Solution Of Cubic Equation Research Articles

An Important Historical Milestone: The Classification of the Cubic Equations

This article investigates the use of the history of mathematics as a pedagogical tool for the teaching and learning of mathematics, using the history of the cubic equation as a specific example.Cubic equations arise intrinsically in many applications in natural sciences and mathematics. For example, in physics, the solutions of the equations of state in thermodynamics, or the computation of the speed of seismic Rayleigh waves require the solutions of cubic equations. In mathematics, they are instrumental in solving the quartic equations, for in the process, these are reduced to cubic equations. The impossibility of trisecting an angle or doubling a cube using only a straightedge and compass is equivalent to solving some cubic equations. As the name implies, the cubic spline approximation, an important tool in numerical analysis, also entails working with cubic functions. Although cubic equations were explored by the ancient Babylonian, Greek, Chinese, Indian, and Egyptian scholars, it took the collective work of many well-known mathematicians such as Diophantus, Archimedes, Fibonacci, del Ferro, Khayyam, Tartaglia, Cardano, Viète, Descartes, and Lagrange to finally obtain a full solution. Our goal in this paper is to investigate one of the most formidable steps in this extensive and prolific history, namely the complete classification of the cubic equations by Omar Khayyam in eleventh century, who was the first scholar to classify cubic equation and hence facilitate a methodical and logical approach to obtaining a general solution.

ПРИМЕНЕНИЕ ПОНЯТИЯ «КОРЕНЬ N-ОЙ СТЕПЕНИ» ПРИ РЕШЕНИИ КУБИЧЕСКИХ УРАВНЕНИЙ. ФОРМУЛА КАРДАНО

The main task of teaching mathematics in secondary schools is to ensure that students have a solid and conscious mastery of the system of mathematical knowledge and skills necessary in everyday life and work for each member of modern society, sufficient to study related disciplines and continue their education. Along with solving the main problem, advanced study of mathematics provides for the formation of students ' sustained interest in the subject, the identification and development of their mathematical abilities. In-depth study of the subject should provide preparation for entering the UNIVERSITY and continuing education, as well as for professional activities that require a sufficiently high mathematical culture. The topic "Root of the nth degree" is necessary for the study of many sections of algebra, namely, to find the derivative and primitive. The article deals with the solution of cubic equations using the Cardano formula. The algorithm for applying this feature is described.

A note on the solutions of cubic equations of state in low temperature region

The cubic equations of state (CEOS) are still much used in the chemical and petroleum industry for process modeling, simulation, and optimization. Calculating the compressibility factor/molar volumes from CEOS is an important and frequently invoked operation that involves analytical, numerical methods of solution of the cubic equation. A simple analytical method that is computationally inexpensive and provides solutions with high accuracy is proposed in the work. The method is first tested against other cubic solvers from the literature for its stability and accuracy. It is then applied to compute the roots of a CEOS in the low-temperature region. The results demonstrate the superiority of the method over other analytical, numerical solvers. The analysis of the different methods and their formulae can aid the selection of an appropriate method in process optimization.

Вільні коливання пружно нелінійного осцилятора з сухим тертям

Методом енергетичного балансу виведено рекурентні співвідношення для розрахунку послідовності спадаючих амплітуд розмахів дисипативного осцилятора з сухим тертям Кулона. Розглянуто різні варіанти нелінійної пружності коливальної системи. У статті розглянуті випадки коливань осцилятора з наступними показниками нелінійності коефіцієнту пружності: степенево-нелінійної пружності з варіантами лінійного пружного осцилятора, осцилятора з м’якою характеристиками пружності при різних її значеннях, квадратичною та кубічною нелінійністю; коливання осцилятора за наявності в виразі сили пружності лінійної складової з різними показниками нелінійності. Виділено випадки, коли виведені рекурентні співвідношення мають замкнені аналітичні розв’язки і побудовано їх. Складені рекурентні співвідношення між амплітудами розмахів та запропоновано їх числове розв’язання методом ітерацій Ньютона. Проведено порівняння числових результатів, до яких призводять такі розв’язки та комп’ютерне інтегрування диференціального рівняння руху. Показано повну узгодженість результатів, одержаних різними способами. Запропоновано компактні формули для розрахунку ширини зони застою при різних нелінійностях. Викладений спосіб розрахунку простий в реалізації , бо не потребує розв’язування нелінійного диференціального рівняння вільних коливань осцилятора. При виведенні рекурентних співвідношень задіяно точний аналітичний розв’язок кубічного рівняння, запропонований Кардано. Виявлено варіанти нелінійностей, коли розрахунок зводиться до використання явних рекурентних співвідношень, пов’язаних з розв’язками квадратного та кубічного рівнянь. У випадку довільного показника нелінійності розрахунок амплітуд доводиться проводити методом ітерацій.

