Problem solving in the borderland between mathematics and physics

The article addresses the problématique of where mathematization is taught in the educational system, and who teaches it. Mathematization is usually not a part of mathematics programs at the upper secondary level, but we argue that physics teaching has something to offer in this respect, if it focuses on solving so-called unformalized problems, where a major challenge is to formalize the problems in mathematics and physics terms. We analyse four concrete examples of unformalized problems for which the formalization involves different order of mathematization and applying physics to the problem, but all require mathematization. The analysis leads to the formulation of a model by which we attempt to capture the important steps of the process of solving unformalized problems by means of mathematization and physicalization.     

Game and decision theory in mathematics education: epistemological, cognitive and didactical perspectives

In the 1950s, game and decision theoretic modeling emerged—based on applications in the social sciences—both as a domain of mathematics and interdisciplinary fields. Mathematics educators, such as Hans Georg Steiner, utilized game theoretical modeling to demonstrate processes of mathematization of real world situations that required only elementary intuitive understanding of sets and operations. When dealing with n-person games or voting bodies, even students of the 11th and 12th grade became involved in what Steiner called the evolution of mathematics from situations, building of mathematical models of given realities, mathematization, local organization and axiomatization. Thus, the students could participate in processes of epistemological evolutions in the small scale. This paper introduces and discusses the epistemological, cognitive and didactical aspects of the process and the roles these activities can play in the learning and understanding of mathematics and mathematical modeling. It is suggested that a project oriented study of game and decision theory can develop situational literacy, which can be of interest for both mathematics education and general education.     

On the efficiency of Newton and Broyden numerical methods in the investigation of the regular polygon problem of (N + 1) bodies

Numerical methods of finding the roots of a system of non-linear algebraic equations are treated in this paper. This paper attempts to give an answer to the selection of the most efficient method in a complex problem of Celestial Dynamics, the so-called ring problem of (N + 1) bodies. We apply Newton and Broyden’s method to these problems and we investigate, by means of their use, the planar equilibrium points, the five equilibrium zones, which are symbolized by A₁, A₂, B, C₂, and C₁ (by order of appearance from the center O to the periphery of the imaginary circle on which the primaries lie) [T.J. Kalvouridis, A planar case of the N + 1 body problem: the ring problem. Astrophys. Space Sci. 260 (3) (1999) 309-325], and the attracting regions of the system. The efficiency of these methods is studied through a comparative process. The obtained results are demonstrated in figures and are discussed.     

Investigating high-school students’ reasoning strategies when they solve linear equations

A cross-curricular structured-probe task-based clinical interview study with 44 pairs of third-year high-school mathematics students, most of whom were high achieving, was conducted to investigate their approaches to a variety of algebra problems. This paper presents results from one problem that involved solving a set of three linear equations of the form ax ± b = cx ± d. One of the equations had a unique solution, another was an identity, and the third resulted in a contradiction. Most pairs of students were able to solve the equation resulting in a unique solution using symbol-manipulation algorithms or by a graphical approach. On the equations resulting in an identity or a contradiction, most student pairs did not know how to interpret the results of their symbol manipulation, and few turned to another representation when symbol manipulation failed them. The constructs of versatility and adaptability [Sfard, A., & Linchevski, L. (1994). The gains and the pitfalls of reification—The case of algebra. Educational Studies in Mathematics, 26, 191-228] guided our interpretation of the findings.     

Simple thinking using complex math vs. complex thinking using simple math—A study using model eliciting activities to compare students' abilities in standardized tests to their modelling abilities

Traditional mathematics assessments often fail to identify students who can powerfully and effectively apply mathematics to real-world problems, and many students who excel on traditional assessments often struggle to implement their mathematical knowledge in real-world settings (Lesh & Sriraman, 2005a). This study employs multi-tier design-based research methodologies to explore this phenomenon from a models and modeling perspective. At the researcher level, a Model Eliciting Activity MEA) was developed as a means to measure student performance on a complex real-world task. Student performance data on this activity and on traditional pre- and post-tests were collected from approximately 200 students enrolled in a second semester calculus course in the Science and Engineering department of the University of Southern Denmark during the winter of 2005. The researchers then used the student solutions to the MEA to develop tools for capturing and assessing the strengths and weaknesses of the mathematical models present in these solutions. Performance on the MEA, pre- and post-test were then analyzed both quantitatively and qualitatively to identify trends in the subgroups corresponding to those described by lesh and Sriraman.     

