Extended spectral breakdown theory

Historically, work on multiple roots for the internal rate of return (PW = 0) equation focuses on tests for the root's uniqueness and interpreting the meaningfulness of computed roots. Instead, this article starts with problems that consist of P, A, and F cash flows. Then for all possible values of these cash flows, this article develops the boundary conditions for no, single, and multiple real roots of either positive or negative sign. The results of characterizing these roots are applied to two examples as a foundation for conclusions about those multiple root problems and hypotheses about other problems with multiple roots. The first example is an important class of resource development projects that may require significant expenditures for environmental remediation at the conclusion of operation. The second example of buying versus leasing a home is often used to demonstrate the value of economic analysis to students.

Asymptotic distributions of the latent roots of the covariance matrix with multiple population roots

Prevalence and morphology of multiple roots, root canals and C-shaped canals in mandibular premolars from cone-beam computed tomography images in a Thai population

RANK ADDITIVITY AND MATRIX POLYNOMIALS

A full integral of a polynomial is defined as its integral with the property that any multiple root of the polynomial is a root of this integral. The paper investigates relationships between the existence of a full integral and the form of a polynomial. In particular, it is proved that a full integral exists if the polynomial has no more than one multiple root. On the other hand, if the number of multiple roots of a polynomial strictly exceeds the number of its simple roots increased by one, then the polynomial has no full integral. Bibliography: 7 titles.

This paper presents a general analysis of all the quartic equations with real coefficients and multiple roots; this analysis revealed some unknown formulae to solve each kind of these equations and some precisions about the relation between these ones and the Resolvent Cubic; for example, it is well-known that any quartic equation has multiple roots whenever its Resolvent Cubic also has multiple roots; however, this analysis reveals that any non-biquadratic quartic equation and its Resolvent Cubic always have the same number of multiple roots; additionally, the four roots of any quartic equation with multiple roots are real whenever some specific forms of its Resolvent Cubic have three non-negative real roots. This analysis also proves that any method to solve third-degree equations is unnecessary to solve quartic equations with multiple roots, despite the existence of the Resolvent Cubic; finally, here is developed a generalized variation of the Ferrari Method and the Descartes Method, which help to avoid complex arithmetic operations during the resolution of any quartic equation with real coefficients, even though this equation has non-real roots; and a new, more simplified form of the discriminant of the quartic equations is also featured here.

The minimum root separation of a complex polynomial A is defined as the minimum of the distances between distinct roots of A. For polynomials with Gaussian integer coefficients and no multiple roots, three lower bounds are derived for the root separation.In each case, the bound is a function of the degree n of A and the sum d of the absolute values of the coefficients of A. The notion of a seminorm for a commutative ring is defined, and it is shown how any seminorm can be extended to polynomial rings and matrix rings, obtaining a very general analogue of Hadamard's determinant theorem. Introduction.Let A(x) be a polynomial of degree n > 0 with complex coefficients a and complex roots a" so thatWe define sepiA), the minimum root separation of A, by(2) sepiA) = min \a -ak\, a i*ak with the convention that sep(A) = in case A has only one distinct root.The computing times required by known algorithms for isolating the zeros of A depend inversely on sep(A).Hence, we are interested in easily computable functions f(a0, , a") of the coefficients such thatHeindel [3], in analyzing the computing time of an algorithm based on Sturm's theorem for isolating the real zeros of any polynomial with integer coefficients, used a weak lower bound for sep(/l) due to Collins.Pinkert [9], presents an analogous algorithm for isolating all zeros, real and complex, of any polynomial with Gaussian integer coefficients.His algorithm is based on Sturm's theorem and the Routh-Hurwitz theorem and uses a stronger lower bound for sepiA), obtained more recently by Collins.Horowitz, using another simpler approach, has recently obtained a third lower bound, intermediate in strength, but just slightly weaker than the stronger bound of Collins.In the following, these three bounds are all derived, with the hope of stimulating further research on the problem.

Durand-Kerner's method for simultaneous rootfinding of a polynomial is locally second order convergent if all the zeros are simple. If this condition is violated numerical experiences still show linear convergence. For this case of multiple roots, Fraigniaud [4] proves that the means of clustering approximants for a multiple root is a better approximant for the zero and called this Quadratic-Like-Convergence of the Means. This note gives a new proof and a refinement of this property. The proof is based on the related Grau's method for simultaneous factoring of a polynomial. A similar property of some coefficients of the third order method due to Borsch-Supan, Maehly, Ehrlich, Aberth and others is proved.

