Exploring Diverse Accounts of Teacher Knowledge

This article explores the nature of teacher knowledge as it is portrayed by Schoenfeld's model of teaching. We attempt to situate Schoenfeld's work in the field of teacher knowledge and to elucidate the contribution that he makes to the growing body of research in this area. Towards this end, we explore two related issues. First, we distinguish between claims about the form of teacher knowledge and claims about the content of teacher knowledge. Second, we propose two families of theories of teacher knowledge, where each family shares common phenomena, methods, and theoretical forms. We argue that these two families capture much of the diversity that exists in the literature on teacher knowledge today. Our goal is to begin to develop a theoretical approach that will not only allow us to situate Schoenfeld's research, but that will also help us to compare existing theories with each other.     

Lower bounds for nonlinear assignment problems using many body interactions

This paper concerns lower bounding techniques for the general α-adic assignment problem. The nonlinear objective function is linearized by the introduction of additional variables and constraints, thus yielding a mixed integer linear programming formulation of the problem. The concept of many body interactions is introduced to strengthen this formulation and incorporated in a modified formulation obtained by lifting the original representation to a higher dimensional space. This process involves two steps — (i) addition of new variables and constraints and (ii) incorporation of the new variables in the objective function. If this lifting process is repeated β times on an α-adic assignment problem along with the incorporation of higher order interactions, it results in the mixed-integer formulation of an equivalent (α + β)-adic assignment problem. The incorporation of many body interactions in the higher dimensional formulation improves its degeneracy properties and is also critical to the derivation of decomposition methods for the solution of these large scale mathematical programs in the higher dimensional space. It is shown that a lower bound to the optimal solution of the corresponding linear programming relaxation can be obtained by dualizing a subset of constraints in this formulation and solving O(N^2(α+β−1)) linear assignment problems, whose coefficients depend on the dual values. Moreover, it is proved that the optimal solution to the LP relaxation is obtained if we use the optimal duals for the solution of the linear assignment problems. This concept of many body interactions could be applied in designing algorithms for the solution of formulations obtained by lifting general MILP's. We illustrate all these concepts on the quadratic assignment problems With these decomposition bounds, we have found the provably optimal solutions of two unsolved QAP's of size 32 and have also improved upon existing lower bounds for other QAP's.     

Discovering and addressing errors during mathematics problem-solving—A productive struggle?

The present study investigates students’ struggles when encountering errors in problem-solving. The focus is students’ problem-solving activities that lead to productive struggle and what the students might gain therefrom. Twenty-four students between the ages of 16 and 17 worked in pairs to solve a linear function problem using GeoGebra, a dynamic software application. Data in the form of recorded conversations, computer activities and post-interviews were analyzed using Hiebert and Grouws’ (2007. Second handbook of research on mathematics teaching and learning (Vol. 1). 404) concept of productive struggles and Schoenfeld's (1985. Mathematical problem solving: ERIC) framework for problem-solving. The study showed that all students made errors concerning incorrect prior knowledge and erroneously constructed new knowledge. All participants engaged in superficial, unproductive struggles moving between a couple of Schoenfeld's episodes. However, a majority of the students managed to transform their efforts into productive struggle. They engaged in several of Schoenfeld's episodes and succeeded in reconstructing useful prior knowledge and constructing correct new knowledge—i.e., solving the problem.     

A stable approach to Newton's method for general mathematical programming problems inR n

The usual approach to Newton's method for mathematical programming problems with equality constraints leads to the solution of linear systems ofn +m equations inn +m unknowns, wheren is the dimension of the space andm is the number of constraints. Moreover, these linear systems are never positive definite. It is our feeling that this approach is somewhat artificial, since in the unconstrained case the linear systems are very often positive definite. With this in mind, we present an alternate Newton-like approach for the constrained problem in which all the linear systems are of order less than or equal ton. Furthermore, when the Hessian of the Lagrangian at the solution is positive definite (a situation frequently occurring), all our systems will be positive definite. Hence, in all cases, our Newton-like method offers greater numerical stability. We demonstrate that the convergence properties of this Newton-like method are superior to those of the standard approach to Newton's method. The operation count for the new method using Gaussian elimination is of the same order as the operation count for the standard method. However, if the Hessian of the Lagrangian at the solution is positive definite and we use Cholesky decomposition, then the order of the operation count for the new method is half that for the standard approach to Newton's method. This theory is generalized to problems with both equality and inequality constraints.     

A class of binary matrices preserving rank under matrix addition and its application

An open problem proposed by Safavi-Naini and Seberry in IEEE transactions on information theory (1991) can be reduced to a combinatorial problem on partitioning a subset of binary matrices. We solve the generalized Naini-Seberry's open problem by considering a certain class of binary matrices. Thus a subliminal channel ofr>1 bit capacity is systematically established for Naini-Seberry's authentication schemes. We also construct concrete examples.     

