families of fullerenes. Finally, correlations between these graph measures are established to quantify their relationship.

On the Bernoulli–Euler–Lagrange–Aitken numerical method for roots of polynomials

The article presents a development of the Euler–Lagrange method for calculation of all roots of an arbitrary polynomial P(z) with complex coefficients based on the calculation of the limits of ratios of determinants (as in the Bernoulli–Aitken–Nikiporets methods) built by means of the Taylor and Laurent series coefficients for the function P′(z) / P(z).

Alternating Roots of Polynomials over Cayley-Dickson Algebras

We introduce the notions of alternating roots of polynomials and alternating polynomials over a Cayley-Dickson algebra, and prove a connection between the alternating roots of a given polynomial and the roots of the corresponding alternating polynomial over the Cayley-Dickson doubling of the algebra. We also include a detailed Octave code for the computation of alternating roots over Hamilton's quaternions.

Landau singularities and higher-order polynomial roots

Landau singularities and higher-order polynomial roots

Expected Number of Roots of Random Orthogonal Polynomials

Expected Number of Roots of Random Orthogonal Polynomials

The Interrelationship between Economic Growth and Tax Revenues in Cambodia

The relationship between economic growth and the growth rate of tax revenues on goods and services, tax revenue on income, profit, and capital gain, and tax revenue on international trade and transaction was analyzed using a VAR model. All variables in this study were found to be integrated of order one, therefore the model was run using first differences. The lag length of the model was determined to be optimal at lag-two based on the information criterion. The estimated results of the model successfully passed all diagnostic tests, including tests for residual normality, serial correlation, and heteroscedasticity. Since all the inverse roots of the AR characteristic polynomial were within the unit circle, the model was deemed stable. The empirical findings from the VAR model indicated that the growth rate of tax revenue on income, profit, and capital had the most significant impact on economic growth, ranging from 2.3505% to 2.7155%. This was followed by tax revenue on goods and services, ranging from 0.5776% to 0.5954%, and tax revenue on international trade and transaction, ranging from 0.2747% to 0.5930%. Furthermore, the response of the growth rate of all tax revenues to changes in the economic growth rate exhibited a cyclical pattern around its mean.

On Geometric Interpretations of Euler’s Substitutions

We consider a classical case of integrals containing an irrational integrand in the form of a square root of a quadratic polynomial. It is known that such “irrational integrals” can be expressed in terms of elementary functions by one of three of Euler’s substitutions. It is less well known that the Euler substitutions have a geometric interpretation. In the framework of this interpretation, one can see that the number 3 is not the most suitable. We show that it is natural to introduce a fourth Euler substitution. In his original treatise, Leonhard Euler used two substitutions which are sufficient to cover all cases.

The Roots of the Reliability Polynomials of Circular Consecutive-k-out-of-n:F Systems

The zeros of the reliability polynomials of circular consecutive-k-out-of-n:F systems are studied. We prove that, for any fixed k≥2, the set of the roots of all the reliability polynomials (for all n≥k) is unbounded in the complex plane. In the particular case k=2, we show that all the nonzero roots are real, distinct numbers and find the closure of the set of roots. For every n≥k, the expressions of the minimum root and the maximum root are given, both for circular as well as for linear systems.

Laplacian Spectrum of Linear Generalized Phenylene-Quadrilateral Networks and Its Applications

Topological indices have gained significant attention in the field of chemical graph theory. These indices offer quantitative measures that accurately represent the topology of molecular graphs, which are used to model chemical compounds. Generally, their physical properties are closely linked to the geometric structures of these compounds. In this paper, we introduce a new family of phenylene-quadrilateral networks that exhibit unique features. These studied topological structures can be seen as generalizations of the phenylenes. To analyze the generalized phenylene-quadrilateral networks, we propose a recursive method for calculating their Kirchhoff index and the number of spanning trees. This method is based on the relationship between the coefficients and roots of the characteristic polynomial.

Discriminants of fields generated by polynomials of given height

We obtain upper bounds for the number of monic irreducible polynomials over Z\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb{Z}$$\\end{document} of a fixed degree n and a growing height H for which the field generated by one of its roots has a given discriminant. We approach it via counting square-free parts of polynomial discriminants via two complementing approaches. In turn, this leads to a lower bound on the number of distinct discriminants of fields generated by roots of polynomials of degree n and height at most H. We also give an upper bound for the number of trinomials of bounded height with given square-free part of the discriminant, improving previous results of I. E. Shparlinski (2010).

On the Roots of (Signless) Laplacian Permanental Polynomials of Graphs

On the Roots of (Signless) Laplacian Permanental Polynomials of Graphs

Offset-Control Plane Coexisting Behaviors in Two-Memristor-Based Hopfield Neural Network

Memristor synapse with activated synaptic plasticity can be taken as an adaptive connection synaptic weight. To demonstrate its kinetic effects, this article presents an improved Hopfield neural network with two memristive self-connection synaptic weights. The two-memristor-based Hopfield neural network (TM-HNN) has a plane equilibrium set related to two memristor initial conditions and its stability distributions are analyzed by two non-zero roots of the eigenvalue polynomial. Afterwards, the parameter-related bifurcation behaviors are investigated using bifurcation plots and phase portraits. Emphatically, the kinetic effects of memristor synapses are demonstrated by taking the memristor initial conditions as two invariant measures. The theoretical and numerical results show that the TM-HNN can exhibit the wondrous offset-control plane coexisting behaviors and its plane coexisting attractors can be controlled by switching the two memristor initial conditions. Besides, a digital hardware device is developed and the offset-control plane coexisting attractors are experimentally reproduced to verify the numerical ones.

