Solving equations with fractions: An analysis of prospective teachers’ solution pathways and errors

We analyzed the solution pathways and errors found in the written responses of 469 prospective teachers solving an equation containing fractions. The majority (332, or 70%) used an algebraic method; 141 of the 332 (42%) were correct, and 22% of the algebraic methods were abandoned before a solution was obtained. We identified the steps in the written solutions, determined which solution pathways led to the correct solution, and identified common errors in the solution pathways of respondents who incorrectly solved the equation. Respondents initially attempted different methods. The most common method was solving by using fractions, but the majority of respondents who solved by using mixed numbers were able to correctly solve the problem. Common errors related to fraction arithmetic and the distributive property. Nearly all of the abandoned pathways contained no errors, but ended with a step that likely would precede an operation with fractions. Our findings suggest that the ability to solve an arithmetic equation with no fractions was necessary, but not sufficient, to solve an arithmetic equation involving fractions, and that the problem of solving equations with fractions was more closely tied to one's difficulties with rational number arithmetic and less with one's understanding of algebra.     

A regularization method for the numerical solution of elliptic boundary control problems with pointwise state constraints

A Lavrentiev type regularization technique for solving elliptic boundary control problems with pointwise state constraints is considered. The main concept behind this regularization is to look for controls in the range of the adjoint control-to-state mapping. After investigating the analysis of the method, a semismooth Newton method based on the optimality conditions is presented. The theoretical results are confirmed by numerical tests. Moreover, they are validated by comparing the regularization technique with standard numerical codes based on the discretize-then-optimize concept. The authors acknowledge support through DFG Research Center “Mathematics for Key Technologies” (FZT 86) in Berlin.     

A study of the convergence of and error estimation for the homotopy perturbation method for the Volterra–Fredholm integral equations

In this work the solution of the Volterra–Fredholm integral equations of the second kind is presented. The proposed method is based on the homotopy perturbation method, which consists in constructing the series whose sum is the solution of the problem considered. The problem of the convergence of the series constructed is formulated and a proof of the formulation is given in the work. Additionally, the estimation of the approximate solution errors obtained by taking the partial sums of the series is elaborated on.     

An advanced study on the solution of nanofluid flow problems via Adomian’s method

Nanofluid flow is one of the most important areas of research at the present time due to its wide and significant applications in industry and several scientific fields. The boundary layer flow of nanofluids is usually described by a system of nonlinear differential equations with boundary conditions at infinity. These boundary conditions at infinity cause difficulties for any of the series method, such as Adomian’s method, the variational iteration method and others.The objective of the present work is to introduce a reliable method to overcome such difficulties that arise due to an infinite domain. The proposed scheme, that we will introduce, is based on Adomian’s decomposition method, where we will solve a system of nonlinear differential equations describing the boundary layer flow of a nanofluid past a stretching sheet.     

Numerical solution of vector Sturm–Liouville problems with Dirichlet conditions and nonlinear dependence on the spectral parameter

A numerical-analytical iterative method is proposed for solving generalized self-adjoint regular vector Sturm–Liouville problems with Dirichlet boundary conditions. The method is based on eigenvalue (spectral) correction. The matrix coefficients of the equations are assumed to be nonlinear functions of the spectral parameter. For a relatively close initial approximation, the method is shown to have second-order convergence with respect to a small parameter. Test examples are considered, and the model problem of transverse vibrations of a hinged rod with a variable cross section is solved taking into account its rotational inertia.     

Modified Runge–Kutta Verner methods for the numerical solution of initial and boundary-value problems with engineering applications

Two new modified Runge–Kutta methods with minimal phase-lag are developed for the numerical solution of Ordinary Differential Equations with engineering applications. These methods are based on the well-known Runge–Kutta method of Verner RK6(5)9b (see J.H. Verner, some Runge–Kutta formula pairs, SIAM J. Numer. Anal 28 (1991) 496–511) of order six. Numerical and theoretical results in some problems of the plate deflection theory show that this new approach is more efficient compared with the well-known classical sixth order Runge–Kutta Verner method.     

Exponentially-fitted Runge-Kutta-Nyström method for the numerical solution of initial-value problems with oscillating solutions

A new approach for constructing efficient Runge-Kutta-Nyström methods is introduced in this paper. Based on this new approach a new exponentially-fitted Runge-Kutta-Nyström fourth-algebraic-order method is obtained for the numerical solution of initial-value problems with oscillating solutions. The new method has an extended interval of periodicity. Numerical illustrations on well-known initial-value problems with oscillating solutions indicate that the new method is more efficient than other ones.     

