Ways in which engaging with someone else’s reasoning is productive

Classrooms which involve students in mathematical discourse are becoming ever more prominent for the simple reason that they have been shown to support student learning and affinity for content. While support for outcomes has been shown, less is known about how or why such strategies benefit students. In this paper, we report on one such finding: namely that when students engage with another’s reasoning, as necessitated by interactive conversation, it supports their own conceptual growth and change. This qualitative analysis of 10 university students provides insight into what engaging with another’s reasoning entails and suggests that higher levels of engagement support higher levels of conceptual growth. We conclude with implications for instructional practice and future research.     

Crystal and magnetic structures of Y2CrS4

Chromium(II) sulfide, Y₂CrS₄, prepared by a solid-state reaction of Y₂S₃ and CrS, showed an antiferromagnetic transition at 65 K. The neutron diffraction patterns at 10 and 90 K were both well refined with the space group Pca2₁. At 90 K, cell parameters were a=12.5518(13) Å, b=7.5245(8) Å, and c=12.4918(13) Å. At 10 K, magnetic peaks were observed, which could be indexed on the same unit cell. Magnetic moments of chromium ions were parallel to the b-axis and antiferromagnetically ordered in each set of the 4a sites.     

Spectroscopic and photoluminescence characteristics of Dy ions in lead containing sodium fluoroborate glasses for laser materials

Lead containing calcium zinc sodium fluoroborate (LCZSFB) glasses doped with different concentrations of trivalent dysprosium ions were prepared and investigated by the XRD, FTIR, optical absorption, photoluminescence and decay curve analysis. The experimentally determined oscillator strengths have been determined by measuring the areas under the absorption peaks and the Judd–Ofelt (J–O) intensity parameters were calculated using the least squares fit method. From the evaluated J–O parameters the radiative transition probability rates, radiative lifetimes and branching ratios were calculated for ⁴F_9/2 excited level. Room temperature photoluminescence spectra for different concentrations of Dy³⁺-doped LCZSFB glasses were obtained by exciting the glass samples at 386 nm. The intensity of Dy³⁺ emission spectra increases with increasing concentration of 0.1, 0.25, 0.5 and 1.0 mol% and beyond 1.0 mol% the concentration quenching is observed. The measuring branching ratios are reasonably high for transitions ⁴F_9/2→⁶H_15/2 and ⁶H_13/2, suggesting that the emission at 484 and 576 nm, respectively, can give rise to lasing action in the visible region. From the visible emission spectra, yellow–blue (Y/B) intensity ratios and chromaticity color coordinates were also estimated. The lifetimes of ⁴F_9/2 metastable state for the samples with different concentrations were also measured and discussed.     

Photoluminescence of CdGeP2 and (Cd,Mn)GeP2

Photoluminescence of CdGeP₂ (112) single crystal and CdGeP₂ epitaxial film grown on GaAs (001) substrate have been studied and their spectral similarity found. Spectral bands associated with donor/acceptor transitions peak at close energies for both substances and all are lower than the energy gap of the chalcopyrite crystal.On the other hand, the growth of (Cd,Mn)GeP₂ ferromagnetic layer on CdGeP₂ (112) single crystal was performed to make it possible observation of PL from both the ferromagnetic layer and substrate. The green laser excitation (514, 532 nm) produces a proper photoluminescence similar to that in the undoped CdGeP₂ crystal and film. An extra emission from the ferromagnetic-nonmagnetic heterojunction occurs to extend up to photon energies exceeding E_g of the host semiconductor. The short wavelength photoluminescence is to be due to (Cd,Mn)GeP₂ dilute magnetic semiconductor (DMS). This fact states that Mn-doped II-IV-V₂ chalcopyrites are closer to II-VI DMS than to another group III-V DMS, where the heavy Mn-doping suppresses photoluminescence at all. Features of the observed short wavelength emission are discussing based on the temperature and spectral analyses.     