Square area integral estimates for subharmonic functions in NTA domains

In this paper, we prove a good-λ inequality between the nontangential maximal function and the square area integral of a subharmonic functionu in a bounded NTA domainD inR ⁿ . We achieve this by showing that a weighted Riesz measure ofu is a Carleson measure, with the Carleson norm bounded by a constant independent ofu. As consequences of the good-λ inequality, we obtain McConnell-Uchiyama's inequality and an analogue of Murai-Uchiyama's inequality for subharmonic functions inD.     

An analysis of parallel routing algorithm on hypercube network by employing covering problem and assignment problem

The application of Hadamard matrix to the parallel routings on the hypercube network was presented by Rabin. In this matrix, every two rows differ from each other by exactlyn/2 positions. A set ofn disjoint paths onn-dimensional hypercube network was designed using this peculiar property of Hadamard matrix. Then, the data is dispersed into n packets and these n packet are transmitted along thesen disjoint paths. In this paper, Rabin’s routing algorithm is analyzed in terms of covering problem and assignment problem. Finally, we conclude thatn packets dispersed are placed in well-distributed positions during transmission, and the randomly selected paths are almost a set ofn edge-disjoint paths with high probability.     

Incorporating the Decision-maker's Preferences in the Goal-programming Model

Many algorithms have been developed for multiple-criteria decision-making problems. Goal programming (GP) is one of these algorithms. This model is a special extension of linear programming. Usually, it is not easy for the decision-maker to choose his aspiration levels a priori. Moreover, the incommensurability of the measurement units of the various objectives creates an aggregation problem. However, in the standard GP formulation, the decision-maker is not required to arbitrate among conflicting objectives. To deal with these difficulties, we explicitly introduce the structure of the decision-maker's preferences into the GP model in order to evaluate the impact of deviations from the decision-maker's aspirations levels. Easily and naturally, the idea of a generalized criterion, as introduced in the Promethee outranking method, will be used to build this structure of preferences.     

Semiinvariants for hopf algebra actions

LetH be a Hopf algebra andM a representation or a corepresentation ofH. In this paper we study semiinvariants ofM. This notion generalizes the known concept of weight spaces in the context of representations of Lie algebras. Our best results are attained for pointed Hopf algebras, and semiinvariants which are related to the coradical filtration ofH.     

The coefficients of the Laurent series expansions of real analytic functions

Letf be a real analytic function of a real variable such that 0 is an isolated (possibly essential) singularity off. In the existing literature the coefficients of the Laurent series expansion off around 0 are obtained by applying Cauchy's integral formula to the analytic continuation off on the complex plane. Here by means of a conformal mapping we derive a formula which determines the Laurent coefficients off solely in terms of the values off and the derivatives off at a real point of analyticity off. Using a more complicated mapping, we similarly determine the coefficients of the Laurent expansion off around 0 where now 0 is a singularity off which is not necessarily isolated.     

A formal theory of generalized intermediate syllogisms

In this paper, we continue developing the formal theory of intermediate quantifiers (expressions such as most, few, almost all, a lot of, many, a great deal of, a large part of, a small part of). The theory is a fuzzy-logic formalization of the concept introduced by Peterson in his book. We will syntactically prove that 105 generalized Aristotle's syllogisms introduced in this book are valid in our theory. At the same time, we will also prove that syllogisms listed there as invalid are invalid also in our theory. Therefore, we believe that our theory provides a reasonable mathematical model of the generalized syllogistics.     

Optimal retailer's ordering policies in the EOQ model under trade credit financing

The main purpose of this note is to modify the assumption of the trade credit policy in previously published results to reflect the real-life situations. All previously published models implicitly assumed that the supplier would offer the retailer a delay period, but the retailer would not offer the trade credit period to his/her customer. In most business transactions, this assumption is debatable. In this note, we assume that the retailer also adopts the trade credit policy to stimulate his/her customer demand to develop the retailer's replenishment model. 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). Under these conditions, we model the retailer's inventory system as a cost minimization problem to determine the retailer's optimal ordering policies. Then a theorem is developed to determine efficiently the optimal ordering policies for the retailer. We deduce some previously published results of other researchers as special cases. Finally, numerical examples are given to illustrate the theorem obtained in this note.     

Banded matrices with banded inverses,II: Locally finite decomposition of spline spaces

Given two function spacesV ₀,V ₁ with compactly supported basis functionsC _i, F_i, i∈Z, respectively, such thatC _i can be written as a finite linear combination of theF _i's, we study the problem of decomposingV ₁ into a direct sum ofV ₀ and some subspaceW ofV ₁ in such a way thatW is spanned by compactly supported functions and that eachF _i can be written as a finite linear combination of the basis functions inV ₀ andW. The problem of finding such locally finite decompositions is shown to be equivalent to solving certain matrix equations involving two-slanted matrices. These relations may be reinterpreted in terms of banded matrices possessing banded inverses. Our approach to solving the matrix equations is based on factorization techniques which work under certain conditions on minors. In particular, we apply these results to univariate splines with arbitrary knot sequences.     

