Formulae For Roots Research Articles

Mode Theory and Propagation of ELF Radio Wave in a Multilayered Oceanic Lithosphere Waveguide

Extremely low-frequency (ELF: below 30 Hz) wave penetration theory in the layered oceanic lithosphere is essential for studying submarine communications. In this article, we investigate the mode theory of ELF wave propagation in a layered oceanic lithosphere waveguide. Based on the real situation of the oceanic lithosphere approximated by the stratified prototype, in which a low-conductivity layer lies between two overlying and underlying high-conductivity regions, a theoretically sound physical model of oceanic lithosphere waveguide is built. The modal equations for transverse magnetic (TM) and transverse electric (TE) polarized guided modes are delivered using surface impedance boundary conditions. By evaluating the reflection coefficients of the upper and lower waveguide boundaries, formulas for the modal equation's roots are derived. Analytical expressions for propagable mode parameters, such as the phase velocity, attenuation rate, excitation factor, and the field strength for different components, are obtained, and the cutoff frequency for high-order modes is given. Computations show that the field component curves exhibit typical peaks and nulls associated with multiple-mode propagation in the waveguide region, with each mode traveling at different phase velocities and attenuation rates. Moreover, it is found that all the high-order modes are evanescent below the cutoff frequency.

Resolvent degree, Hilbert’s 13th Problem and geometry

We develop the theory of resolvent degree, introduced by Brauer [Brau2] in order to study the complexity of formulas for roots of polynomials and to give a precise formulation of Hilbert’s 13th Problem. We extend the context of this theory to enumerative problems in algebraic geometry, and consider it as an intrinsic invariant of a nite group. As one application of this point of view, we prove that Hilbert’s 13th Problem, and his Sextic and Octic Conjectures, are equivalent to various enumerative geometry problems, for example problems of nding lines on a smooth cubic surface or bitangents on a smooth planar quartic.

Skew hook formula for $d$-complete posets via equivariant $K$-theory

Peterson and Proctor obtained a formula which expresses the multivariate generating function for P-partitions on a d-complete poset P as a product in terms of hooks in P. In this paper, we give a skew generalization of Peterson–Proctor’s hook formula, i.e. a formula for the generating function of (P∖F)-partitions for a d-complete poset P and an order filter F of P, by using the notion of excited diagrams. Our proof uses the Billey-type formula and the Chevalley-type formula in the equivariant K-theory of Kac–Moody partial flag varieties. This generalization provides an alternate proof of Peterson–Proctor’s hook formula. As the equivariant cohomology version, we derive a skew generalization of a combinatorial reformulation of Nakada’s colored hook formula for roots.

Rational expressions for multiple roots of algebraic equations

The general polynomial with variable coefficients is considered. In terms of the resultants of this polynomials and its derivatives simple rational expressions in the coefficients of the polynomial are found for its multiple zeros. Similar results are extended to systems of polynomial equations with unknowns. Justifications of the formulae for multiple roots thus obtained are based on the properties of the logarithmic Gauss map of the discriminant variety of a system of equations and on a linearization procedure for the system. The resulting formulae are of interest not only for theoretical aspects of the algebra of polynomials, but also for numerical mathematics and various areas of applied mathematics connected with finding critical points of polynomial maps. Bibliography: 20 titles.

On the global convergence of Schröder’s iterative formulae for real roots of algebraic equations

Schröder’s formulae of the first ( S1 ) and second ( S2 ) kind of order m of convergence are generalizations of Newton’s ( m = 2 ) and Halley’s ( S2 , m = 3 ) iterative formulae for finding zeros of functions. Davies and Dawson show that for entire functions with only real zeros, Halley’s formula converges globally and monotonically to their zeros, independently of the initial value on the real line. We show that the S2 formulae of odd order ≥ 5 enjoy the same convergence feature for polynomials with only real zeros. Numerical examples illustrate this. We illustrate no monotonic convergence of the S1 formulae and of the S2 formulae of even order.

On the matrix pth root functions and generalized Fibonacci sequences

This study is devoted to the polynomial representation of the matrix pth root functions. The Fibonacci-Hörner decomposition of the matrix powers and some techniques arisen from properties of generalized Fibonacci sequences, notably the Binet formula, serves as a triggering factor to provide explicit formulas for the matrix pth roots. Special cases and illustrative numerical examples are given.