A superlinear procedure for finding a multiple root is presented. In it the secant method is applied to the given function divided by a divided difference whose increment shrinks toward zero as the root is approached. Two function evaluations per step are required, but no derivatives need be calculated.

We have a good number of eighth-order iterative methods for simple zeros of nonlinear equations in the available literature. But, unfortunately, we don’t have a single iterative method of eighth-order for multiple zeros with known or unknown multiplicity. Some scholars from the worldwide have tried to present optimal or non-optimal multipoint eighth-order iteration functions. But, unfortunately, none of them get success in this direction and attained maximum sixth-order convergence in the case of multiple zeros with known multiplicity m. Motivated and inspired by this fact, we propose an optimal scheme with eighth-order convergence based on weight function approach. In addition, an extensive convergence study is discussed in order to demonstrate the eighth-order convergence of the present scheme. Moreover, we also show the applicability of our scheme on some real life and academic problems. These problems illustrate that our methods are more efficient among the available multiple root finding techniques.

In a recent paper, (Math Notes 91(3–4): 508–516, 2012) Zarhin proved that each member of a naturally defined family of linear maps \({\mathbb {C}}^n \rightarrow {\mathbb {C}}^n\) has co-rank one. We present a direct proof of Zarhin’s result about complex polynomials with distinct roots; it is rather similar to that of Appendix by Vik.S. Kulikov to Zarhin’s paper but we give explicit constants. We also discuss the case of a polynomial with multiple roots.

The paper deals with structural-parametric models for describing dynamic processes of technical systems of an intelligent building. The task of searching for the transfer function of the synthesized elements and devices of its information-measuring and control systems based on the Mason method is formalized. The components of the transfer function are presented in the form of characteristic polynomials in the structural scheme of the energy-information model of the circuit. The results of a comparative analysis of search methods for multiple real and complex conjugate polynomial roots are presented. To organize their search, an iterative method of unconditional optimization of Fletcher-Reeves was chosen. This conjugate gradient method allows to solve the problem of numerical optimization in a finite number of steps and shows the best convergence in comparison with the methods of the fastest descent, with the same order of difficulty of performing the steps of the algorithm.

This paper is concerned with a classical yet still mystifying problem regarding multiple roots of the angles-only initial orbit determination (IOD) polynomial equations of Lagrange, Laplace, and Gauss of the form: f(x) = x8+ax6+bx3+c=0 where a,c 0 has been extensively treated in the celestial mechanics literature. However, the literature on applied astrodynamics has not treated this multiple-root issue in detail, and not many specific numerical examples with multiple roots are available in the literature. In this paper, a very simple method of determining the correct root from two or three non-spurious roots is presented, which doesn’t utilize any a priori knowledge and/or additional observations of the object. The proposed method exploits a simple approximate polynomial equation of the form: g(x) = x8+ax6=0. An approximate polynomial equation, either g(x) = x8+c=0 or g(x) = x8+ax6=x6(x2+a) = 0, can also be used for quickly estimating an initial guess of the correct root.

The quantum-field renormalization group and operator expansion are used to investigate the infrared asymptotic behavior of the velocity correlation function in the theory of fully developed turbulence. The scaling dimensions of all composite operators constructed from the velocity field and its time derivatives are calculated, and their contributions to the operator expansion are determined. It is shown that the asymptotic behavior of the equal-time correlation function is determined by Galilean-invariant composite operators. The corrections to the Kolmogorov spectrum associated with the operators of canonical dimension 6 are found. The consequences of Galilean invariance for the renormalized composite operators are considered.

We extend the power law phase screen theory for ionospheric scintillation to account for the case where the refractive index irregularities follow a two‐component inverse power law spectrum. The two‐component model includes, as special cases, an unmodified power law and a modified power law with spectral break that may assume the role of an outer scale, intermediate break scale, or inner scale. As such, it provides a framework for investigating the effects of a spectral break on the scintillation statistics. Using this spectral model, we solve the fourth moment equation governing intensity variations following propagation through two‐dimensional field‐aligned irregularities in the ionosphere. A specific normalization is invoked that exploits self‐similar properties of the structure to achieve a universal scaling, such that different combinations of perturbation strength, propagation distance, and frequency produce the same results. The numerical algorithm is validated using new theoretical predictions for the behavior of the scintillation index and intensity correlation length under strong scatter conditions. A series of numerical experiments are conducted to investigate the morphologies of the intensity spectrum, scintillation index, and intensity correlation length as functions of the spectral indices and strength of scatter; retrieve phase screen parameters from intensity scintillation observations; explore the relative contributions to the scintillation due to large‐ and small‐scale ionospheric structures; and quantify the conditions under which a general spectral break will influence the scintillation statistics.