Comparing the development of the multiplication of fractions in Turkish and American textbooks

This study analyzed the methods used to teach the multiplication of fractions in Turkish and American textbooks. Two Turkish textbooks and two American textbooks, Everyday Mathematics (EM) and Connected Mathematics 3 (CM), were analyzed. The analyses focused on the content and the nature of the mathematical problems presented in the textbooks. The findings of the study showed that the American textbooks aimed at developing conceptual understanding first and then procedural fluency, whereas the Turkish textbooks aimed at developing both concurrently. The American textbooks provided more opportunities for different computational strategies. The solutions to most problems in all textbooks required a single computational step, a numerical answer, and procedural knowledge. Furthermore, compared with the Turkish textbooks, the American textbooks contained a greater number of problems that required high-level cognitive skills such as mathematical reasoning.     

The introduction of real numbers in secondary education: an institutional analysis of textbooks

In this paper we analyse the introduction of irrational and real numbers in secondary textbooks, and specifically the propositions on how these should be taught, in a sample of Brazilian textbooks used in state schools and approved by the Ministry of Education. The analyses discussed in this paper follow an institutional perspective (using Chevallard's Anthropological Theory of Didactics). Our results indicate that the notion of irrational number is generally introduced on the basis of the decimal representation of numbers, and that the mathematical need for the construction of the field of real numbers remains unclear in the textbooks. It seems that textbooks used in secondary teaching institutions develop mathematical organisations which focus on the practical block.     

The initial treatment of the concept of function in the selected secondary school mathematics textbooks in the US and China

In order to provide insight into cross-national differences in students’ achievement, this study compares the initial treatment of the concept of function sections of Chinese and US textbooks. The number of lessons, contents, and mathematical problems were analyzed. The results show that the US curricula introduce the concept of function one year earlier than the Chinese curriculum and provide strikingly more problems for students to work on. However, the Chinese curriculum emphasizes developing both concepts and procedures and includes more problems that require explanations, visual representations, and problem solving in worked-out examples that may help students formulate multiple solution methods. This result could indicate that instead of the number of problems and early introduction of the concept, the cognitive demands of textbook problems required for student thinking could be one reason for differences in American and Chinese students’ performances in international comparative studies. Implications of these findings for curriculum developers, teachers, and researchers are discussed.     

Real meromorphic functions and linear differential polynomials

We determine all real meromorphic functions f in the plane such that f has finitely many zeros, the poles of f have bounded multiplicities, and f and F have finitely many non-real zeros, where F is a linear differential polynomial given by F = f (k) +Σk-1j=0ajf(j) , in which k≥2 and the coefficients aj are real numbers with a0≠0.     

Stability analysis of positive switched systems via joint linear copositive Lyapunov functions

In this paper, we study the asymptotic stability of continuous-time positive switched linear systems for the case when each subsystem is only stable. By using the so-called “joint linear copositive Lyapunov function” (JLCLF) generalizing the common linear copositive Lyapunov function, we show that the system remains asymptotically stable under appropriate switching if it has a JLCLF. Then, the main result is extended to positive switched linear systems with time delay.     

Large linear manifolds of noncontinuable boundary-regular holomorphic functions

We prove in this paper that if G is a domain in the complex plane satisfying appropriate topological or geometrical conditions, then there exists a large (dense or closed infinite-dimensional) linear submanifold of boundary-regular holomorphic functions on G all of whose nonzero members are not continuable across any boundary point of G.     

Meromorphic iterative roots of linear fractional functions

Iterative root problem can be regarded as a weak version of the problem of embedding a homeomorphism into a flow. There are many results on iterative roots of monotone functions. However, this problem gets more diffcult in non-monotone cases. Therefore, it is interesting to find iterative roots of linear fractional functions (abbreviated as LFFs), a class of non-monotone functions on ℝ. In this paper, iterative roots of LFFs are studied on ℂ. An equivalence between the iterative functional equation for non-constant LFFs and the matrix equation is given. By means of a method of finding matrix roots, general formulae of all meromorphic iterative roots of LFFs are obtained and the precise number of roots is also determined in various cases. As applications, we present all meromorphic iterative roots for functions z and 1/z. This work was supported by the Youth Fund of Sichuan Provincial Education Department of China (Grant No. 07ZB042)     

Asymptotically linear solutions of differential equations via Lyapunov functions

We discuss the existence of solutions with oblique asymptotes to a class of second order nonlinear ordinary differential equations by means of Lyapunov functions. The approach is new in this field and allows for simpler proofs of general results regarding Emden-Fowler like equations.     

Lower bounds for linear forms in values of certain hypergeometric functions

By using Padé approximations of the first kind, a lower bound for the modulus of a linear form with integer coefficients in the values of certain hypergeometric functions at a rational point are obtained. This estimate depends on all the coefficients of the linear form. Translated fromMatematicheskie Zametki, Vol. 67, No. 3, pp. 441–451, March, 2000.     