Review of wavelet methods for the solution of reaction–diffusion problems in science and engineering

Wavelet method is a recently developed tool in applied mathematics. Investigation of various wavelet methods, for its capability of analyzing various dynamic phenomena through waves gained more and more attention in engineering research. Starting from ‘offering good solution to differential equations’ to capturing the nonlinearity in the data distribution, wavelets are used as appropriate tools at various places to provide good mathematical model for scientific phenomena, which are usually modeled through linear or nonlinear differential equations. Review shows that the wavelet method is efficient and powerful in solving wide class of linear and nonlinear reaction–diffusion equations. This review intends to provide the great utility of wavelets to science and engineering problems which owes its origin to 1919. Also, future scope and directions involved in developing wavelet algorithm for solving reaction–diffusion equations are addressed.     

A boundary value approach to the numerical solution of initial value problems by multistep methos †

A boundary value appraoch to the numerical solution of initial value problems by means of linear multistep methods is presented. This theory is based on the study of linear difference equations when their general solution is computed by imposing boundary conditions. All the main stability and convergence properties of the obtained methods are investigated abd compared to those of the classical multistep methods. Then, as an example, new itegration formulas, called extended trapezoidal rules, are derived. For any order they have the same stability properties (in the sense of the definitions given in this paper) of the trapezoidal rule, which is the first method in this class. Some numerical examples are presented to confirm the theoretical expectations and to allow us to trust a future code based on boundary value methods.     

On the solution of affine generalized Nash equilibrium problems with shared constraints by Lemke’s method

Affine generalized Nash equilibrium problems (AGNEPs) represent a class of non-cooperative games in which players solve convex quadratic programs with a set of (linear) constraints that couple the players’ variables. The generalized Nash equilibria (GNE) associated with such games are given by solutions to a linear complementarity problem (LCP). This paper treats a large subclass of AGNEPs wherein the coupled constraints are shared by, i.e., common to, the players. Specifically, we present several avenues for computing structurally different GNE based on varying consistency requirements on the Lagrange multipliers associated with the shared constraints. Traditionally, variational equilibria (VE) have been amongst the more well-studied GNE and are characterized by a requirement that the shared constraint multipliers be identical across players. We present and analyze a modification to Lemke’s method that allows us to compute GNE that are not necessarily VE. If successful, the modified method computes a partial variational equilibrium characterized by the property that some shared constraints are imposed to have common multipliers across the players while other are not so imposed. Trajectories arising from regularizing the LCP formulations of AGNEPs are shown to converge to a particular type of GNE more general than Rosen’s normalized equilibrium that in turn includes a variational equilibrium as a special case. A third avenue for constructing alternate GNE arises from employing a novel constraint reformulation and parameterization technique. The associated parametric solution method is capable of identifying continuous manifolds of equilibria. Numerical results suggest that the modified Lemke’s method is more robust than the standard version of the method and entails only a modest increase in computational effort on the problems tested. Finally, we show that the conditions for applying the modified Lemke’s scheme are readily satisfied in a breadth of application problems drawn from communication networks, environmental pollution games, and power markets.     

How is cultural influence manifested in the formation of mathematics textbooks? A comparative case study of resource book series between Shanghai and England

Focusing on issues about the development of mathematics textbooks from a cultural perspective, this study examined a widely-used curriculum resource series, One Lesson One Exercise, published in China, and its adapted English series, published in the UK, to explore how cultural influence is manifested in the two series of resource books. For the study we established a conceptual framework classifying culture into six types in relation to people’s beliefs, values and ways of interacting about them, for data collection and analysis. The results indicate that there exist considerable differences between the Chinese and the English series that are related to cultural factors. It appears that, to a large extent, culture plays an essential role in the development of mathematics textbooks. Concerning the different types of culture, the results show that most adaptations between the Chinese series and the English series are related to ‘ways of behaving and customs’ and ‘artifacts, flora and fauna’, followed by ‘identities’ and ‘geography’, and the least are related to ‘organisations’ and ‘history’. Based on the study, we argue that the relevance and importance of culture to the development of mathematics textbooks must not be underestimated, and more research in this direction is needed.     

Uniqueness and Hölder continuity of the solution to multivalued equilibrium problems in metric spaces

Multivalued equilibrium problems in general metric spaces are considered. Uniqueness and Hölder continuity of the solution are established under Hölder continuity and relaxed Hölder-related monotonicity assumptions. The assumptions appear to be weaker and the inclusion to be properly stronger than that of the recent results in the literature. Furthermore, our theorems include completely some known results for variational inequalities in Hilbert spaces, which were demonstrated via geometrical techniques based on the orthogonal projection in Hilbert spaces and the linearity of the canonical pair\(\langle .,.\rangle\).     

