General Polynomial Equation Research Articles

Uniform and Simplified Small-Signal Model for Inductive Power Transfer Systems

This article explores the unification and simplification of the small-signal model for representative inductive power transfer systems. Various famous compensations are included, such as the S–S, LCC–S, and LCC–LCC. Through system decomposition, the uniform model concept is applied to the inverter, rectifier, and resonant tank. Meanwhile, the model complexity and similarity are discussed for various resonant tanks, and proper simplification is used to reduce the model order. The uniform equivalent circuit model is finally proposed for the target systems, based on which the explicit transfer function can be represented by a general polynomial equation. The model order of the S–S, LCC–S, and LCC–LCC systems are five, two, and five, respectively. The experiment shows the model is accurate up to one-fifth of the switching frequency for all the target systems.

Fitting a triaxial ellipsoid to a set of quasi-selenoidal points

Abstract The aim of this work is the determination of the parameters of the triaxial ellipsoid of the Moon, as derived from a quasi-selenoid model. After a detailed description of various quasi-selenoid models of the lunar gravity field, which were proposed in the last twenty years, we prepare suitable data sets of three-dimensional Cartesian coordinates. The mathematical model adopted is the general (polynomial) equation of an ellipsoid functionally related to the nine unknowns: the coordinates of the ellipsoid center, the three rotation angles and the three ellipsoid semiaxes. Furthermore, we adopt mathematical models for one special and two degenerate cases of the triaxial ellipsoid. We implement the least-squares method of indirect observations and we derive results for eighteen data sets of quasi-selenoidal points. From the results, we report the values of the semiaxes of the triaxial ellipsoid of fitting with three unknowns, for the model GL0660B, to be 1,738,256.3 ± 0.2 m, 1,738,023.1 ± 0.2 m and 1,737,603.2 ± 0.2 m, while the other unknowns remain insignificant. This triaxial ellipsoid leads to the improvement in the RMS value of the height anomaly at about 12 per cent in comparison to the oblate spheroid.

A New Method for the Cubic Polynomial Equation

I present a method to solve the general cubic polynomial equation based on six years of research that started back in 1985 when, in the fifth grade, I first learned of Bhaskara's formula for the quadratic equation. I was fascinated by Bhaskara's formula and naively thought I could replicate his method for the third degree equation, but only succeeded in 1990, after countless failed attempts. The solution involves a simple transformation to form a cube and which, by chance, happens to reduce the degree of the equation from three to two (which seems to be the case of all polynomial equations that admit solutions by means of radicals). I also talk about my experiences trying to communicate these results to mathematicians, both at home and abroad.

Unified Functional Method for Solving General Polynomial Equations of Degree Less Than Five

A unified method for solving that incorporate a computational formula that relate the coefficients of the depressed equation and the coefficients of the standard polynomial equation is proposed in this study. This is to ensure that this method is valid for all It shall apply the undetermined parameter method of auxiliary function to obtain solutions to these polynomial equations of degree less than five in one variable. In particular, the result of our work is a unification and improvement on the work of several authors in the sense that only applicable for the case of polynomial equation of degree one. Finally, our results improve and generalize the result by applying standard formula methods for solving higher degree polynomials. It is recommended that the effort should be made toward providing other variant methods that are simpler and friendly.

Classification of All Non-Isomorphic Regular and Cuspidal Arm Anatomies in an Orthogonal Metamorphic Manipulator

This paper proposes a classification of all non-isomorphic anatomies of an orthogonal metamorphic manipulator according to the topology of workspace considering cusps and nodes. Using symbolic algebra, a general kinematics polynomial equation is formulated, and the closed-form parametric solution of the inverse kinematics is obtained for the coming anatomies. The metamorphic design space was disjointed into eight distinct subspaces with the same number of cusps and nodes plotting the bifurcating and strict surfaces in a cartesian coordinate system { θ π 1 , θ π 2 , d 4 } . In addition, several non-singular, smooth and continuous trajectories are simulated to show the importance of this classification.

Optimization of Capacitive Wireless Power Transfer System for Maximum Efficiency

Optimal inductance and capacitance values of a receiver (Rx) LC matching network are proposed in order to maximize the efficiency of capacitive wireless power transfer (CWPT). To maximize the overall Tx-to-load efficiency, a general polynomial equation is proposed that gives an optimum Rx inductance and capacitance value under the given constraints of inductor losses, load impedance, and coupling capacitance. It is found that the efficiency is maximized when the Rx input resistance is approximately equal to the impedance of the coupling capacitance. It is also shown that Tx inductance and capacitance values should be chosen based on the trade-off between efficiency and Tx amplifier operation. The proposed method is applied to wireless charging of an Unmanned Aerial Vehicle (UAV) because the small Rx plate of CWPT can easily be mounted on a UAV. The fabricated system delivers 45 W with overall efficiency of 78.2% across wide lateral misalignments even with a small Rx plate at 6.78 MHz.

