Real Roots Research Articles

A New Technique for Finding Real Roots of Non-Linear Equations Using Arithmetic Mean Formula

A New Technique for Finding Real Roots of Non-Linear Equations Using Arithmetic Mean Formula

A Complete Review of the General Quartic Equation with Real Coefficients and Multiple Roots

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.

Differential Equation for Turbulence Power Losses and Energy Spectra Based on Consolidated Empirical Results

A second order differential equation for the energy dissipation rate of turbulence is presented. The derivation procedure is explained. The obtained governing equation is a Euler equation, which integration naturally conduces to power laws for the energy dissipation rate as a function of the wavenumber, a result that is extended to the energy spectrum of turbulence. Power laws are obtained for the cases of two equal and two different real roots. For the case of two conjugate complex roots, the solution is a sum of sine and cosine functions of the normal logarithm of the wavenumber. The differential equation accrues from a more basic equation obtained through thermodynamic-type steps that joint part of already consolidated empirical and semi-empirical information on turbulence existing in the literature, and is formally analogue to the Thermodynamics equation of thermal radiation. It is also shown that parameters of turbulence like length and velocity scales may be related to this formulation.

On the distribution of real roots of almost-periodical polynomials

On the distribution of real roots of almost-periodical polynomials

Higher-Order Lambert Problem Solution Based on Differential Algebra

Lambert’s problem involves solving the orbits connecting two position vectors with a given flight time. When introducing variations to the input of this problem (that is, terminal positions and the flight time), the output terminal velocities vary in response. The higher-order Lambert problem pursues a higher-order Taylor approximation of the output with respect to the input. Instead of finding a real number root, Householder methods are adapted to find a Taylor series solution of the transfer-time equation. By using explicit derivatives of the transfer-time equation, the newly implemented Householder methods converge faster than the general partial inversion method. In applications such as pork-chop plots and orbital admittance maps that require solving a large number of Lambert’s problems, a higher-order Lambert solution can reduce the running time by more than 80% when compared to solving the original Lambert problems one by one. Furthermore, the investigation of the parameter space of Lambert’s problem yields useful insights to the selection of higher-order expansion points.

Root Finding Methods Through GUI in Spreadsheets

Root finding Methods like the Bisection Method, Newton Raphson Method, Secant Method, and False Position Method have been revisited through a new approach. EACH METHOD’S user-friendly GUI computer programs have been developed on an Excel spreadsheet. A root locator graph is introduced in the spreadsheets, which helps identify the initial guess(es) required to calculate root using these methods. All real roots are now easily calculable by any of the four methods. The detailed steps to calculate roots are also shown on the spreadsheet. The spreadsheet can be made ready for the next calculation with a single click. We have developed a technique for Excel that helps accept polynomial in variable ‘x’ without directly mentioning cell location.

Preservation of classes of entire functions defined in terms of growth restrictions along the real axis under perturbations of their zero sets

Four special subsets of the Schwartz algebra are defined (this algebra consists of all entire functions of exponential type and of polynomial growth on the real axis). Perturbations of the zero sets for functions belonging to each of these subsets are studied. It is shown that the boundedness of the real part of the perturbing sequence is a sufficient and, generally speaking, unimprovable condition for preservation the subset from which the function in question is taken. An application of these results to spectral synthesis problems for differentiation-invariant subspaces of the Schwartz class on an interval of the real line is considered.

The set of zeros of the Riemann zeta function as the point spectrum of an operator

A possible way of proving the Riemann hypothesis consists of constructing a selfadjoint operartor whose spectrum coincides with the set { z : | Im ⁡ z | > 1 2 , ζ ( 1 2 − i z ) = 0 } \{z\,: \, |\operatorname {Im}z|>\frac 12, \ \zeta \big (\frac {1}{2}-iz\big )=0\} . In the paper we construct a rank-one perturbation of a selfadjoint operator related to a certain canonical system for which a similar property is fulfilled.

Extrema of polynomials with real roots and Diophantine equations

There are many results in the literature concerning polynomial values and (shifted) power values of polynomials with consecutive integer roots, or more generally, with roots forming an arithmetic progression. It is an interesting question that how far one can ‘disturb’ the structure of the roots such that the finiteness results still remain valid. Also there are many results into this direction, with adding or removing one or more terms (roots).In this paper we study a case where (part of) the symmetric root structure is preserved, however, we allow (possibly large) increasing gaps between the roots. We prove that the finiteness of the solutions can also be guaranteed under these generalized circumstances. In our proofs we combine Baker's method and the Bilu-Tichy theorem with a new result providing an increasing property of the extremal values of polynomials with distinct real roots satisfying certain symmetry and increasing gap properties.

