Real Roots Research Articles

General approach on the best fitted linear operator and basis function for homotopy methods and application to strongly nonlinear oscillators

In this study, we propose a universal mathematical guideline for the selection of auxiliary linear operator L(u) and the basis function u0(x), which are the principal parts of Homotopy methods: Homotopy Perturbation Method (HPM) and Homotopy Analysis Method (HAM). We assume the coefficients of derivatives involved in L(u) as a function of auxiliary roots of L(u)=0. Based on the residual error minimization we compute unknown roots and thereby obtain the optimal linear operator at an order of approximation. Also, an effort is made to explore the role of the number of unknowns embedded into L(u) (for different choices of real roots of L(u)=0) on the rate of converges of the solution and CPU uses. Hence the best fitted linear-operator and corresponding basis function are proposed for a particular problem. Also, we propose to discretize the exact square residual using Simpson’s 1/3 algorithms, and compare its accessibility in comparison to other existing residual calculation methods. Initially, we applied our algorithms to three IVPs (Logistic, Stiff system and Riccati system) and three singular BVPs, and then finally to a strongly nonlinear oscillator (cubic-quintic Duffing equation). We compare our technique’s accuracy, efficiency and simplicity with other contemporary analytical and numerical methods. It is demonstrated that our optimal linear operator, which is updated at every order of approximation, is much more efficient, important (than the artificial parameters and functions of developed HAM), and self-sufficient for the convergence of series solutions. Our approach is more effective, straightforward and easy to use when applied to many nonlinear problems arising in mechanics and engineering. It is seen for the initial value problems that our technique provides the globally convergent series solution, while the solutions by HPM and optimal HAM are locally convergent. This novel development will make homotopy methods more powerful.

‘Incomplete’ Pál type interpolation problems on zeros of polynomials with complex coefficients

We termed the Pál type interpolation problem as PTIP. Here we studied the regularity of (0,1)-PTIP and (0,2)-PTIP, where we omitted a real or complex node from a set of zeros of polynomials with complex coefficients.

On Descartes' rule of signs for hyperbolic polynomials

We consider univariate real polynomials with all real roots and with two sign changes in the sequence of their coefficients which are all non-vanishing. Assume that one of the changes is between the linear and the constant term. By Descartes' rule of signs, such degree \(d\) polynomials have 2 positive and \(d-2\) negative roots. We consider the possible sequences of the moduli of their roots on the real positive half-axis. When these moduli are distinct, we give the exhaustive answer to the question which positions can the moduli of the two positive roots occupy.

How many real zeros does a random Dirichlet series have?

How many real zeros does a random Dirichlet series have?

NEW CONJECTURES FOR THE ENTIRE FUNCTIONS ASSOCIATED WITH FRACTIONAL CALCULUS

In this paper, we address the entire Fourier sine and cosine integrals related to the Mittag-Leffler function. We guess that the entire functions have the real zeros in the entire complex plane. They can be connected with the well-known conjectures in analytic number theory. They are considered as the special solutions for the time-fractional diffusion equation within the Caputo fractional derivative.

Distribution of deer stones and migration of the Scythians

In the article, in addition to the classification and chronology of deer stones created by the ancient nomadic Scythians (Oghuz) in their ancestral home of Central Asia, the author made an attempt to present the history of the ancient nomads in a new way through the prism of their cultural values. Today, archeology has grown stronger, discovered many heritage sites and is now studying their secrets from new angles. One of these secrets are deer stones, which are already familiar to people, but the author showed that these deer stones are the real roots and material value of the ancient history of the entire Turkic world. In addition, the author shows the chronology of the deer stones by considering the location of their origin, as well as the migration of the ancient Oghuzs following the migration of the ancient Sumerians and Hellenes from Central Asia to Europe.