Critical temperatures and critical concentrations in diluted magnetic systems: General solution of the Ising model in effective-field theory approach

Using the single-site effective-field theory approximation the site-diluted spin-1∕2 Ising model is investigated simultaneously on lattices with arbitrary values of the coordination number. The polynomial equation is derived that drives the positions of the critical temperature of the model simultaneously for arbitrary values of the concentration p of the magnetic atoms as well as for all values of the coordination number z. This equation is used for detailed investigation of the dependence of the critical temperature values on the model parameters. It is shown that the critical temperatures of the model can be approximated with high precision by simple function zp−1 especially for large values of the coordination number. The general polynomial equation for the determination of the critical concentration below which the second order phase transition do not exist at all is obtained and analyzed and the explicit analytical expressions for the critical concentrations are found for coordination numbers z=3 and 4. Moreover, algebraic polynomial equations for the magnetization of the model as well as for the saturated magnetization are also derived and discussed.

A general view on the critical behavior in the effective field theory approximation of the Ising model with arbitrary coordination number

A general polynomial equation is derived for the determination of the critical temperatures of the Ising model on lattices with arbitrary values of the coordination number in the framework of the single-site cluster effective field theory approximation. This equation is used for the investigation of the properties of the critical temperature of the model as the function of the coordination number. It is shown that the value of the critical temperature of the model can be approximated with very high precision by the simple function z−1 already for relatively small values of z. In addition, it is also shown that the critical temperatures only for the two smallest values of the coordination number (z=3 and 4) can be found in the closed analytic form.

Galois theory for general systems of polynomial equations

We prove that the monodromy group of a reduced irreducible square system of general polynomial equations equals the symmetric group. This is a natural first step towards the Galois theory of general systems of polynomial equations, because arbitrary systems split into reduced irreducible ones upon monomial changes of variables. In particular, our result proves the multivariate version of the Abel–Ruffini theorem: the classification of general systems of equations solvable by radicals reduces to the classification of lattice polytopes of mixed volume 4 (which we prove to be finite in every dimension). We also notice that the monodromy of every general system of equations is either symmetric or imprimitive. The proof is based on a new result of independent importance regarding dual defectiveness of systems of equations: the discriminant of a reduced irreducible square system of general polynomial equations is a hypersurface unless the system is linear up to a monomial change of variables.

Picking genuine zeros of cubics in the Tschirnhaus method

In 1683, the German mathematician Ehrenfried Walther von Tschirnhaus introduced a polynomial transformation which, he claimed, would eliminate all intermediate terms in a polynomial equation of any degree, thereby reducing it to a binomial form from which roots can easily be extracted [1]. As mathematicians at that time were struggling to solve quintic equations in radicals with no sign of any success, the Tschirnhaus transformation gave them some hope, and in 1786, Bring was able to reduce the general quintic to the form x5 + ax + b = 0, even though he didn't succeed in his primary mission of solving it. It seems Bring's work got lost in the archives of University of Lund. Unaware of Bring's work, Jerrard (1859) also arrived at the same form of the quintic using a quartic Tschirnhaus transformation [2]. From the works of Abel (1826) and Galois (1832) it is now clear that the general polynomial equation of degree higher than four cannot be solved in radicals.

Assessment of the lubricant behaviour of biodiesel fuels using Stribeck curves

The molecular structure (carbon chain length, degree of unsaturation and branching of the chains) and the presence of contaminants in a biodiesel have a critical influence on their lubricant behaviour. Ball-on-disc tribological tests were performed with two different standard reference materials for biodiesel — NIST SRM 2772 (soybean) and NIST SRM 2773 (animal fat). It is demonstrated that the Stribeck curves, the plots of coefficient of friction (μ) vs. Stribeck parameter S (the product of viscosity with speed divided by the normal contact load), fit to a general polynomial equation of the type μ=μ0 (1−KBL×S+KEHL×Sn). Constants KBL and KEHL reveal the contributions of the boundary lubrication (BL) and the elastohydrodynamic (EHL) phenomena, respectively. The animal fat biodiesel is the most effective fuel as it shows the highest KBL and the lowest KEHL values leading to the most stable coefficient of friction regarding speed variation. This behaviour is attributed to the higher content of contaminants, namely mono- and di-glycerides, and the presence of sulphur in the animal fat biodiesel.

General polynomials over division algebras and left eigenvalues

In this paper, we present an isomorphism between the ring of general polynomials over a division ring of degree $p$ over its center $F$ and the group ring of the free monoid with $p^2$ variables. Using this isomorphism, we define the characteristic polynomial of a matrix over any division algebra, i.e. a general polynomial with one variable over the algebra whose roots are precisely the left eigenvalues. Plus, we show how the left eigenvalues of a $4 \times 4$ matrices over any division algebra can be found by solving a general polynomial equation of degree 6 over that algebra.

On topological invariants of real algebraic functions

A natural covering responsible for the complexity of the ramification of roots of the general real polynomial equation is considered, and the homology groups of its base are calculated; for equations of degree ⩽ 5, a complete description of the topology of this base is given.

