Microcomputer-Assisted Mathematics: Polynomial Equations Revisited

In this paper, we first show that, under certain conditions, the solution of a single quadratic diophantine equation in four variables $Q(x_1,\,x_2,\,x_3,\,x_4)=0$ can be expressed in terms of bilinear forms in four parameters. We use this result to establish a necessary, though not sufficient, condition for the solvability of the simultaneous quadratic diophantine equations \[ Q_j(x_1,\ x_2,\ x_3,\ x_4)=0,\quad j=1,\ 2, \] and give a method of obtaining their complete solution. In general, when these two equations have a rational solution, they represent an elliptic curve, but we show that there are several cases in which their complete solution may be expressed by a finite number of parametric solutions and/or a finite number of primitive integer solutions. Finally, we relate the solutions of the quartic equation \[ y^2=t^4+a_1t^3+a_2t^2+a_3t+a_4 \] to the solutions of a pair of quadratic diophantine equations, and thereby obtain new formulae for deriving rational solutions of the aforementioned quartic equation starting from one or two known solutions.

In this chapter we begin our study of Problem IV, which is about solving polynomial equations, that is, finding the zeros of polynomials. We begin in familiar territory, namely solving quadratic equations, which you learned to do in high school. While thousands of years ago the Babylonians, Chinese, Egyptians, and Greeks knew how to find some solutions of some quadratic equations, it was not until the year 628 that an explicit, but not completely general, solution of \(ax^2+bx=c\) was produced by the Indian mathematician and astronomer Brahmagupta (590–668).

(1996). A Few Expository Mini-Essays. The American Mathematical Monthly: Vol. 103, No. 3, pp. 257-263.

It is proved in this paper that Abel’s and Galois’s proofs that the quintic equations have no radical solutions are invalid. Due to Abel’s and Galois’s work about two hundred years ago, it was generally accepted that general quintic equations had no radical solutions. However, Tang Jianer et al. recently prove that there are radical solutions for some quintic equations with special forms. The theories of Abel and Galois cannot explain these results. On the other hand, Gauss et al. proved the fundamental theorem of algebra. The theorem declared that there were n solutions for the n degree equations, including the radical and non-radical solutions. The theories of Abel and Galois contradicted with the fundamental theorem of algebra. Due to the reasons above, the proofs of Abel and Galois should be re-examined and re-evaluated. The author carefully analyzed the Abel’s original paper and found some serious mistakes. In order to prove that the general solution of algebraic equation he proposed was effective for the cubic equation, Abel took the known solutions of cubic equation as a premise to calculate the parameters of his equation. Therefore, Abel’s proof is a logical circular argument and invalid. Besides, Abel confused the variables with the coefficients (constants) of algebraic equations. An expansion with 14 terms was written as 7 terms, 7 terms were missing. We prefer to consider Galois’s theory as a hypothesis rather than a proof. Based on that permutation group S5 had no true normal subgroup, Galois concluded that the quintic equations had no radical solutions, but these two problems had no inevitable logic connection actually. In order to prove the effectiveness of radical extension group of automorphism mapping for the cubic and quartic equations, in the Galois’s theory, some algebraic relations among the roots of equations were used to replace the root itself. This violated the original definition of automorphism mapping group, led to the confusion of concepts and arbitrariness. For the general cubic and quartic algebraic equations, the actual solving processes do not satisfy the tower structure of Galois’s solvable group. The resolvents of cubic and quartic equations are proved to have no symmetries of Galois’s soluble group actually. It is invalid to use the solvable group theory to judge whether the high degree equation has a radical solution. The conclusion of this paper is that there is only the Sn symmetry for the n degree algebraic equations. The symmetry of Galois’s solvable group does not exist. Mathematicians should get rid of the constraints of Abel and Galois’s theories, keep looking for the radical solutions of high degree equations.