A note on the complexity of flow shop scheduling with transportation constraints

This note investigates two-machine flow shop scheduling with transportation constraints to minimize makespan. Recently, Soukhal et al. [A. Soukhal, A. Oulamara, P. Martineau, Complexity of flow shop scheduling problems with transportation constraints, European Journal of Operational Research 161 (2005) 32–41] proved that this problem is strongly NP-hard when the capacity of the truck is limited to two or three parts. The considered problem with blocking constraints is also proved to be strongly NP-hard by Soukhal et al. Unfortunately, their proofs contain mistakes. We point out their proofs’ invalidity and then show that, when the capacity of the truck is limited to two parts, the problem is binary NP-hard, and when the capacity of the truck is limited to three parts the problem is strongly NP-hard even if the jobs have a common processing time on machine one and all jobs have the same transportation time. We show also that the last result can be generalized to any fixed c (c ? 3) parts.     

Beyond realistic considerations: modeling conceptions and controls in task examples with simple word problems

Using simple word problems, we analyze possible teacher conceptions on the process of problem solving, its goals and the choices that a problem solver can make in problem mathematization. We identify several possible teacher conceptions that would be responsible for the different didactical contracts that teachers create in the mathematics class. Using especially chosen and designed task examples, we demonstrate the diagnosis of teacher own controls in solving problems and in evaluating problem solutions. We also discuss characteristics of task examples that might promote a shift from a problem solving perspective to a modeling perspective that goes beyond merely accepting alternative solutions due to realistic considerations. This shift in perspective would be exhibited through a new understanding of the process of fitting mathematical models in problem situations.     

Resolution, in L-spaces, of transmission problems set in an unbounded domains

The goal of this work is to give a complete study of some abstract transmission problems (P^δ), for every δ > 0, set in unbounded domain composed of a half-line ]−∞, 0[ and a thin layer ]0, δ[. Existence and uniqueness results are obtained for strict solutions in UMD Banach spaces, by using essentially the semigroup theory and the Dore-Venni’s Theorem given in [8].     

Ekeland’s variational principle for vectorial multivalued mappings in a uniform space

In this paper, we establish Ekeland’s variational principle and an equilibrium version of Ekeland’s variational principle for vectorial multivalued mappings in the setting of separated, sequentially complete uniform spaces. Our approaches and results are different from those in Chen et al. (2008), Hamel (2005), and Lin and Chuang (2010) [13], [14] and [15]. As applications of our results, we study vectorial Caristi’s fixed point theorems and Takahashi’s nonconvex minimization theorems for multivalued mappings and their equivalent forms in a separated, sequentially complete uniform space. We also apply our results to study maximal element theorems, which are unified methods of several variational inclusion problems. Our results contain many known results in the literature Fang (1996) [21], and will have many applications in nonlinear analysis.     

Inventory models for perishable items with inventory level dependent demand rate

This paper presents inventory models for perishable items with inventory level dependent demand rate. The models with and without backlogging are studied. In the backlogging model, it is assumed that the backlogging rate is dependent on the waiting time and the amount of products already backlogged simultaneously. Two cases that holding inventory is profitable or not are studied, respectively. The smallest shelf space to ensure shortage not occur when holding inventory is not profitable is obtained. In the model without backlogging, it is assumed that the remaining stock at the end of the inventory cycle is disposed of with salvage value. The necessary and sufficient conditions for the existence and uniqueness of the optimal solution of these models are investigated. At last, some numerical examples are presented to illustrate the effectiveness of the proposed model. The model in this paper is generalization of present ones. In particularly, the model is reduced to Padmanabhan and Vrat’s when δ₁ = 0, and Dye and Ouyang’s when δ₂ = 0. If S = s and δ₂ = 0, it is Chang, Goyal and Teng’s model.     