Effect of substrate texture on the growth of hematite nanowires

The effect of texture of iron foil substrate on the growth of hematite nanowires by annealing method has been investigated in detail. Three substrates of different textures were prepared from a [2 0 0] oriented iron foil by some simple processes. The hematite nanowires on these substrates were synthesized by annealing iron foil at 700 °C in moist oxygen. The growth pattern of nanowires on these substrates showed that the growth of hematite nanowires depends strongly on the iron substrate texture and [1 1 0] oriented iron grains are necessary for their growth. The samples were characterized by Scanning Electron Microscope (SEM), Transmission Electron Microscope (TEM), X-ray diffraction (XRD), Electron Back Scatter Diffraction (EBSD) and Raman Spectroscopy. We have also tried to explain the various observations on the mechanism of growth. Mainly, the presence of water vapor significantly enhanced the formation of hematite nanowires which resulted in a very dense and aligned growth of nanowires on the substrate areas of favorable texture. Finally, the study proved the substrate texture to be a powerful tool to control growth of nanowires and can be used efficiently for patterning and large scale synthesis of the nanowires.     

Poincaré duality algebras mod two

We study Poincaré duality algebras over the field F₂ of two elements. After introducing a connected sum operation for such algebras we compute the corresponding Grothendieck group of surface algebras (i.e., Poincaré algebras of formal dimension 2). We show that the corresponding group for 3-folds (i.e., algebras of formal dimension 3) is not finitely generated, but does have a Krull-Schmidt property.We then examine the isomorphism classes of 3-folds with at most three generators of degree 3, provide a complete classification, settle which such occur as the cohomology of a smooth 3-manifold, and list separating invariants.The closing section and Appendix A provide several different means of constructing connected sum indecomposable 3-folds.     

A system of partial differential equations modeling the competition for two complementary resources in flowing habitats

This paper examines a system of reaction-diffusion equations arising from a flowing water habitat. In this habitat, one or two microorganisms grow while consuming two growth-limiting, complementary (essential) resources. For the single population model, the existence and uniqueness of a positive steady-state solution is proved. Furthermore, the unique positive solution is globally attracting for the system with regard to nontrivial nonnegative initial values. Mathematical analysis for the two competing populations is carried out. More precisely, the long-time behavior is determined by using the monotone dynamical system theory when the semi-trivial solutions are both unstable. It is also shown that coexistence solutions exist by using the fixed point index theory when the semi-trivial solutions are both (asymptotically) stable.     

A multi-frequency inverse source problem

This paper is concerned with an inverse source problem that determines the source from measurements of the radiated fields away at multiple frequencies. Rigorous stability estimates are established when the background medium is homogeneous. It is shown that the ill-posedness of the inverse problem decreases as the frequency increases. Under some regularity assumptions on the source function, it is further proven that by increasing the frequency, the logarithmic stability converts to a linear one for the inverse source problem.     

Covering and radius-covering arrays: Constructions and classification

The minimum number of rows in covering arrays (equivalently, surjective codes) and radius-covering arrays (equivalently, surjective codes with a radius) has been determined precisely only in special cases. In this paper, explicit constructions for numerous best known covering arrays (upper bounds) are found by a combination of combinatorial and computational methods. For radius-covering arrays, explicit constructions from covering codes are developed. Lower bounds are improved upon using connections to orthogonal arrays, partition matrices, and covering codes, and in specific cases by computation. Consequently for some parameter sets the minimum size of a covering array is determined precisely. For some of these, a complete classification of all inequivalent covering arrays is determined, again using computational techniques. Existence tables for up to 10 columns, up to 8 symbols, and all possible strengths are presented to report the best current lower and upper bounds, and classifications of inequivalent arrays.     

Authentication schemes from actions on graphs, groups, or rings

We propose a couple of general ways of constructing authentication schemes from actions of a semigroup on a set, without exploiting any specific algebraic properties of the set acted upon. Then we give several concrete realizations of this general idea, and in particular, we describe several authentication schemes with long-term private keys where forgery (a.k.a. impersonation) is NP-hard. Computationally hard problems that can be employed in these realizations include the Graph Colorability problem, the Diophantine problem, and many others.