A Modeling Perspective on Interpreting Rates of Change in Context

Functions provide powerful tools for describing change, but research has shown that students find difficulty in using functions to create and interpret models of changing phenomena. In this study, we drew on a models and modeling perspective to design an instructional approach to develop students’ abilities to describe and interpret rates of change in the context of exponential decay. In this article, we elaborate the characteristics of the model development sequence and we examine how students interpreted and described non-constant rates of change in context. We provide evidence for how a focus on the context made visible students’ reasoning about rates of change, including difficulties related to the use of language when describing changes in the negative direction. We argue that context and the use of language, forefronted in a modeling approach, should play an important role in supporting the development of students’ reasoning about changing phenomena.     

QM/MM calculation of solvent effects on absorption spectra of guanine

Electronic spectra of guanine in the gas phase and in water were studied by quantum mechanical/molecular mechanical (QM/MM) methods. Geometries for the excited‐state calculations were extracted from ground‐state molecular dynamics (MD) simulations using the self‐consistent‐charge density functional tight binding (SCC‐DFTB) method for the QM region and the TIP3P force field for the water environment. Theoretical absorption spectra were generated from excitation energies and oscillator strengths calculated for 50 to 500 MD snapshots of guanine in the gas phase (QM) and in solution (QM/MM). The excited‐state calculations used time‐dependent density functional theory (TDDFT) and the DFT‐based multireference configuration interaction (DFT/MRCI) method of Grimme and Waletzke, in combination with two basis sets. Our investigation covered keto‐N7H and keto‐N9H guanine, with particular focus on solvent effects in the low‐energy spectrum of the keto‐N9H tautomer. When compared with the vertical excitation energies of gas‐phase guanine at the optimized DFT (B3LYP/TZVP) geometry, the maxima in the computed solution spectra are shifted by several tenths of an eV. Three effects contribute: the use of SCC‐DFTB‐based rather than B3LYP‐based geometries in the MD snapshots (red shift of ca. 0.1 eV), explicit inclusion of nuclear motion through the MD snapshots (red shift of ca. 0.1 eV), and intrinsic solvent effects (differences in the absorption maxima in the computed gas‐phase and solution spectra, typically ca. 0.1–0.3 eV). A detailed analysis of the results indicates that the intrinsic solvent effects arise both from solvent‐induced structural changes and from electrostatic solute–solvent interactions, the latter being dominant. © 2009 Wiley Periodicals, Inc. J Comput Chem 2010     

Deterministic random walks on regular trees

Jim Propp's rotor–router model is a deterministic analog of a random walk on a graph. Instead of distributing chips randomly, each vertex serves its neighbors in a fixed order. Cooper and Spencer (Comb Probab Comput 15 (2006) 815–822) show a remarkable similarity of both models. If an (almost) arbitrary population of chips is placed on the vertices of a grid ?^d and does a simultaneous walk in the Propp model, then at all times and on each vertex, the number of chips on this vertex deviates from the expected number the random walk would have gotten there by at most a constant. This constant is independent of the starting configuration and the order in which each vertex serves its neighbors. This result raises the question if all graphs do have this property. With quite some effort, we are now able to answer this question negatively. For the graph being an infinite k‐ary tree (k ≥ 3), we show that for any deviation D there is an initial configuration of chips such that after running the Propp model for a certain time there is a vertex with at least D more chips than expected in the random walk model. However, to achieve a deviation of D it is necessary that at least exp(Ω(D²)) vertices contribute by being occupied by a number of chips not divisible by k at a certain time. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

Phase relations in the system V/Nb/O. V. Investigation of mixed crystals V1-xNbxO2

Mixed crystals V_1-xNb_xO₂ exist over the whole area of the quasibinary line VO₂-NbO₂. The existence of Nb_{5+ beside V3+ and V4+ on the V-rich side and V3+ beside Nb5+ and Nb4+ on the Nb-rich side of the mixed crystals is demonstrated by XANES-measurements. The compound VNbO}4_(V0.5_Nb0.5_O2_{) is described as a double oxide with vanadium only as V3+ and niobium only as Nb5+. At this point the electric resistivity of the solid solution shows a maximum.}     

Undergraduate Students’ Quantitative Reasoning in Economic Contexts

Contributing to a growing body of research on undergraduate students’ quantitative reasoning, the study reported in this article used task-based interviews to investigate business calculus students’ quantitative reasoning when solving two optimization tasks situated in the context of revenue and profit maximization. Analysis of verbal responses and work written by 12 pairs of students during the task-based interviews revealed that nearly all pairs of students created new quantities (e.g., diminishing marginal returns). Students used these new quantities to reason about relationships among computer sales, sales discount, and total revenue in a revenue maximization task. The creation of these quantities helped the students to solve the problem posed in the task. Ten pairs of students interpreted marginal cost as total cost and marginal revenue as total revenue in a profit maximization task. Implications for business calculus instruction and directions for future research are discussed.     

Error Propagation in Game Trees

Game tree search is the core of most attempts to make computers play games. We present a fairly general theoretical analysis on how leaf evaluation errors influence the value estimation of a game position at the root. By an approach using prime factorization arguments in the ring of polynomials, we show that in this setting the maximum number of leaf-disjoint strategies proving a particular property is a key notion. This number precisely describes the quality of the heuristic game value in terms of the quality of the leaf evaluation heuristics. We extend this model to include random nodes (rolls of a die). Surprisingly, this changes the situation: utill the number of leaf-disjoint strategies ensures robustness against leaf evaluation errors, but the converse is not true. An average node may produce additional robustness similar to additional leaf-disjoint strategies. This work extends earlier ones which only deal with 0, 1 valued nodes, or without randomness.The first author was partially supported (associate member) by the graduate school ‘Effiziente Algorithmen und Mehrskalenmethoden’, Deutsche Forschungsgemeinschaft. The second author was partially supported by the Future and Emerging Technologies programme of the EU under contract numbers IST-1999-14186 (ALCOM-FT) and IST-2001-33116 (FLAGS).