Control decisions and personal beliefs: their effect on solving mathematical problems

The main purpose of this paper is to discuss how college students enrolled in a college level elementary algebra course exercised control decisions while working on routine and non-routine problems, and how their personal belief systems shaped those control decisions. In order to prepare students for success in mathematics we as educators need to understand the process steps they use to solve homework or examination questions, in other words, understand how they “do” mathematics. The findings in this study suggest that an individual’s belief system impacts how they approach a problem. Lack of confidence and previous lack of success combined to prompt swift decisions to stop working. Further findings indicate that students continue with unsuccessful strategies when working on unfamiliar problems due to a perceived dependence of solution strategies to specific problem types. In this situation, the students persisted in an inappropriate solution strategy, never reaching a correct solution. Control decisions concerning the pursuit of alternative strategies are not an issue if the students are unaware that they might need to make different choices during their solutions. More successful control decisions were made when working with familiar problems.     

On the solutions of a functional equation arising from multiplication of quantum integers

This paper is the first of several papers in which we prove, for the case where the fields of coefficients are of characteristic zero, four open problems posed in the work of Melvyn Nathanson (2003) [1] concerning the solutions of a functional equation arising from multiplication of quantum integers q[n]=qn−1+qn−2+?+q+1. In this paper, we prove one of the problems. The next papers, namely [002], [003] and [004] by Lan Nguyen, contain the solutions to the other 3 problems.     

Use of words and visuals in modelling context of annual plant

This study looks at the various verbal and non-verbal representations used in a process of modelling the number of annual plants over time. Analysis focuses on how various representations such as words, diagrams, letters and mathematical equations evolve in the mathematization process of the modelling context. Our results show that (1) visual representations such as flowcharts are used not only in the process to symbolization, but also used in the justification of symbols, (2) some of the visual representations serve as a bridge between the words in the problem context and the symbols that represent the mathematical equations of the number of annual plants and (3) words and context help to introduce visual representations and symbols. Also, once students come up with the visual representations and symbols, they show better understanding about words used in the problem context. These observations imply that the modelling and mathematization process is not just one-directional and linear from words describing real-life situations to the symbols in mathematical equations and expressions. Rather, the mathematization can be promoted through using other visuals that help make this transition smooth by organizing the given information in a way that can be used towards mathematization.     

Exploring college students’ mental representations of inferential statistics

We report a case study that explored how three college students mentally represented the knowledge they held of inferential statistics, how this knowledge was connected, and how it was applied in two problem solving situations. A concept map task and two problem categorization tasks were used along with interviews to gather the data. We found that the students’ representations were based on incomplete statistical understanding. Although they grasped various concepts and inferential tests, the students rarely linked key concepts together or to tests nor did they accurately apply that knowledge to categorize word problems. We suggest that one reason the students had difficulty applying their knowledge is that it was not sufficiently integrated. In addition, we found that varying the instruction for the categorization task elicited different mental representations. One instruction was particularly effective in revealing students’ partial understandings. This finding suggests that modifying the task format as we have done could be a useful diagnostic tool.     

An O(mn log (nU)) time algorithm to solve the feasibility problem

The phase I maximum flow and most positive cut methods are used to solve the feasibility problem. Both of these methods take one maximum flow computation. Thus the feasibility problem can be solved using maximum flow algorithms. Let n and m be the number of nodes and arcs, respectively. In this paper, we present an algorithm to solve the feasibility problem with integer lower and upper bounds. The running time of our algorithm is O(mn log (nU)), where U is the value of maximum upper bound. Our algorithm improves the O(m² log (nU))-time algorithm in [12]. Hence the current algorithm improves the running time in [12] by a factor of n. Sleator and Goldberg’s algorithm is one of the well-known maximum flow algorithms, which runs in O(mn log n) time, see [5]. Under similarity assumption [11], our algorithm runs in O(mn log n) time, which is the running time of Sleator and Goldberg’s algorithm. The merit of our algorithm is that, in the case of infeasibility of the given network, it not only diagnoses infeasibility but also presents some information that is useful to modeler in estimating the maximum cost of adjusting the infeasible network.     