Formulas for roots of L-functions of the airy exponential sums

We study the L-function of the Airy exponential sum associated to the polynomial f(X)=Xd+λX using Dwork's dual theory on cohomology level. We get explicit formulas for all the roots of the L-function in the ordinary case in terms of p-adic GKZ-hypergeometric functions. Our method can also be used to give explicit formulas for all the roots of the L-function of the exponential sum associated to f(X)=Xd+λd−1Xd−1+⋯+λ0.

Three kinds of particles on a single rationally parameterized world line

We consider the light cone (`retardation') equation (LCE) of an inertially moving observer and a single worldline parameterized by arbitrary rational functions. Then a set of apparent copies, R- or C-particles, defined by the (real or complex conjugate) roots of the LCE will be detected by the observer. For any rational worldline the collective R-C dynamics is manifestly Lorentz-invariant and conservative; the latter property follows directly from the structure of Vieta formulas for the LCE roots. In particular, two Lorentz invariants, the square of total 4-momentum and total rest mass, are distinct and both integer-valued. Asymptotically, at large values of the observer's proper time, one distinguishes three types of the LCE roots and associated R-C particles, with specific locations and evolutions; each of three kinds of particles can assemble into compact large groups - clusters. Throughout the paper, we make no use of differential equations of motion, field equations, etc.: the collective R-C dynamics is purely algebraic

Algorithms for quaternion polynomial root-finding

In 1941 Niven pioneered root-finding for a quaternion polynomial P(x), proving the fundamental theorem of algebra (FTA) and proposing an algorithm, practical if the norm and trace of a solution are known. We present novel results on theory, algorithms and applications of quaternion root-finding. Firstly, we give a new proof of the FTA resulting in explicit formulas for both exact and approximate quaternion roots of P(x) in terms of exact and approximate complex roots of the real polynomial F(x)=P(x)P¯(x), where P¯(x) is the conjugate polynomial. In particular, if |F(c)|≤ϵ, then for a computable quaternion conjugate q of c, |P(q)|≤ϵ. Consequences of these include relevance of root-finding methods for complex polynomials, computation of bounds on zeros, and algebraic solution of special quaternion equations. Secondly, working directly in the quaternion space, we develop Newton and Halley methods and analyze their local behavior. Surprisingly, even for a quadratic quaternion polynomial Newton’s method may not converge locally. Finally, we derive an analogue of the Bernoulli method in the quaternion space for computing the dominant root in certain cases. This requires the development of an independent theory for the solution of quaternion homogeneous linear recurrence relations. These results also lay a foundation for quaternion polynomiography.

Classifying zeros of two-sided quaternionic polynomials and computing zeros of two-sided polynomials with complex coefficients

We improve the method of Janovska and Opfer for computing the zeros on the surface of a given sphere for a quaternionic two-sided polynomial. We classify the zeros of quaternionic two-sided polynomials into three types — isolated, spherical and circular — and characterize each type. We provide a method to find all quaternion zeros for two-sided polynomials with complex coefficients. We also establish standard formulae for roots of a quadratic two-sided polynomial with complex coefficients, which yields a simpler and more efficient algorithm to produce all zeros in the quadratic case.

A Review on Development Course of Formula for Roots of Equation

This essay starting with the origin of thought about equation, discusses the development course of radical formula for equation from simple to higher degree, introduces the derivation method of radical formula for cubic equation and quartic equation, and compares two methods of radical formula for quartic equation(Ferrali method and Descartes method). There is no general formula of root for equation of 5-th order or higher degree. Eventually, this essay expounds the development process of equation’s contributions to algebra field.

An alternative presentation of the method of variation of parameters for higher-order equations

An alternative approach to Lagrange’s method of variation of parameters is presented. Explicit formulas for solutions of arbitrary initial value problems for linear equations of the first, second and third order are provided. These formulas are as simple for practical using as the formula for roots of quadratic equations.

Some fourth-order nonlinear solvers with closed formulae for multiple roots

In this paper, we present six new fourth-order methods with closed formulae for finding multiple roots of nonlinear equations. The first four of them require one-function and three-derivative evaluation per iteration. The last two require one-function and two-derivative evaluation per iteration. Several numerical examples are given to show the performance of the presented methods compared with some known methods.

Variation of the unit root along the Dwork family of Calabi–Yau varieties

We study the variation of the unit roots of members of the Dwork families of Calabi–Yau varieties over a finite field by the method of Dwork–Katz and also from the point of view of formal group laws. A p-adic analytic formula for the unit roots away from the Hasse locus is obtained.