Some remarks on the construction of higher order algorithms in convex optimization

Consider the problem of minimizing a real functionalf. A Newton-like method requires first an approximationD(d) off(x + d)?f(x) at the current iteratex, valid for smalld to an order higher than 1, and consists in minimizingD. In this paper, we will introduce a new concept of such an approximation for convex functions without differentiability assumptions. The connections with the classical concept based on the Taylor development of a differentiablef are exhibited. The material is used to study a conceptual (nonimplementable) algorithm. Some hints are given which could lead to an implementable version.     

On the uniqueness of the Shapley value

L.S. Shapley [1953] showed that there is a unique value defined on the classD of all superadditive cooperative games in characteristic function form (over a finite player setN) which satisfies certain intuitively plausible axioms. Moreover, he raised the question whether an axiomatic foundation could be obtained for a value (not necessarily theShapley value) in the context of the subclassC (respectivelyC′, C″) of simple (respectively simple monotonic, simple superadditive) gamesalone. This paper shows that it is possible to do this. Theorem I gives a new simple proof ofShapley's theorem for the classG ofall games (not necessarily superadditive) overN. The proof contains a procedure for showing that the axioms also uniquely specify theShapley value when they are restricted to certain subclasses ofG, e.g.,C. In addition it provides insight intoShapley's theorem forD itself. Restricted toC′ orC″, Shapley's axioms donot specify a unique value. However it is shown in theorem II that, with a reasonable variant of one of his axioms, a unique value is obtained and, fortunately, it is just theShapley value again.     

Newton and the Notion of Limit

We investigate Newton's understanding of the limit concept through a study of certain proofs appearing in the Principia. We find that Newton, not Cauchy, was the first to present an epsilon argument, and that, in general, Newton's understanding of limits was clearer than commonly thought. We observe Newton's distinction between two properties easily confused, namely f/g→1 and f - g→0, we resolve a problem created by a spurious translation appearing in Cajori's revision of Motte's original translation, and we come to a deeper understanding of the well-known but less well understood Lemma XI of Section I, Book I. Copyright 2001 Academic Press.Nous examinons la notion newtonienne du concept de limite en étudiant certaines preuves qui apparaissent dans les Principia. Nous découvrons que Newton, et non pas Cauchy, a été le premier à présenter un argument d'epsilon et que, en général, la compréhension newtonienne de la limite était bien plus lucide qu'on ne le pense communément. Nous observons la distinction que fait Newton entre deux propriétes qui se confondent facilement, à savoir f/g→1 et f - g→0, et nous résolvons un problème né d'une traduction incorrecte parue dans la révision par Cajori de la traduction de Motte. Nous parvenons ainsi à comprendre plus complètement le Lemma XI, Section I, Livre I, bien connu mais moins bien compris.     

New lower bounds for Hopcroft's problem

We establish new lower bounds on the complexity of the following basic geometric problem, attributed to John Hopcroft: Given a set ofn points andm hyperplanes in $\mathbb{R}^d $ , is any point contained in any hyperplane? We define a general class ofpartitioning algorithms, and show that in the worst case, for allm andn, any such algorithm requires time Ω(n logm + n ^2/3m^2/3 + m logn) in two dimensions, or Ω(n logm + n ^5/6m^1/2 + n^1/2m^5/6 + m logn) in three or more dimensions. We obtain slightly higher bounds for the counting version of Hopcroft's problem in four or more dimensions. Our planar lower bound is within a factor of 2O(log^*(n+m)) of the best known upper bound, due to Matou?ek. Previously, the best known lower bound, in any dimension, was Ω(n logm + m logn). We develop our lower bounds in two stages. First we define a combinatorial representation of the relative order type of a set of points and hyperplanes, called amonochromatic cover, and derive lower bounds on its size in the worst case. We then show that the running time of any partitioning algorithm is bounded below by the size of some monochromatic cover. As a related result, using a straightforward adversary argument, we derive aquadratic lower bound on the complexity of Hopcroft's problem in a surprisingly powerful decision tree model of computation.     

Solution of a system of nonstrictly hyperbolic conservation laws

In this paper we study a special case of the initial value problem for a 2×2 system of nonstrictly hyperbolic conservation laws studied by Lefloch, whose solution does not belong to the class ofL ^∞ functions always but may contain δ-measures as well: Lefloch's theory leaves open the possibility of nonuniqueness for some initial data. We give here a uniqueness criteria to select the entropy solution for the Riemann problem. We write the system in a matrix form and use a finite difference scheme of Lax to the initial value problem and obtain an explicit formula for the approximate solution. Then the solution of initial value problem is obtained as the limit of this approximate solution.