Solution representations of percentage change problems: the pre-service primary teachers’ mathematical thinking and reasoning

Solution representations can reveal how problem solvers communicate mathematical thinking and reasoning in problem-solving process. The present study examined the solution representations used by 20 pre-service teachers for the percentage change problems. The pre-service teachers were invited to solve a combination of simple and complex percentage change problems. The score for the majority of simple problems was 75% or above, but the score for the complex problems was below 75%. The highest percentage error occurred when the pre-service teachers encountered a percentage greater than 100% in the percentage change problems. Irrespective of their level of mathematics qualifications, the equation approach demonstrating two-step problem-solving process was the predominant strategy adopted by the pre-service teachers. The equation approach imposes low cognitive load and, therefore, is more accessible and efficient than the unitary approach. A few pre-service teachers used the unitary approach. The findings indicate that the pre-service teachers possessed relevant mathematical knowledge for percentage change problems. Furthermore, the inclusion of the equation approach in mathematics textbooks would provide an alternative perspective regarding the teaching and learning of percentage change problems.     

How well prepared are the teachers of tomorrow? An examination of prospective mathematics teachers' pedagogical content knowledge

There is a growing emphasis in the teaching profession on pedagogical content knowledge (PCK) as an important knowledge component. The study reported in this article investigates Turkish prospective mathematics teachers’ mathematics teaching knowledge in the numbers content domain. A series of 10 open-ended scenario-type questions were adopted to challenge 83 prospective mathematics teachers’ knowledge of the learner and presentation of content in the context of PCK. The participants’ responses were analysed by means of rubrics and scoring guides developed by the researchers. The results showed that many of the future teachers performed well in determining what misconceptions students might express in the given scenarios. However, a majority of the participants performed poorly on presentation of content in terms of instructional strategies. In line with these results, the authors offer some suggestions for teacher training programmes.     

A family of new smoothing functions and?a?nonmonotone smoothing Newton method for?the?nonlinear complementarity problems

In this paper, based on a p-norm with p being any fixed real number in the interval (1,+??), we introduce a family of new smoothing functions, which include the smoothing symmetric perturbed Fischer function as a special case. We also show that the functions have several favorable properties. Based on the new smoothing functions, we propose a nonmonotone smoothing Newton algorithm for solving nonlinear complementarity problems. The proposed algorithm only need to solve one linear system of equations. We show that the proposed algorithm is globally and locally superlinearly convergent under suitable assumptions. Numerical experiments indicate that the method associated with a smaller p, for example p=1.1, usually has better numerical performance than the smoothing symmetric perturbed Fischer function, which exactly corresponds to p=2.     

Local solvability and blowup of the solution of the Rosenau-Bürgers equation with different boundary conditions

We consider several models of initial boundary-value problems for the Rosenau-Bürgers equation with different boundary conditions. For each of the problems, we prove the unique local solvability in the classical sense, obtain a sufficient condition for the blowup regime, and estimate the time of the solution decay. The proof is based on the well-known test-function method.     

A closed-form solution of the Black–Litterman model with conditional value at risk

We consider a portfolio optimization problem of the Black–Litterman type, in which we use the conditional value-at-risk (CVaR) as the risk measure and we use the multi-variate elliptical distributions, instead of the multi-variate normal distribution, to model the financial asset returns. We propose an approximation algorithm and establish the convergence results. Based on the approximation algorithm, we derive a closed-form solution of the portfolio optimization problems of the Black–Litterman type with CVaR.     

A study about the solution of convection–diffusion–reaction equation with Danckwerts boundary conditions by analytical,method of lines and Crank–Nicholson techniques

We compare different solutions of the convection–diffusion–reaction problem with Danckwerts boundary conditions. Analytical solution is found, and method of lines and Crank–Nicholson method are described, applied, and compared for different values of Péclet and Damköhler numbers. The eigenvalues and eigenfunctions have been obtained for all the possible values of the dimensionless parameters. And the analytical expression of the concentration has been calculated with the optimum number of terms in the Fourier expansion.     

Error estimate of the homogenization solution for elliptic problems with small periodic coefficients on L~∞(Ω)

In this paper,we discuss the multi-scale homogenization theory for the second order elliptic problems with small periodic coefficients of the form xi(aij(xε) uεx(jx)) = f(x).Assuming n = 2 and u0 ∈ W 1,∞(Ω),we present an error estimate between the homogenization solution u0(x) and the exact solution uε(x) on the Sobolev space L∞(Ω).     

A study of the influence of center conditions on the domain of parameters of Newton’s method by using recurrence relations

This paper focuses on the importance of center conditions on the first derivative of the operator involved in the solution of nonlinear equations by Newton’s method when the semilocal convergence of the method is established from the technique of recurrence relations.