There have been many conflicting observations between the linear or curvilinear decline in maximal heart rate (HRmax) with age. The aim of this study was to determine if linear or curvilinear equations would better describe the decline in HRmax with age in individuals of differing cardiorespiratory fitness (CRF) levels. Treadmill cardiopulmonary exercise test (CPX) results from participants (1510 men and 1134 women; 18–76 years) free of overt cardiovascular disease were retrospectively examined using cross-sectional and longitudinal study designs. Participants completing ≥2 CPX with ≥1 year between test dates were included in the longitudinal analysis (325 men and 150 women). Linear and quadratic regressions were applied to age and HRmax for the whole cohort and respective CRF groups (high, moderate, and low, relative to age and gender normative values). To test for differences among linear, quadratic, and polynomial equations, the change in R 2 (cross-sectional analysis) and Bayesian information criterion (BIC) (longitudinal analysis) from the linear to the more complex models were calculated. The quadratic or polynomial regression in the cross-sectional analysis, marginally improved the variance in HRmax explained by age compared to the linear regression for the whole cohort (0.2%), moderate fit group (0.3%), and low fit group (0.8%). With no improvements in the high fit group. BIC did not improve for any CRF category in the longitudinal analysis. In conclusion, the minimal differences among linear, quadratic, and polynomial equations in the respective CRF groups, emphasizes the use of linear prediction equations to estimate HRmax.

Chapter 1 - The characteristic set method and its applications

This paper develops a general method for solving reservoir simulation equations, which is based upon an implicit finite difference formulation of the partial differential equations of compositional thermal simulation, permitting the simulation of nearly all enhanced oil recovery techniques. The general method is used to derive 10 solution schemes, which include the well-known IMPES, Simultaneous Solution (SS), and Sequential Solution (SEQ) methods as sub-cases. The novelty of the general method lies in showing the interrelations of these methods, and also affording the possibility of deriving still other methods. The theoretical development of the general method is given in detail, and the special methods are derived in reference to the reservoir problems for which they were employed. The starting point is a system of nonlinear finite difference equations, typical of compositional thermal simulation. Newtonian iteration then leads to a system of equations. The partitioning of the left-hand side (containing the unknown vector) in a particular manner permits the derivation of 10 solution approaches, which are identified in detail. The proposed general method bridges the gap in the development of the reported solution methods, and also points to several new approaches. The procedure described can be utilized for a variety of simulation problems, with fully implicit source-sink terms (including heat loss, thermal cracking, etc.), for both the standard and multipoint difference schemes in multi dimensions.

Polynomial equations, relating the growth rate of Yersinia enterocolitica to temperature (0–25°C) and pH (4.5–6‐5) in a liquid medium were constructed for four different acidulants. The logarithm of the time for a 100‐fold increase in bacterial numbers could be represented by a quadratic response surface function of pH and temperature. The interactions between pH and temperature on growth rate were found to be additive. Values for a 2 log cycle increase in growth derived from the model were in good agreement with experimental values. Predictions from the quadratic model and from a square root model were compared with experimental values in laboratory media and UHT milk. The mean square error (MSE) for the quadratic response surface model was smaller than that for the square root model in 81% of cases. In UHT milk the square root model increasingly underestimated growth rate, as the temperature decreased and would ‘fail dangerous’ if used for predictive purposes. This indicated that the response surface model is more reliable for predicting the growth of Y. enterocolitica under conditions of sub‐optimal temperature and pH.

In this paper the rate of stability of solutions of matrix polynomial equations of the typeA0+A1X+A2X2+...+A m X m =0 is studied. Particular attention is given to the case where the matrix polynomialL(λ):=A0+A1λ+A2λ2+...+A m λ m is weakly hyperbolic, i.e., for every non-zero vectorx the scalar polynomial 〈L(λ)x, x〉 has only real roots. Also the rate of stability of solutions of matrix quadratic equations of the typeXBX+XA-DX-C=0 is studied. Here the special case that is of interest to continuous-time optimal control theory, that is, the case whereB=B* is positive semidefinite andC=C*,A=−D*, is discussed in detail. The analogous theory for the discrete-time optimal control leads to the equation $$X = A^* XA + Q - (B^* XA)^* (R + B^* XB)^{ - 1} B^* XA,$$ and the rate of stability of solutions of this equation is also studied. Most of the problems are discussed in both real and complex settings.

To model the operating cost of thermal generating units, it is common to use polynomial relations between their power output and fuel input. These mathematical relations are known as fuel-cost functions, which are the heart of optimization algorithms. These functions could be modeled as first, second, or third order polynomial equations. The first order or linear equation is weak to explain the variability of units' operating cost. Also, the third order or cubic polynomial equation is rarely used in the literature, because its third element does not have any significant contribution to add. Thus, the second order or quadratic polynomial equation becomes the most popular fuel-cost function. Sometimes, different linear equations grouped as a piecewise function are used to accelerate the computational speed and linear programming algorithms can be directly involved. This study tries to achieve the goal of the last approach without using any piecewise function. That is, improving the preceding single linear equation to be a competitive fuel-cost function.