Frequency-Variant Double-Zero Single-Pole Reactive Coupling Networks for Coupled-Resonator Microwave Bandpass Filters

In this work, a family of frequency-variant reactive coupling (FVRCAFVC) networks is introduced and discussed as new building blocks for the synthesis of coupled-resonator bandpass filters with real or complex transmission zeros (TZs). The FVRCAFVC is a type of nonideal frequency-dependent inverter that has nonzero elements on the diagonal of the impedance matrix, along with a nonlinear frequency-variation profile of its transimpedance parameter. The distinctive feature of these new FVRCsAFVC is that they can be modeled with a simple bridged-T network consisting of four reactive elements. Such FVRCsAFVCs can introduce one pole and up to two TZs—so that they are referred to as double-zero single-pole (DZSP) networks. DZSP networks allow the pole to be located above, below, or in-between the TZs. Depending on the choice of the elements in the bridged-T circuit, the position of the TZs and the pole can be controlled independently. The coupling matrix for filters with DZSP networks can be found by solving an inverse-structured nonlinear eigenvalue problem (ISNEVP). Examples of possible implementations of DZSP couplingAFVC networks are provided for both lumped and quasi-lumped-element circuits, as well as for transmission-line-based and waveguide technologies. The application of DZSP coupling networks to the design of microwave bandpass filters with generalized Chebyshev-type characteristics is illustrated by means of the synthesis of three microstrips and two waveguide coupled-resonator bandpass filters. The waveguide filters are verified through electromagnetic (EM) simulations and the microstrip filters are manufactured and characterized for proof-of-concept demonstration purposes, showing a fairly close agreement between EM simulations and measurements.

NUMERICAL AND ANALYTICAL INVESTIGATION OF THE J-INTEGRAL VALUE IN A DOUBLE CANTILEVER BEAM SAMPLE WITH AN EXTREMELY THIN ELASTIC-PLASTIC ADHESION LAYER

The loading of a double cantilever beam sample, which is a composition of cantilevers and an adhesive layer, is considered, which leads to deformation of the layer by normal rupture. The basis of the mathematical model describing the equilibrium of the composite is the variational equilibrium equation connecting the stresses averaged over the thickness of the layer and the stress-strain state of the cantilever by a linear parameter. The behavior of the cantilever is considered within the framework of a linearly elastic body, the adhesive layer is considered elastic-plastic without hardening. The composite is considered in a state of plane deformation. Assuming the fulfillment of the condition of complete plasticity of the equality of two principal stresses acting orthogonally to separation and elastic volumetric deformation, the constitutive relations of the behavior of the layer at the stage of elastoplastic deformation are obtained. Using the distribution of the displacement field according to the theory of Mindlin-Reisner plates, a system of ordinary differential equations, conjugation conditions for solutions, and boundary conditions are obtained from the variational formulation of the problem. It is shown that, depending on the ratio of the linear parameter and the height of the console, the characteristic equation of the corresponding homogeneous system of differential equations has both real and complex roots. The corresponding general solutions of the system are found. Based on the consideration of one region of elastoplastic deformations, an iterative procedure for finding the length of the plastic zone is proposed, taking into account the satisfaction of boundary conditions. From the solution of the problem for a given value of the linear parameter and a critical external load, the stress-strain state of the layer is found, on the basis of which the J-integral is calculated. It is shown that as the linear parameter tends to zero, the J-integral asymptotically tends to the value found in the framework of the linearly elastic behavior of the layer material. For a certain range of values of the linear parameter, the obtained solution was compared with the finite element solution of the 2D problem.

The number of real zeros of elliptic polynomials

The number of real zeros of elliptic polynomials

麦克劳林关于方程实根最大绝对值下界的法则

麦克劳林关于方程实根最大绝对值下界的法则

On Real Roots of Systems of Trancendental Equations with Real Coefficients

On Real Roots of Systems of Trancendental Equations with Real Coefficients

On The Algebraic Classification of the 4-Cyclic Refined Neutrosophic Real Roots of Unity Group

This paper is dedicated to finding all 4-cyclic refined neutrosophic real solutions of the equation 𝑋𝑛=1 which are called 4-cyclic refined real roots of unity. Also, we classify the algebraic group of these solutions as a direct product of some familiar finite cyclic groups. On the other hand, we illustrate many examples to clarify the validity of our work.

On The 3-Cyclic Refined Neutrosophic Real Roots of Unity and Their Algebraic Classification

The objective of this paper is to find all formulas that describe the 3-cyclic refined neutrosophic real solutions of the equation 𝑋𝑛=1 which are called 3-cyclic refined real roots of unity. Also, we classify the algebraic group represented by these solutions as a direct product of some familiar finite abelian groups. On the other hand, we illustrate many examples to clarify the validity of our work.