An analytical comparison of the LP relaxations of integer models for the k-club problem

Given an undirected graph G = (V, E), a k-club is a subset of nodes that induces a subgraph with diameter at most k. The k-club problem is to find a maximum cardinality k-club. In this study, we use a linear programming relaxation standpoint to compare integer formulations for the k-club problem. The comparisons involve formulations known from the literature and new formulations, built in different variable spaces. For the case k = 3, we propose two enhanced compact formulations. From the LP relaxation standpoint these formulations dominate all other compact formulations in the literature and are equivalent to a formulation with a non-polynomial number of constraints. Also for k = 3, we compare the relative strength of LP relaxations for all formulations examined in the study (new and known from the literature). Based on insights obtained from the comparative study, we devise a strengthened version of a recursive compact formulation in the literature for the k-club problem (k > 1) and show how to modify one of the new formulations for the case k = 3 in order to accommodate additional constraints recently proposed in the literature.     

Problem F2∥Cmax with forbidden jobs in the first or last position is easy

Saadani et al. [N.E.H. Saadani, P. Baptiste, M. Moalla, The simple F2∥C_max with forbidden tasks in first or last position: A problem more complex that it seems, European Journal of Operational Research 161 (2005) 21–31] studied the classical n-job flow shop scheduling problem F2∥C_max with an additional constraint that some jobs cannot be placed in the first or last position. There exists an optimal job sequence for this problem, in which at most one job in the first or last position is deferred from its position in Johnson’s [S.M. Johnson, Optimal two- and three-stage production schedules with setup times included, Naval Research Logistics Quarterly 1 (1954) 61–68] permutation. The problem was solved in O(n²) time by enumerating all candidate job sequences. We suggest a simple O(n) algorithm for this problem provided that Johnson’s permutation is given. Since Johnson’s permutation can be obtained in O(n log n) time, the problem in Saadani et al. (2005) can be solved in O(n log n) time as well.     

A model for heuristic coordination of real life distribution inventory systems with lumpy demand

This paper presents an approximation model for optimizing reorder points in one-warehouse N-retailer inventory systems subject to highly variable lumpy demand. The motivation for this work stems from close cooperation with a supply chain management software company, Syncron International, and one of their customers, a global spare parts provider. The model heuristically coordinates the inventory system using a near optimal induced backorder cost at the central warehouse. This induced backorder cost captures the impact that a reorder point decision at the warehouse has on the retailers’ costs, and decomposes the multi-echelon problem into solving N + 1 single-echelon problems. The decomposition framework renders a flexible model that is computationally and conceptually simple enough to be implemented in practice.     

Finite difference methods for half inverse Sturm-Liouville problems

We investigate finite difference solution of the Hochstadt-Lieberman problem for a Sturm-Liouville operator defined on (0, π): given the value of the potential q on (c, π), where 0 < c < π, use eigenvalues to estimate q on (0, c). Our methods use an asymptotic correction technique of Paine, de Hoog and Anderssen, and its extension to Numerov’s method for various boundary conditions. In the classical case c = π/2, Numerov’s method is found to be particularly effective. Since eigenvalue data is scarce in applications, we also examine stability problems associated with the use of the extra information on q when c < π/2, and give some suggestions for further research.     

Optimal retailer’s replenishment decisions in the EPQ model under two levels of trade credit policy

The main purpose of this paper is to investigate the optimal retailer’s replenishment decisions under two levels of trade credit policy within the economic production quantity (EPQ) framework. We assume that the supplier would offer the retailer a delay period and the retailer also adopts the trade credit policy to stimulate his/her customer demand to develop the retailer’s replenishment model under the replenishment rate is finite. Furthermore, we assume that the retailer’s trade credit period offered by supplier M is not shorter than the customer’s trade credit period offered by retailer N (M ? N). Since the retailer cannot earn any interest in this situation, M < N.