A New Understanding on the Problem That the Quintic Equation Has No Radical Solutions

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.

An appraisal of Lalanne's nomogram

A number of methods are available for the solution of cubic equations. Amongst these, and possibly less well known than those based on algebraic or trigonometrical procedures, is an elegant graphical solution devised by Lalanne [1]. His approach and the resulting nomogram are appraised in this paper.The general cubic equation in the variable X

A further analysis of Cardano’s main tool in the De Regula Aliza: on the origins of the splittings

In the framework of the De Regula Aliza (1570), Cardano paid much attention to the so-called splittings for the family of equations $$x^3 = a_1x + a_0$$ ; my previous article (Confalonieri in Arch Hist Exact Sci 69:257–289, 2015a) deals at length with them and, especially, with their role in the Ars Magna in relation to the solution methods for cubic equations. Significantly, the method of the splittings in the De Regula Aliza helps to account for how Cardano dealt with equations, which cannot be inferred from his other algebraic treatises. In the present paper, this topic is further developed, the focus now being directed to the origins of the splittings. First, we investigate Cardano’s research in the Ars Magna Arithmeticae on the shapes for irrational solutions of cubic equations with rational coefficients and on the general shapes for the solutions of any cubic equation. It turns out that these inquiries pre-exist Cardano’s research on substitutions and cubic formulae, which will later be the privileged methods for dealing with cubic equations; at an earlier time, Cardano had hoped to gather information on the general case by exploiting analogies with the particular case of irrational solutions. Accordingly, the Ars Magna Arithmeticae is revealed to be truly a treatise on the shapes of solutions of cubic equations. Afterwards, we consider the temporary patch given by Cardano in the Ars Magna to overcome the problem entailed by the casus irreducibilis as it emerges once the complete picture of the solution methods for all families of cubic equations has been outlined. When Cardano had to face the difficulty that appears if one deals with cubic equations using the brand-new methods of substitutions and cubic formulae, he reverted back to the well-known inquiries on the shape of solutions. In this way, the relation between the splittings and the older inquiries on the shape of solutions comes to light; furthermore, this enables the splittings to be dated 1542 or later. The last section of the present paper then expounds the passages from the Aliza that allow us to trace back the origins of the substitution $$x = y + z$$ , which is fundamental not only to the method of the splittings but also to the discovery of the cubic formulae. In this way, an insight into Cardano’s way of dealing with equations using quadratic irrational numbers and other selected kinds of binomials and trinomials will be provided; moreover, this will display the role of the analysis of the shapes of solutions in the framework of Cardano’s algebraic works.

Energy-Based Yield Criteria for Orthotropic Materials, Exhibiting Strength-Differential Effect. Specification for Sheets under Plane Stress State

AbstractA general proposition of an energy-based limit condition for anisotropic materials exhibiting strength-differential effect (SDE) based on spectral decomposition of elasticity tensors and the use of scaling pressure-dependent functions is specified for the case of orthotropic materials. A detailed algorithm (based on classical solutions of cubic equations) for the determination of elastic eigenstates and eigenvalues of the orthotropic stiffness tensor is presented. A yield condition is formulated for both two-dimensional and three-dimensional cases. Explicit formulas based on simple strength tests are derived for parameters of criterion in the plane case. The application of both criteria for the description of yielding and plastic deformation of metal sheets is discussed in detail. The plane case criterion is verified with experimental results from the literature.

English

English

Continued proportions and Tartaglia's solution of cubic equations

We analyze Tartaglia's account, in 1546, of the circumstances leading to his breakthrough regarding the solution of cubic equations. He claims that he solved x3+rx2=q in 1530, well before he could handle, in 1535, equations with a linear term px (and no quadratic term). This claim is at variance with Cardano's narrative as well as with later treatments of the problem, in which the solution of equations of the latter type provides the basis for the solution of all the other types of cubic equations. We show that Tartaglia's claim is supported in his text by the use of the theory of continued proportions, that occurs as a Leitmotiv. We show that relations on continued proportions stressed by Pacioli as basic “keys” provide a simple derivation of the results given by Tartaglia, that is consistent with their chronological order. Thus, his narrative contains not only priority claims, but also proposes an account of the mathematical steps that led him to his results.

Some Contributions to the Solution of Cubic Equations

In many applications, the solution of a cubic equation is requi red. This study improves on the existing methods of solving cubic equations. The general expres sion for the quadratic factor of a cubic equation is derived. The discriminant of the cubic equa tion, adopted in this study as D, comprises of two terms, 1

An olympiadic system of equations

Solution of an olympiadic system with three variables is considered. Solution of cubic equations is discussed. Some advices for young olympiadists are given.