Ambient data-driven SSO online monitoring of type-3 wind turbine generator integrated power systems based on MMPF-KF method

Series compensation grids connected with type-3 wind turbine generator (WTG)-based wind farms have suffered numerous subsynchronous oscillation (SSO) events worldwide. For early alerting of SSO and effective development of protection and control strategies, it is critical to monitor and identify SSO accurately and quickly. Ambient data is continuously available, which is useful for online monitoring. This paper proposes an ambient data-driven SSO online monitoring method based on the Kalman filter (KF) combined with the multi-model partitioning filter (MMPF). The KF is utilized to fit the measured ambient data with an auto regressive (AR) model. Then, the damping factor (or damping ratio) and frequency in the SSO mode can be acquired by solving the roots of the characteristic polynomial corresponding to the AR model. Moreover, the MMPF is an effective model order selection method applied to the KF for better identification. The performance of the MMPF-KF method is demonstrated by simulations and real-time experiments. The results of case studies validate the effectiveness of the proposed method under various conditions.

An integral representation for quantum amplitudes

The central impediment to reducing multidimensional integrals of transition amplitudes to analytic form, or at least to a fewer number of integral dimensions, is the presence of magnitudes of coordinate vector differences (square roots of polynomials) in disjoint products of functions. Fourier transforms circumvent this by introducing a three-dimensional momentum integral for each of those products, followed in many cases by another set of integral representations to move all of the resulting denominators into a single quadratic form in one denominator whose square my be completed. Gaussian transforms introduce a one-dimensional integral for each such function while squaring the square roots of coordinate vector differences and moving them into a common exponential. Addition theorems may also be used for extracting the angular variables, and sometimes direct integration is even possible. Each method has its strengths and weaknesses. An integral representation is derived herein that stands as an alternative to these four approaches. A number of consequent integrals of Macdonald functions, hypergeometric functions, and Meijer G-functions with complicated arguments are given.

Spectral analysis of the linearized triple-quantum-dot shuttle in the adiabatic regime

In this paper we seek to perform a spectral analysis of the linearized Hamiltonian of the triple-quantum-dot shuttle in the adiabatic regime. This system consists of a linear array of three quantum dots where a central quantum dot oscillates between two quantum dots that remain fixed, the linearization of the Hamiltonian is through tunneling rates. For our analysis we visualize the problem from the point of view of the displaced oscillator picture, which allows us to build a convenient composite base to rewrite HˆTQDS and by approximating the evolution of the system to an adiabatic evolution it is possible to find the energies of the system through a simple analytic diagonalization process and a search for the roots of a cubic polynomial, as well as the corresponding eigenvectors. The Hamiltonian of the shuttle can be interpreted as a particular case of a more general Hamiltonian in a perturbative scheme, but the mathematical treatment does not proceed in the formalism of perturbation theory. The analytical results obtained were compared with the exact results from numerical diagonalization for different levels of the adiabatic regime, showing that the more strictly the adiabatic condition is fulfilled, the analytical results tend to be exact quickly. In addition, we were able to reproduce some isolated exact solutions from the analytical solution model developed by Kus et al. in 1986 and taken up by Peralta et al. in 2022 to apply it to the shuttle.

Roots of Polynomials and Umbilics of Surfaces

Roots of Polynomials and Umbilics of Surfaces

Metric theory of diophantine approximation and asymptotic estimates for the number of polynomials with given discriminants divisible by a large power of a prime number

Discriminants of polynomials characterize the distribution of roots of polynomials in the complex plane. In recent years, for integer polynomials, exact lower-bound estimates have been obtained for the number of polynomials of a given degree and height. The method of obtaining these estimates is based on Minkowski’s theorems in the geometry of numbers and the metric theory of Diophantine approximation. A new method is proposed and allows one to obtain upperbound estimates for the number of polynomials with bounded discriminants in Archimedean and non-Archimedean metrics. The method generalizes the ideas of H. Davenport, B. Volkman, and V. Sprindzuk that allowed them to obtain significant advances in solving Mahler’s problem.

On the Zeros of Exponential Polynomials

We consider the problem of deciding the existence of real roots of real-valued exponential polynomials with algebraic coefficients. Such functions arise as solutions of linear differential equations with real algebraic coefficients. We focus on two problems: theZero Problem, which asks whether an exponential polynomial has a real root, and theInfinite Zeros Problem, which asks whether such a function has infinitely many real roots. Our main result is that for differential equations of order at most 8 the Zero Problem is decidable, subject to Schanuel’s Conjecture, while the Infinite Zeros Problem is decidable unconditionally. We show moreover that a decision procedure for the Infinite Zeros Problem at order 9 would yield an algorithm for computing the Lagrange constant of any given real algebraic number to arbitrary precision, indicating that it will be very difficult to extend our decidability results to higher orders.