A non-singular one-loop wave function of the universe from a new eigenvalue asymptotics in quantum gravity

Recent work on euclidean quantum gravity on the four-ball has proved regularity at the origin of the generalized ζ-function built from eigenvalues for metric and ghost modes, when diffeomorphism-invariant boundary conditions are imposed in the de Donder gauge. The hardest part of the analysis involves one of the four sectors for scalar-type perturbations, the eigenvalues of which are obtained by squaring up roots of a linear combination of Bessel functions of integer adjacent orders, with a coefficient of linear combination depending on the unknown roots. This paper obtains, first, approximate analytic formulae for such roots for all values of the order of Bessel functions. For this purpose, both the descending series for Bessel functions and their uniform asymptotic expansion at large order are used. The resulting generalized ζ-function is also built, and another check of regularity at the origin is obtained. For the first time in the literature on quantum gravity on manifolds with boundary, a vanishing one-loop wave function of the Universe is found in the limit of small three-geometry, which suggests a quantum avoidance of the cosmological singularity driven by full diffeomorphism invariance of the boundary-value problem for one-loop quantum theory.

Power series expansion of the roots of a secular equation containing symbolic elements: Computer algebra and Moseley’s law

We use computer algebra to expand the Pekeris secular determinant for two-electron atoms symbolically, to produce an explicit polynomial in the energy parameter ε, with coefficients that are polynomials in the nuclear charge Z. Repeated differentiation of the polynomial, followed by a simple transformation, gives a series for ε in decreasing powers of Z. The leading term is linear, consistent with well-known behavior that corresponds to the approximate quadratic dependence of ionization potential on atomic number (Moseley’s law). Evaluating the 12-term series for individual Z gives the roots to a precision of 10 or more digits for Z⩾2. This suggests the use of similar tactics to construct formulas for roots vs atomic, molecular, and variational parameters in other eigenvalue problems, in accordance with the general objectives of gradient theory. Matrix elements can be represented by symbols in the secular determinants, enabling the use of analytical expressions for the molecular integrals in the differentiation of the explicit polynomials. The mathematical and computational techniques include modular arithmetic to handle matrix and polynomial operations, and unrestricted precision arithmetic to overcome severe digital erosion. These are likely to find many further applications in computational chemistry.

Eliminating multiple root problems in estimation (with comments by John J. Hanfelt, C. C. Heyde and Bing Li, and a rejoinder by the authors)

Estimating functions, such as the score or quasiscore,can have more than one root. In many of these cases, theory tells us that there is a unique consistent root of the estimating function. However, in practice, there may be considerable doubt as to which root is appropriate as a parameter estimate. The problem is of practical importance to data analysts and theoretically challenging as well. In this paper, we review the literature on this problem. A variety of examples are provided to illustrate the diversity of situations in which multiple roots can arise. Some methods are suggested to investigate the possibility of multiple roots, search for all roots and compute the distributions of the roots. Various approaches are discussed for selecting among the roots. These methods include (1) iterating from consistent estimators, (2) examining the asymptotics when explicit formulas for roots are available, (3) testing the consistency of each root, (4) selecting by bootstrapping and (5) using information-theoretic methods for certain parametric models. As an alternative approach to the problem, we consider how an estimating function can be modified to reduce the number of roots. Finally, we survey some techniques of artificial likelihoods for semiparametric models and discuss their relationship to the multiple root problem.

Analytical Relations Connecting Linear Interfaces and Volume Fractions in Rectangular Grids

The computational-geometric problems arising when a linear interface cuts a cube are considered. They are of interest in particular for the calculation of volume fractions or interface positions in three-dimensional interface calculations in the Volume of Fluid (VOF) methods. Typically, the normal vector is known. One then wants to compute the volume fraction knowing the interface position, or conversely the interface position knowing the volume fraction. Explicit expressions of general use are given, and the algorithms used to search for solutions are described in detail. Explicit formulas for cubic roots are found to be less than two thirds as time consuming as Newton–Raphson iterations.

Are There Field Dependent Formulas For the Roots to Polynomials?

Are There Field Dependent Formulas For the Roots to Polynomials?

Apéry numbers, Jacobi sums, and special values of generalized p-adic hypergeometric functions

We present some summation formulae for some special values of ratios of generalized p-adic hypergeometric functions in terms of the p-adic gamma function. By well-known methods such formulae yield expressions for roots of congruence zeta-functions in terms of Jacobi sums in the nonsingular case. We also express various formal-group congruences, including those involving Apéry numbers, in terms of p-adic hypergeometric functions at singular and nonsingular points of the associated differential equation. These congruences yield p-adic analytic formulae for unit roots of certain Hecke polynomials.