A Note on the Analytical Solution of Cubic Equations of State in Process Simulation

This work address the problem of round-off errors in the analytical solution (Cardano’s Method) of cubic equations of state inside the simulation of chemical processes in the low-temperature region, as it is the case of cryogenic processes involving hydrates calculations. It is proposed as a strategy that can be taken as an iterative refinement of the solution obtained by the analytical method and allows one to take advantage of the calculations fulfilled in the application of the analytical solution (Cardano’s Method). Numerical experimentations are presented in order to show the application of the proposed strategy.

Some further bounds for the Q-index of nested split graphs

The Q-index of a simple graph is the largest eigenvalue of its signless Laplacian, or Q-matrix. In our previous paper [1] we gave three lower and three upper bounds for the Q-index of nested split graphs, also known as threshold graphs. In this paper, we give another two upper bounds, which are expressed as solutions of cubic equations (in contrast to quadratics from [1]). Some computational results are also included.

Network inference using asynchronously updated kinetic Ising model

Network structures are reconstructed from dynamical data by respectively naive mean field (nMF) and Thouless-Anderson-Palmer (TAP) approximations. TAP approximation adds simple corrections to the nMF approximation, taking into account the effect of the focused spin on itself via its influence on other neighboring spins. For TAP approximation, we use two methods to reconstruct the network: (a) iterative method; (b) casting the inference formula to a set of cubic equations and solving it directly. We investigate inference of the asymmetric Sherrington-Kirkpatrick (aS-K) model using asynchronous update. The solutions of the set of cubic equations depend on temperature T in the aS-K model, and a critical temperature T(c)≈2.1 is found. The two methods for TAP approximation produce the same results when the iterative method is convergent. Compared to nMF, TAP is somewhat better at low temperatures, but approaches the same performance as temperature increases. Both nMF and TAP approximation reconstruct better for longer data length L, but for the degree of improvement, TAP performs better than nMF.

Solution of cubic equations: An alternative method

The exact solution of polynomial equations has long been an area of interest. Students are familiar with the solution of linear and quadratic equations, but when they move to cubic and quartic equations, they find that the solution involves a non-intuitive transformation. Cardan’s method of solving the cubic equation is well known (see [1] and [2]; see also [3] for a geometrical interpretation of the method). This article suggests a different and simpler formulation to solve cubic equations.

Solution of Cubic and Quartic Equations

Solution of Cubic and Quartic Equations In this article, the principal n-th root of a complex number is defined, the Vieta's formulas for polynomial equations of degree 2, 3 and 4 are formalized. The solution of quadratic equations, the Cardan's solution of cubic equations and the Descartes-Euler solution of quartic equations in terms of their complex coefficients are also presented [5].

Khayyam with Cabri: experiences of pre-service mathematics teachers with Khayyam's solution of cubic equations in dynamic geometry environment

The study reported in this article deals with the observed actions of Turkish pre-service mathematics teachers in dynamic geometry environment (DGE) as they were learning Khayyam's method for solving cubic equations formed as x3 + ax = b. Having learned the method, modelled it in DGE and verified the correctness of the solution, students generated their own methods for solving different types of cubic equations such as x3 + ax2 = b and x3 + a = bx in the light of Khayyam's method. With the presented teaching experiment, students realized that Khayyam's mathematics is different from theirs. We consider that this gave them an opportunity to have an insight about the cultural and social aspects of mathematics. In addition, the teaching experiment showed that dynamic geometry software is an excellent tool for doing mathematics because of their dynamic nature and accurate constructions. And, it can be easily concluded that the history of mathematics is useful resource for enriching mathematics learning environment.

Relations between roots and coefficients of cubic equations with one root negative the reciprocal of another

Under predetermined conditions on the roots and coefficients, necessary and sufficient conditions relating the coefficients of a given cubic equation x 3 + ax 2 + bx + c = 0 can be established so that the roots possess desired properties. In this note, the condition for one root of a cubic equation to be the negative reciprocal of another one is obtained. Given that the coefficients a, b, c of the cubic equation are in arithmetical or geometrical progression, further conditions are deived for one root to be the negative reciprocal of another. These results provide useful means for checking calculated roots of cubic equations and could serve the needs of teachers and students of Mathematical Sciences in tertiary institutions when the solution of cubic equations are first studied.

Small solutions of cubic equations with prime variables in arithmetic progressions

Small solutions of cubic equations with prime variables in arithmetic progressions