Relative equilibria of point vortices and the fundamental theorem of algebra

Relative equilibria of identical point vortices may be associated with a generating polynomial that has the vortex positions as its roots. A formula is derived that relates the first and second derivatives of this polynomial evaluated at a vortex position. Using this formula, along with the fundamental theorem of algebra, one can sometimes write a general polynomial equation. In this way, results about relative equilibria of point vortices may be proved in a compact and elegant way. For example, the classical result of Stieltjes, that if the vortices are on a line they must be situated at the zeros of the N th Hermite polynomial, follows easily. It is also shown that if in a relative equilibrium the vortices are all situated on a circle, they must form a regular N -gon. Several other results are proved using this approach. An ordinary differential equation for the generating polynomial when the vortices are situated on two perpendicular lines is derived. The method is extended to vortex systems where all the vortices have the same magnitude but may be of either sign. Derivations of the equation of Tkachenko for completely stationary configurations and its extension to translating relative equilibria are given.

92.66 Parametric forms for general polynomial equations

92.66 Parametric forms for general polynomial equations

Solving SAT by algorithm transform of Wu’s method

Recently algorithms for solving propositional satisfiability problem, or SAT, have aroused great interest, and more attention has been paid to transformation problem solving. The commonly used transformation is representation transform, but since its intermediate computing procedure is a black box from the viewpoint of the original problem, this approach has many limitations. In this paper, a new approach called algorithm transform is proposed and applied to solving SAT by Wu’s method, a general algorithm for solving polynomial equations. By establishing the correspondence between the primitive operation in Wu’s method and clause resolution in SAT, it is shown that Wu’s method, when used for solving SAT, is primarily a restricted clause resolution procedure. While Wu’s method introduces entirely new concepts, e.g. characteristic set of clauses, to resolution procedure, the complexity result of resolution procedure suggests an exponential lower bound to Wu’s method for solving general polynomial equations. Moreover, this algorithm transform can help achieve a more efficient implementation of Wu’s method since it can avoid the complex manipulation of polynomials and can make the best use of domain specific knowledge.

LQG Feedback and Feedforward Self-Tuning Control

Abstract A new LQG self-tuning control algorithm which includes feedforward action is derived using a very general polynomial equation system model. The algorithm is derived for use in advanced or complex industrial applications where the simple ARMAX model (used successfully in process control applications) is not adequate. The main contribution of the paper is the derivation of an identification algorithm (based on the recursive prediction-error method) for estimation of the parameters of the general system model. Calculation of the control law requires spectral factorization and the solution of diophantine equations. Although the full algorithm is quite complex some optional simplifications are suggested which make the computations more straightforward.

On the solution of algebraic equations by the decomposition method

The decomposition method ( G. Adomian, “Stochastic Systems,” Academic Press, New York, 1983) developed to solve nonlinear stochastic differential equations has recently been generalized to nonlinear (and/or) stochastic partial differential equations, systems of equations, and delay equations and applied to diverse applications. As pointed out previously (see reference above) the methodology is an operator method which can be used for nondifferential operators as well. Extension has also been made to algebraic equations involving real or complex coefficients. This paper deals specifically with quadratic, cubic, and general higher-order polynomial equations and negative, or nonintegral powers, and random algebraic equations. Further work on this general subject appears elsewhere (G. Adomian, “Stochastic Systems II,” Academic Press, New York, in press).

Parameter plane approach to the analysis of optimal linear control systems

The paper shows how the parameter plane technique can be applied to the characteristic equation of the optimal linear control system chosen to minimize a quadratic performance criterion. The technique, as applied to a general polynomial equation, is described paying particular attention to the plotting of constant relative damping, constant undamped natural frequency and constant real pole position characteristic curves in the parameter plane which has two variable parameters as co-ordinates. The method is then applied to an optimal linear control system characteristic equation which is written in a convenient form for determining the structure of the characteristic equation coefficients in terms of the variable parameters. Particular attention is given to two parameters contained within a diagonal control weighting matrix and by observing the characteristic curves in the parameter plane the effect of different values of these two parameters on the optimal transient performance can be investigated without...

Soil reaction in relation to horticulture

The molecular structure (carbon chain length, degree of unsaturation and branching of the chains) and the presence of contaminants in a biodiesel have a critical influence on their lubricant behaviour. Ball-on-disc tribological tests were performed with two different standard reference materials for biodiesel — NIST SRM 2772 (soybean) and NIST SRM 2773 (animal fat). It is demonstrated that the Stribeck curves, the plots of coefficient of friction (μ) vs. Stribeck parameter S (the product of viscosity with speed divided by the normal contact load), fit to a general polynomial equation of the type μ = μ0 (1 − KBL × S + KEHL × Sn). Constants KBL and KEHL reveal the contributions of the boundary lubrication (BL) and the elastohydrodynamic (EHL) phenomena, respectively. The animal fat biodiesel is the most effective fuel as it shows the highest KBL and the lowest KEHL values leading to the most stable coefficient of friction regarding speed variation. This behaviour is attributed to the higher content of contaminants, namely mono- and di-glycerides, and the presence of sulphur in the animal fat biodiesel.