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Analysis of Stress
Published in Mumtaz Kassir, Applied Elasticity and Plasticity, 2017
An algebraic solution of the cubic equation in Equation 2.41 may be obtained by following the procedure used in solving Equation 2.29 for the ordinary stress state. Making use of the substitution, S = 2(R/3)1/2 Cos θ, 0 < θ < 60°, it is readily confirmed that the angle θ is governed by cos3θ=−332QR3/2
Numerical Methods
Published in Ching Jen Chen, Richard Bernatz, Kent D. Carlson, Wanlai Lin, Finite Analytic Method in Flows and Heat Transfer, 2020
Ching Jen Chen, Richard Bernatz, Kent D. Carlson, Wanlai Lin
The accuracy of the numerical solution depends, primarily, on each of these steps. That is, the number and arrangement of nodes (grid generation),the derivation of the algebraic equation (numerical method), andthe solution to the linear system of algebraic equations (algebraic solution method).
Elasticity problems and the finite difference method
Published in R.C. Coates, M.G. Coutie, F.K. Kong, Structural Analysis, 2018
R.C. Coates, M.G. Coutie, F.K. Kong
This cubic equation can be conveniently solved graphically, or else by systematic trial and error (which takes much less time than many people think, and which is preferable, for practical purposes, to the formal algebraic solution) giving the three values of O"j as before. To determine the direction corresponding to 0"1, use Eqn 11.5-1:
General solution of spin-1 Ising model in the effective field theory approximation: critical temperatures and spontaneous magnetization
Published in Phase Transitions, 2022
In conclusion, in the present paper, we have performed a general analysis of the spin-1 Ising model on the lattices with arbitrary values of the coordination number in the framework of the single-site EFT cluster approximation. The system of two fully algebraic polynomial equations for the determination of the reduced critical temperatures of the model simultaneously for all values of the coordination number is derived. Moreover, the possibility of further reduction of this system of equations into the single polynomial equation for the smallest values of the coordination number and 5 is demonstrated. The corresponding single polynomial equation for the critical temperature is explicitly derived for z=3. The existence of the fully algebraic solution of the problem, which gives nontrivial possibility to determine the values of the critical temperature for arbitrary large value of the coordination number, has allowed us to find a simple linear formula that approximates the critical temperatures of the model with very high precision. Thus one can estimate the critical temperature of the model with high precision but without any calculations. In addition, the algebraic solution of the model in the form of the system of two algebraic polynomial equations valid for all values of the coordination number is also found for the spontaneous magnetization.
Identification of the Area of Vulnerability to Voltage Sags Based on Galerkin Method
Published in Electric Power Components and Systems, 2019
Yongzhi Zhou, Hao Wu, Boliang Lou, Hui Deng, Yonghua Song, Wen Hua, Yijun Shen
According to Abel–Ruffini theorem, there is no algebraic solution to the general polynomial equation of degree 5 or higher with arbitrary coefficients [22]. Thus, the higher order polynomial equation with over 4th-degree is solved by numerical approaches, where good initial points are desired to guarantee the convergence of iteration. In the proposed method, the equation roots of i-th order approximation are used as the initial points of i + 1-th order. Because the equation roots of two successive approximations are very close, the initial points provided by the lower approximation can guarantee the convergence in most cases.
The investigation of prospective mathematics teachers’ non-algebraic solution strategies for word problems
Published in International Journal of Mathematical Education in Science and Technology, 2020
Mehmet Fatih Öçal, Ceylan Şen, Gürsel Güler, Tuğrul Kar
In mathematics education, problem solving has a long history as part of the school curriculum and as an instructional goal [41]. Taking this importance into account, special attention is given to the investigations of situations that contribute to the development of problem-solving abilities. Various researchers and instructional documents [7,8,28] emphasized that improvement in students’ problem-solving skills relies on doing algebraic and non-algebraic solutions together and fulfil the relation between them. Moreover, Shimizu [42] indicated that it is an important characteristic of Japanese lessons that mathematics courses at elementary and high schools are organized around creating multiple solutions for a few problems. Multiple solution approaches in problem-solving processes and emphasizing the relationships among them improve students’ creativity skills by contributing to their flexible thinking [43,44]. Besides, non-algebraic solution approaches contribute to explain the meaning given to the variables of algebraic solution approaches and the operations performed. These supports students’ abilities of cultivating transition from arithmetic to algebraic. However, in this study it was found that prospective teachers were insufficient in producing non-algebraic solutions for the word problems, failed to present alternative solutions because of their limited understanding for interpreting algebraic equation, and had limited knowledge on algebraic and non-algebraic solution approaches. Research has shown that the teachers’ knowledge about a course subject impacts the knowledge that students develop [4,15,16,45]. Cai et al. [6] assert that arithmetic and algebra can be seen as inseparable situations if teachers can help students develop arithmetic and algebraic thinking skills together at elementary education level. The shortcomings identified in this study indicate that the prospective teachers may not be able to develop this approach in teaching environments.