Slice regular functions in several variables

In this paper, we lay the foundations of the theory of slice regular functions in several (non-commuting) variables ranging in any real alternative ^*-algebra, including quaternions, octonions and Clifford algebras. This higher dimensional function theory is an extension of the classical theory of holomorphic functions of several complex variables. It is based on the construction of a family of commuting complex structures on {mathbb {R}}^{2^n}. One of the relevant aspects of the theory is the validity of a Cauchy-type integral formula and the existence of ordered power series expansions. The theory includes all polynomials and power series with ordered variables and right coefficients in the algebra. We study the real dimension of the zero set of polynomials in the quaternionic and octonionic cases and give some results about the zero set of polynomials with Clifford coefficients. In particular, we show that a nonconstant polynomial always has a non empty zero set.

Context-Free Grammars for Several Triangular Arrays

In this paper, we present a unified grammatical interpretation of the numbers that satisfy a kind of four-term recurrence relation, including the Bell triangle, the coefficients of modified Hermite polynomials, and the Bessel polynomials. Additionally, as an application, a criterion for real zeros of row-generating polynomials is also presented.

Zero-range potentials for Dirac particles: Bound-state problems

A model in which a massive Dirac particle in R3 is bound by N⩾1 spatially distributed zero-range potentials is presented. Interactions between the particle and the potentials are modeled by subjecting a particle’s bispinor wave function to certain limiting conditions at the potential centers. Each of these conditions is parametrized by a 2 × 2 Hermitian matrix (or, equivalently, a real scalar and a real vector) and mixes the upper and the lower components of the wave function. The problem of determining particle’s bound-state eigenenergies is reduced to the problem of finding real zeros of a determinant of a certain 2N×2N matrix. As the lower component of the particle’s wave function is inverse-square singular at each of the potential centers, the wave function itself is not square-integrable. Nevertheless, one can define a scalar pseudo-product with the property that wave functions belonging to different eigenenergies are orthogonal with respect to it. The wave functions may then be normalized so that their self-pseudo-products are plus one, minus one or zero. An auxiliary set of Sturmian functions is constructed and used to derive an explicit representation of particle’s matrix Green’s function. For illustration purposes, two particular systems are studied in detail: (1) a particle bound in a field of a single zero-range potential, (2) a particle bound in a field of two identical zero-range potentials.

Notes on efficient computation of Ramanujan cubic equations

This paper considers properties of a theorem of Ramanujan to develop properties and algorithms related to cubic equations. The Ramanujan cubics are related to the Cardano cubics and Padovan recurrence relations. These generate cubic identities related to heptagonal triangles and third order recurrence relations, as well as an algorithm for finding the real root of the relevant Ramanujan cubic equation. The algorithm is applied to, and analyzed for, some of the earlier examples in the paper.

A Story of Computational Science: Colonel Titus’ Problem from the 17th Century

Experimentation and the evaluation of algorithms have a long history in algebra. In this paper we follow a single test example over more than 250 years. In 1685, John Wallis published A treatise of algebra, both historical and practical, containing a solution of Colonel Titus’ problem that was proposed to him around 1650. The Colonel Titus problem consists of three algebraic quadratic equations in three unknowns, which Wallis transformed into the problem of finding the roots of a fourth-order (quartic) polynomial. When Joseph Raphson published his method in 1690, he demonstrated the method on 32 algebraic equations and one of the examples was this quartic equation. Edmund Halley later used the same polynomial as an example for his new methods in 1694. Although Wallis used the method of Vietè, which is a digit–by–digit method, the more efficient methods of Halley and Raphson are clearly demonstrated in the works by Raphson and Halley. For more than 250 years the quartic equation has been used as an example in a wide range of solution methods for nonlinear equations. This paper provides an overview of the Colonel Titus problem and the equation first derived by Wallis. The quartic equation has four positive roots and the equation has been found to be very useful for analyzing the number of roots and finding intervals for the individual roots, in the Cardan–Ferrari direct approach for solving quartic equations, and in Sturm’s method of determining the number of real roots of an algebraic equation. The quartic equation, together with two other algebraic equations, have likely been the first set of test examples used to compare different iteration methods of solving algebraic equations.