On accurate asymptotic approximations of roots for polynomial equations containing a small, but fixed parameter

<p>In this paper, polynomial equations with real coefficients and in one variable were considered which contained a small, positive but specified and fixed parameter $ \varepsilon_0 \neq 0 $. By using the classical asymptotic method, roots of the polynomial equations have been constructed in the literature, which were proved to be valid for sufficiently small $ \varepsilon $-values (or equivalently for $ \varepsilon \to 0 $). In this paper, it was assumed that for some or all roots of a polynomial equation, the first few terms in a Taylor or Laurent series in a small parameter depending on $ \varepsilon $ exist and can be constructed. We also assumed that at least two approximations $ x_1(\varepsilon) $ and $ x_2(\varepsilon) $ for the real roots exist and can be constructed. For a complex root, we assumed that at least two real approximations $ a_1(\varepsilon) $ and $ a_2(\varepsilon) $ for the real part of this root, and that at least two real approximations $ b_1(\varepsilon) $ and $ b_2(\varepsilon) $ for the imaginary part of this root, exist and can be constructed. Usually it was not clear whether for $ \varepsilon = \varepsilon_0 $ the approximations were valid or not. It was shown in this paper how the classical asymptotic method in combination with the bisection method could be used to prove how accurate the constructed approximations of the roots were for a given interval in $ \varepsilon $ (usually including the specified and fixed value $ \varepsilon_0 \neq 0 $). The method was illustrated by studying a polynomial equation of degree five with a small but fixed parameter $ \varepsilon_0 = 0.1 $. It was shown how (absolute and relative) error estimates for the real and imaginary parts of the roots could be obtained for all values of the small parameter in the interval $ (0, \varepsilon_0] $.</p>

ON THE BEHAVIORS OF SOLUTIONS IN LINEAR NONHOMOGENEOUS DELAY DIFFERENTIAL EQUATIONS WITH PERIODIC COEFFICIENTS

This paper deals with the behaviors of solutions for linear nonhomogeneous delay differential equations. In this study, a periodic solution, an asymptotic result and a useful exponential estimate of the solutions are established. Our results are obtained by the use of real roots of the corresponding characteristic equation.

The hybrid method for finding the roots of a polynomial over finite fields based on affine expansion

This paper proposes a hybrid method to find the roots of a polynomial in a finite field based on combining the polynomial decomposition into the sum of multiples of the affine polynomial with the analytical method to find the roots of low-order polynomials. The proposed method allows a significant reduction in complexity and is applicable in the design of low-latency BCH and Reed-Solomon decoders.

On real zeros of the Hurwitz zeta function

In this paper, we present results on the uniqueness of the real zeros of the Hurwitz zeta function in given intervals. The uniqueness in question, if the zeros exist, has already been proved for the intervals (0,1) and (−N,−N+1) for N≥5 by Endo-Suzuki and Matsusaka, respectively. We prove the uniqueness of the real zeros in the remaining intervals by examining the behavior of certain associated polynomials.

Construction of Polynomial Eigenfunctions of a Second-Order Linear Differential Equation

A system of third-order recurrence relations for the coefficients of polynomial eigenfunctions (PEFs) of a differential equation is solved. A recurrence relation for three consecutive PEFs and a formula for differentiating PEFs are obtained. We consider differential equations one of which generalizes the Hermite and Laguerre differential equations and the other is a generalization of the Jacobi differential equation. For these equations, we construct functions bringing them to self-adjoint form and find conditions under which these functions become weight functions. Examples are given where the PEFs for nonweight functions do not have real zeros.

Numerical Solution of Fractional Order Integro-Differential Equations via Müntz Orthogonal Functions

In this paper, we derive a spectral collocation method for solving fractional-order integro-differential equations by using a kind of Müntz orthogonal functions that are defined on 0,1 and have simple and real roots in this interval. To this end, we first construct the operator of Riemann–Liouville fractional integral corresponding to this kind of Müntz functions. Then, using the Gauss–Legendre quadrature rule and by employing the roots of Müntz functions as the collocation points, we arrive at a system of algebraic equations. By solving this system, an approximate solution for the fractional-order integro-differential equation is obtained. We also construct an upper bound for the truncation error of Müntz orthogonal functions, and we analyze the error of the proposed collocation method. Numerical examples are included to demonstrate the validity and accuracy of the method.

Numerical radius and spectral radius inequalities with an estimation for roots of a polynomial

Numerical radius and spectral radius inequalities with an estimation for roots of a polynomial