On the theoretical, conceptual, and philosophical foundations for research in mathematics education

The current infatuation in the U.S. with “what works” studies seems to leave education researchers with less latitude to conduct studies to advance theoretical and model-building goals and they are expected to adopt philosophical perspectives that often run counter to their own. Three basic questions are addressed in this article:What is the role of theory in education research? How does one's philosophical stance influence the sort of research one does? And,What should be the goals of mathematics education research? Special attention is paid to the importance of having a conceptual framework to guide one's research and to the value of acknowledging one's philosophical stance in considering what counts as evidence.     

The millimeter-wave spectrum of acetaldehyde in its two lowest torsional states

A large number of new millimeter-wave spectral lines of gaseous acetaldehyde have been measured at frequencies up to 250 GHz. These lines arise from rotational transitions of acetaldehyde in its two lowest (v_t = 0,1) torsional states and involve angular momentum quantum numbers J ≤ 12 and K ≤ 10. A global data set consisting of 562 lines has been obtained by combining the millimeter-wave lines with previously measured lower frequency data involving the two lowest torsional states. This data set has been analyzed via an internal axis method previously used to study the spectra of CH₃OH and CH₃SH. The root-mean-square deviation of the fit is only 685 kHz. An analogous least-squares fit to 335v_t = 0 lines yields a root-mean-square deviation of 269 kHz.     

The competing roles of topology and spin density in the magnetic behavior of spin-delocalized radicals: Donor–acceptor annelated nitronyl nitroxides

The effects of acceptor–donor interactions in thienyl substituted benzimidazole-nitronyl nitroxides (TBNN) on the absorption spectroscopy, spin density distribution, magnetic behavior, and crystallographic packing were explored through spectroscopy, computation, and characterization of structure and magnetic properties in the crystalline phase. The electronic spectra of the radicals exhibit a strong broad absorption in the NIR (λ_max ～ 1000 nm) that exhibits solvatochromism consistent with charge transfer between the thienyl (donor) and benzonitronyl nitroxide (acceptor) dyads. Computational analysis allowed assignment of the transition as a HOMO–SOMO transition (TD-DFT UB3LYP/6-31G7). The TBNN radicals form highly disordered slipped π-stacks in the solid state that give rise to antiferromagnetic interactions consistent with 1D chain interactions. The magnetic behavior was well-fit to a Bonner–Fisher model to give exchange parameters of J = ?2 to ?10 cm^?1 depending on substitution. The weak exchange parameters are attributed to the degree of solid-state disorder, and the observed properties can be rationalized by the effects of substitution on the electronic structure and topology of the radicals.     

Global well-posedness,scattering and blow-up for the energy-critical focusing non-linear wave equation

We study the energy-critical focusing non-linear wave equation, with data in the energy space, in dimensions 3, 4 and 5. We prove that for Cauchy data of energy smaller than the one of the static solution W which gives the best constant in the Sobolev embedding, the following alternative holds. If the initial data has smaller norm in the homogeneous Sobolev space H ¹ than the one of W, then we have global well-posedness and scattering. If the norm is larger than the one of W, then we have break-down in finite time.     

Effects of molecular mass and surface treatment on adsorbed poly(methyl methacrylate) on silica

The behavior of relatively monodisperse adsorbed poly(methyl methacrylate) (PMMA) samples, from 19 to 587 kDa on silica, was studied using modulated differential scanning calorimetry and FTIR. On untreated Cab? O? Sil silica, the glass transition temperatures (T_gs) were higher (by around 30 °C), and the transitions were significantly broader (by a factor of 5–6) than those for the corresponding bulk samples. While the T_gs for the bulk polymers showed the expected dependence on molecular mass, the polymers on untreated silica showed little dependence, i.e., at the same adsorbed amounts, the glass transitions were very similar. The FTIR spectra of the adsorbed PMMA (on untreated silica) showed the presence of at least two resonances, one for the bound (hydrogen bonded to surface silanols) and another for free carbonyls. Fitting of the spectra allowed the estimation of the bound fractions of carbonyls that were dependent on the adsorbed amount, but not molecular mass. On Cab? O? Sil treated with hexamethyldisilizane (HMDS), the adsorbed PMMA exhibited glass transition behavior with little molecular‐mass dependence; the T_gs for the different PMMA samples were very similar to those of the high‐molecular mass bulk polymer, but with additional broadening of about a factor of 2. FTIR spectra for the PMMA samples on the treated silica did not show significant amounts of any of the hydrogen‐bonded carbonyl groups. © 2008 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 46: 649–658, 2008     

Flächenabbildungen mit gemeinsamem Invariantensystem

Zusammenfassung und Ausblick Unsere Betrachtungen zeigen, daß die Differentialinvarianten, die zu einem Paar aufeinander abgebildeter Flächen gehören, dieses, auch wenn man triviale Verschiedenheiten nicht als Unterschiede rechnet, durchaus nicht immer eindeutig zu bestimmen brauchen, trotzdem sie die affine Beziehung zwischen den Umgebungen je zweier entsprechender Flächenpunkte vollkommen im Raume festlegen.Beispiele, die dies näher belegen, sollen in einer besonderen Arbeït mitgeteilt werden.Naturgemäß stellt sich uns nun als nächstes Problem das folgende:Welche Bedingungen müssen vier Funktionen i₁,i₂,i,j von zwei Veränderlichenu, v erfüllen, damit zwei Vektoren und (u, v) die den Gleichungen (1) und (1) genügen, existieren?Herrn Geheimrat Prof. Dr.C. Carathéodory gewidmet.