Discovering the Roots: Uniform Closure Results for Algebraic Classes Under Factoring

Newton iteration is an almost 350-year-old recursive formula that approximates a simple root of a polynomial quite rapidly. We generalize it to a matrix recurrence (allRootsNI) that approximates all roots simultaneously. In this form, the process yields better circuit complexity in the case when the number of rootsris small but the multiplicities are exponentially large. Our method sets up a linear system inrunknowns and iteratively builds the roots as formal power series. For an algebraic circuit\( f(x_1,\ldots ,x_n) \)of sizes, we prove that each factor has size at most a polynomial insand the degree of the squarefree part off. Consequently, if\( f_1 \)is a\( 2^{\Omega (n)} \)-hard polynomial, then any nonzero multiple\( \prod _{i} f_i^{e_i} \)is equally hard for arbitrary positive\( e_i \)’s, assuming that\( \sum _i\deg (f_i) \)is at most\( 2^{O(n)} \).It is an old open question whether the class of poly(n) size formulas (respectively, algebraic branching programs) is closed under factoring. We show that given a polynomialfof degree\( n^{O(1)} \)and formula (respectively, algebraic branching program) size\( n^{O(\log n)} \), we can find a similar-size formula (respectively, algebraic branching program) factor in randomized poly(\( n^{\log n} \)) time. Consequently, if the determinant requires an\( n^{\Omega (\log n)} \)size formula, then the same can be said about any of its nonzero multiples.In all of our proofs, we exploit the following property of multivariate polynomial factorization. Under a random linear transformation\( \tau \), the polynomial\( f(\tau \overline{x}) \)completely factors via power series roots. Moreover, the factorization adapts well to circuit complexity analysis. Therefore, with the help of the strong mathematical characterizations and the ‘allRootsNI’ technique, we make significant progress towards the old open problems; supplementing the vast body of classical results and concepts in algebraic circuit factorization (e.g., [17,51,54,111]).

On the Liouville and strong Liouville properties for a class of non-local operators

We prove a necessary and sufficient condition for the Liouville and strong Liouville properties of the infinitesimal generator of a Lévy process and subordinate Lévy processes. Combining our criterion with the necessary and sufficient condition obtained by Alibaud et al., we obtain a characterization of (orthogonal subgroup of) the set of zeros of the characteristic exponent of the Lévy process.

Global Stability and Canard Explosions of the Predator-Prey Model with the Sigmoid Functional Response

For the predator-prey model with the sigmoid functional response whose denominator has no real roots, the dynamics has already been characterized. Whereas, when the denominator has a real root, the model is singular there, and its dynamics has not been classified. This paper focuses on the case that the denominator has a repeated real root and obtains the next results. The positive equilibrium (if exists) is globally stable provided that it is locally stable. If it is a weak focus, it must be of order one and stable. When the positive equilibrium is unstable, the system has always a limit cycle, which could come from a singular Hopf bifurcation, and undergoes two consecutive canard explosions via relaxation oscillations. In addition, numerical simulations reveal that the curvature of the critical curve at the canard point affects the period of the canard cycle.

Stability problem in a set of Lebesgue measure zero of bi-additive functional equation

Let X be a vector space and Y be a Banach space. Our aim in this paper is to investigate the Hyers-Ulam stability problem of the following bi-additive functional equation f(x + y, s − t) + f(x − y, s + t)=2f(x, s) − 2f(y, t), x, y, s, t ∈ X, where f : X × X → Y . As a consequence, we discuss the stability of the considered functional equation in a restricted domain and in the set of Lebesgue measure zero.

Determining a Particular Solution for n-th Order Linear Differential Equations with Constant Coefficients that Have the Free Term in the Following form P(x). ln x and the Characteristic Equation Has Distinct Real Roots

Abstract For n-th order linear differential equations with constant coefficients the problem of determining the general solution involves determining the general solution of the attached homogeneous equation and, on the other hand, determining a particular solution. This paper aims to analyze the conditions under which the general solution can be obtained for a n-th order linear differential equation with constant coefficients which have the free term in the following form P(x). ln x under the condition that the attached characteristic equation has real and distinct roots, meaning the order of multiplicity for each root is 1. For this we will use the method of variation of constants which leads us to a system whose solution we will gain by using the Upper incomplete Gamma function.

Closed-Form Oscillatory Condition in Electrical Circuits Containing Two Fractional Order Elements

Oscillatory condition in LTI dynamic systems is generally expressed as possessing purely imaginary solutions by their characteristic equations. Dealing with the class of fractional order systems, such a condition is equivalently restated as owing complex roots with specific arguments by a polynomial defined based on the system characteristic equation. The degree of this polynomial can be unboundedly high. Consequently, such statement for the oscillatory condition in fractional order systems, which is based on arguments of the roots of a polynomial with an unbounded degree, cannot be viewed as a closed-form expression. To tackle this challenge, this brief introduces an approach to obtain a closed-form oscillatory condition for electrical circuits containing two fractional order elements. It is shown that the obtained results are helpful in analytically finding the oscillatory curves in parametric regions. Moreover, it is discussed whether the existence of more than one frequency in the frequency content of steady-state oscillations produced by the considered class of fractional order circuits is possible.