Entire functions, analytic continuation, and the fractional parts of a linear function

The main result of the paper is as follows.Theorem. Suppose that G(z) is an entire function satisfying the following conditions: 1) the Taylor coefficients of the function G(z) are nonnegative: 2) for some fixed C>0 and A>0 and for |z|>R₀, the following inequality holds:

Generalized prolate spheroidal wave functions for offset linear canonical transform in Clifford analysis

Prolate spheroidal wave functions (PSWFs) possess many remarkable properties. They are orthogonal basis of both square integrable space of finite interval and the Paley–Wiener space of bandlimited functions on the real line. No other system of classical orthogonal functions is known to obey this unique property. This raises the question of whether they possess these properties in Clifford analysis. The aim of the article is to answer this question and extend the results to more flexible integral transforms, such as offset linear canonical transform. We also illustrate how to use the generalized Clifford PSWFs (for offset Clifford linear canonical transform) we derive to analyze the energy preservation problems. Clifford PSWFs is new in literature and has some consequences that are now under investigation. Copyright © 2012 John Wiley & Sons, Ltd.     

An analysis of the amount and characteristics of writing prompts in Grade 3 mathematics student books

Curriculum guidelines and professional organizations’ recommendations lack details about how often and how much students should write in mathematics and what characteristics should define their writing. This study presents an analytic framework that addresses how often students are prompted in student mathematics books to write, how much they may be encouraged to write, and the characteristics of the writing prompts. Consequently, 2,095 writing prompts in student books across 10 comprehensive Grade 3 resources were analyzed. Findings indicate a marked variation in how often and how much students are positioned to write. Most prompts have students explain what they did to solve a problem and why about number concepts, with most pressing for procedures. The greatest percentage of prompts had students write about their own solutions and do not urge them to include specific writing features.     

A unifying theory of exactness of linear penalty functions

In this article, we develop a theory of exact linear penalty functions that generalizes and unifies most of the results on exact penalization existing in the literature. We discuss several approaches to the study of both locally and globally exact linear penalty functions, and obtain various necessary and sufficient conditions for the exactness of a linear penalty function. We pay more attention than usual to necessary conditions, which allows us to deeply understand the exact penalty technique.     

Quadratic approximation of penalty functions for solving large-scale linear programs

Using the least squares, modified Lagrangian function, and some other methods as examples, the capabilities of the new optimization technique based on the quadratic approximation of penalty functions that has been recently proposed by O. Mangasarian for a special class of linear programming problems are demonstrated. The application of this technique makes it possible to use unified matrix operations and standard linear algebra packages (including parallel ones) for solving large-scale problems with sparse strongly structured constraint matrices. With this technique, the computational schemes of some well-known algorithms can take an unexpected form.     

Cognitive opportunities in textbooks: the cases of grade four and eight textbooks in Israel

The cognitive domain in mathematics, defined as thinking and understanding in the process of learning mathematics, is a main focus of curricula in many countries. This study explores breadth and depth of understanding as addressed in mathematics textbooks certified as aligned to Israeli national mathematics curricula. We compare opportunities for students to engage with mathematics requiring different types and levels of understanding provided by the tasks in mathematics textbooks. Comparison of two fourth grade and two eighth grade mathematics textbooks showed significant differences in the opportunities to learn in the cognitive domain that each provides. These differences can be quantified; the quantification defines the cognitive demand of the textbook. The cognitive demand of the four textbooks varies. This reveals a potential source of inequity in students’ opportunities to learn mathematics. Results should prompt discussion around standardization and alignment of textbooks to the cognitive goals of the curriculum.     

A comparative analysis of linear fitting for non-linear functions on optimization: A case study: Air pollution problems

A very frequent problem in advanced mathematical programming models is the linear approximation of convex and non-convex non-linear functions in either the constraints or the objective function of an otherwise linear programming problem. In this paper, based on a model that has been developed for the evaluation and selection of pollutant emission control policies and standards, we shall study several ways of representing non-linear functions of a single argument in mixed integer, separable and related programming terms. Thus we shall study the approximations based on piecewise constant, piecewise adjacent, piecewise non-adjacent additional and piecewise non-adjacent segmented functions. In each type of modelization we show the problem size and optimization results of using the following techniques: separable programming, mixed integer programming with Special Ordered Sets of type 1, linear programming with Special Ordered Sets of type 2 and mixed integer programming using strategies based on the quasi-integrality of the binary variables.     

Delayed perturbation of Mittag‐Leffler functions and their applications to fractional linear delay differential equations

In this paper, we propose a delayed perturbation of Mittag‐Leffler type matrix function, which is an extension of the classical Mittag‐Leffler type matrix function and delayed Mittag‐Leffler type matrix function. With the help of the delayed perturbation of Mittag‐Leffler type matrix function, we give an explicit formula of solutions to linear nonhomogeneous fractional delay differential equations.     

MATHEMATICAL ANALYSIS OF SOME NEURAL NETWORKS FOR SOLVING LINEAR AND QUADRATIC PROGRAMMING

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