Intended Treatments of Arithmetic Average in U.S. and Asian School Mathematics Textbooks

This study examined how selected U.S. and Asian mathematics curricula are designed to facilitate students' understanding of the arithmetic average. There is a consistency regarding the learning goals among these curriculum series, but the focuses are different between the Asian series and the U.S. reform series. The Asian series and the U.S. commercial series focus the arithmetic average more on conceptual and procedural understanding of the concept as a computational algorithm than on understanding the concept as a representative of a data set; however, the two U.S. reform series focus the concept more on the latter. Because of the different focuses, the Asian and the U.S. curriculum series treat the concept differently. In the Asian series, the concept is first introduced in the context of “equal‐sharing” or “per‐unit‐quantity,” and the averaging formula is formally introduced at a very early stage. In the U.S. reform series, the concept is discussed as a measure of central tendency, and after students have some intuitive ideas of the statistical aspect of the concept, the averaging algorithm is briefly introduced.     

Ergodic Properties and Julia Sets of Weierstrass Elliptic Functions

 We discuss properties of the Julia and Fatou sets of Weierstrass elliptic ℘ functions arising from real lattices. We give sufficient conditions for the Julia sets to be the whole sphere and for the maps to be ergodic, exact, and conservative. We also give examples for which the Julia set is not the whole sphere.     

Quantum theory of the structure and bonding in proteins: Part 5. Further studies on the C10 hydrogen bond of the tripeptide

The existence of two additional C₁₀ hydrogen bonds in the tripeptide molecule is demonstrated, so that altogether there are four such hydrogen bonds in this molecule. The accuracy with which the hydrogen bond can be located is examined, and this accuracy is sufficient for chemical and biological purposes. Some further comparisons with experimental facts, including an enkephalin fragment, are reported.     

Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints

In this paper we consider a mathematical program with equilibrium constraints (MPEC) formulated as a mathematical program with complementarity constraints. Various stationary conditions for MPECs exist in literature due to different reformulations. We give a simple proof to the M-stationary condition and show that it is sufficient for global or local optimality under some MPEC generalized convexity assumptions. Moreover, we propose new constraint qualifications for M-stationary conditions to hold. These new constraint qualifications include piecewise MFCQ, piecewise Slater condition, MPEC weak reverse convex constraint qualification, MPEC Arrow-Hurwicz-Uzawa constraint qualification, MPEC Zangwill constraint qualification, MPEC Kuhn-Tucker constraint qualification, and MPEC Abadie constraint qualification.     

Single-machine common due window assignment and scheduling to minimize the total cost

In this paper we consider a single-machine common due window assignment and scheduling problem with batch delivery cost. The starting time and size of the due window are decision variables. Finished jobs are delivered in batches. There is no capacity limit on each delivery batch, and the cost per batch delivery is fixed and independent of the number of jobs in the batch. The objective is to find a job sequence, a delivery date for each job, and a starting time and a size for the due window that jointly minimize the total cost comprising earliness, weighted number of tardy jobs, job holding, due window starting time and size, and batch delivery. We provide some properties of the optimal solution and present polynomial-time algorithms for the problem.     

A Class of Quadratic Programs with Linear Complementarity Constraints

We consider a class of quadratic programs with linear complementarity constraints (QPLCC) which belong to mathematical programs with equilibrium constraints (MPEC). We investigate various stationary conditions and present new and strong necessary and sufficient conditions for global and local optimality. Furthermore, we propose a Newton-like method to find an M-stationary point in finite steps without MEPC linear independence constraint qualification. The research of this author is partially supported by NSERC, and Research Grand Council of Hong Kong.     

A delay-dependent model with HIV drug resistance during therapy

The use of combination antiretroviral therapy has proven remarkably effective in controlling HIV disease progression and prolonging survival. However, the emergence of drug resistance can occur. It is necessary that we gain a greater understanding of the evolution of drug resistance. Here, we consider an HIV viral dynamical model with general form of target cell density, drug resistance and intracellular delay incorporating antiretroviral therapy. The model includes two strains: wild-type and drug-resistant. The basic reproductive ratio for each strain is obtained for the existence of steady states. Qualitative analysis of the model such as the well-posedness of the solutions and the equilibrium stability is provided. Global asymptotic stability of the disease-free and drug-resistant steady states is shown by constructing Lyapunov functions. Furthermore, sufficient conditions related to the properties of the target cell density are obtained for the local asymptotic stability of the positive steady state. Numerical simulations are conducted to study the impact of target cell density and intracellular delay focusing on the stability of the positive steady state. The occurrence of Hopf bifurcation of periodic solutions is shown to depend on the target cell density.     

Boundary of Anosov dynamics and evolution equations for surfaces

We show that a compact surface of genus greater than one, without focal points and a finite number of bubbles (“good” shaped regions of positive curvature) is in the closure of Anosov metrics. Compact surfaces of nonpositive curvature and genus greater than one are in the closure of Anosov metrics, by Hamilton's work about the Ricci flow. We generalize this fact to the above surfaces without focal points admitting regions of positive curvature using a “magnetic” version of the Ricci flow, the so‐called Ricci Yang‐Mills flow.     

Two-Way Tables: Issues at the Heart of Statistics and Probability for Students and Teachers

Some problems exist at the intersection of statistics and probability, creating a dilemma in relation to the best approach to assist student understanding. Such is the case with problems presented in two-way tables representing conditional information. The difficulty can be confounded if the context within which the problem is set is one where students have preconceived opinions on the direction of the potential association present. This article considers school students’ responses to two problems of association, with data presented in 2 × 2 tables. A hierarchical rubric is presented to document students’ understandings. Teachers’ pedagogical content knowledge is also considered in relation to the same two problems. Findings include a surprising relationship of outcomes for students across the problem contexts and some concern about teachers’ pedagogical content knowledge in this area of the curriculum.