A major problem in algebra was to find a formula for the roots of polynomial equations of degree 5 or more, and it was proved by Abel and Ruffini that there was no formula for the roots. Lill developed the ‘graphical method’ by exploring the roots of polynomial equations using a geometric method rather than an algebraic method. To solve arbitrary polynomial equations, certain conditions of the ‘graphical method’ must be satisfied, but Lill did not provide a specific method for this. Beloch expressed quadratic and cubic equations using Lill's method and then discovered a method of satisfying specific conditions using origami. However, when the degree is 4 or more, several conditions must be satisfied simultaneously to solve it with origami, so there has been no research on solving the roots of arbitrary polynomial equations. Therefore, this study selected the following research questions. First, we develop a program that automatically solves quadratic and cubic equations using Lill's method, geogebra, and python. Second, we develop a program that automatically solves arbitrary polynomial equations of degree 4 or higher using Lill's method and python. As a result of the study, we were able to manually find the paths of quadratic and cubic equations using Geogebra and automatically find them using Python and develop a program to solve them using Lill's method. To find the path of an arbitrary nth degree polynomial equation of degree 4 or higher, we used object-oriented programming from the Pygame library to solve it. The significance of this research is that it visualizes the process of approximating the roots of an arbitrary polynomial equation, approaches it with a geometric method rather than an algebraic method, and develops a program to solve arbitrary polynomial equations in a generalized way.

The classic method of solving the cubic and the quartic equations using Tschirnhaus transformation yields true as well as false solutions. Recently some papers on this topic are published, in which methods are given to get only the true solutions of cubic and quartic equations. However these methods have some limitations. In this paper the author presents a method of solving cubic and quartic equations using Tschirnhaus transformation, which yields only the true solutions. The proposed method is much simpler than the methods published earlier.

A numerical method for the solution of the kinetic equation of a reactor

The method of solution is based on the following considerations: In order to clamp a supported plate, it is necessary to act on its edge with bending couples that will produce, along the edge, rotations equal but contrary to those produced by the action of external loads which are taken as known. If now at one edge of a supported rectangular plate a distribution of bending couples is applied, along this edge a rotation is produced as a function of the point which we call a « direct rotation ». At the same time there appear on the other edges of the plate « indirect rotations » which are connected with the « direct » rotation by functional relations of integral nature. When the couples act along the whole perimeter we obtain for each edge a comprehensive rotation equal to the sum of the direct rotation produced by the couples on the edge in question and of the indirect ones produced by the couples applied to the other edges. From what has been said, this sum is a function of the four direct rotations, and if we wish to clamp the edge, it must take on a known value. The same considerations apply to the remaining edges, so that if the direct rotations are presumed to be principal unknowns -and here lies the main feature of the procedure — we obtain a system of four integral equations of the second kind with fixed limits. Assuming a means of separating the unknowns based on the consideration of the symmetrical and antisymmetrical components of the quantities in question, we come to tackle the real solution of the integral equations by means of the iteration method. From this the recurrent formulas are deduced and the convergence is analysed. When the direct rotations are known, we come to know the additional deformations which clamp the supported plate. At the end, two examples illustrate the procedure.

There are several measures for comparing methods for solving a single nonlinear equation. The first is the order of convergence, then the cost to achieve such rate. This cost is measured by counting the number of functions (and derivatives) evaluated at each step. After that, efficiency is defined as a function of the order of convergence and cost. Lately, the idea of basin of attraction is used. This shows how far one can start and still converge to the root. It also shows the symmetry/asymmetry of the method. It was shown that even methods that show symmetry when solving polynomial equations are not so when solving nonpolynomial ones. We will see here that the Euler–Cauchy method (a member of the Laguerre family of methods for multiple roots) is best in the sense that the boundaries of the basins have no lobes. The symmetry in solving a polynomial equation having two roots at ±1 with any multiplicity is obvious. In fact, the Euler–Cauchy method converges very fast in this case. We compare one member of a family of fifth-order methods for multiple roots with several well-known lower-order and efficient methods. We will show using a basin of attraction that the fifth-order method cannot compete with most of those lower-order methods.