Exploring students' mathematical performance,metacognitive experiences and skills in relation to fundamental theorem of calculus

Several studies have explored students’ understanding of the relationships between definite integrals and areas under curve(s). So far, however, there has been less attention to students’ understanding of the Fundamental Theorem of Calculus (FTC). In addition, students’ metacognitive experiences and skills whilst solving FTC questions have not previously been explored. This paper explored students’ mathematical performance, metacognitive experiences and metacognitive skills in relation to FTC questions by interviewing nine university and eight Year 13 students. The findings show that several students had difficulty solving questions related to the FTC and that students’ metacognitive experiences and skills could be further developed.     

Revised Bloom's taxonomy and integral calculus: unpacking the knowledge dimension

In this paper, the knowledge dimension for Revised Bloom's taxonomy (RBT) is unpacked for integral calculus. As part of this work, the 11 subtypes of the knowledge dimension are introduced, and through document analysis of chapter 4 of the RBT handbook, these subtypes are defined. Then, by consulting materials frequently used for teaching integral calculus, each subtype is exemplified. The developed dimension may enable or enhance opportunities for dialogue between lecturers, teachers, and researchers about how to develop and align educational objectives, teaching activities, and assessments in integral calculus, or how metacognition and metacognitive knowledge could be used to support teaching and learning.     

On smoothing,regularization, and averaging in stochastic approximation methods for stochastic variational inequality problems

Traditionally, most stochastic approximation (SA) schemes for stochastic variational inequality (SVI) problems have required the underlying mapping to be either strongly monotone or monotone and Lipschitz continuous. In contrast, we consider SVIs with merely monotone and non-Lipschitzian maps. We develop a regularized smoothed SA (RSSA) scheme wherein the stepsize, smoothing, and regularization parameters are reduced after every iteration at a prescribed rate. Under suitable assumptions on the sequences, we show that the algorithm generates iterates that converge to the least norm solution in an almost sure sense, extending the results in Koshal et al. (IEEE Trans Autom Control 58(3):594–609, 2013) to the non-Lipschitzian regime. Additionally, we provide rate estimates that relate iterates to their counterparts derived from a smoothed Tikhonov trajectory associated with a deterministic problem. To derive non-asymptotic rate statements, we develop a variant of the RSSA scheme, denoted by aRSSA\(_r\), in which we employ a weighted iterate-averaging, parameterized by a scalar r where \(r = 1\) provides us with the standard averaging scheme. The main contributions are threefold: (i) when \(r<1\) and the parameter sequences are chosen appropriately, we show that the averaged sequence converges to the least norm solution almost surely and a suitably defined gap function diminishes at an approximate rate \(\mathcal{O}({1}\slash {\root 6 \of {k}})\) after k steps; (ii) when \(r<1\), and smoothing and regularization are suppressed, the gap function admits the rate \(\mathcal{O}({1}\slash {\sqrt{k}})\), thus improving the rate \(\mathcal{O}(\ln (k)/\sqrt{k})\) under standard averaging; and (iii) we develop a window-based variant of this scheme that also displays the optimal rate for \(r < 1\). Notably, we prove the superiority of the scheme with \(r < 1\) with its counterpart with \(r=1\) in terms of the constant factor of the error bound when the size of the averaging window is sufficiently large. We present the performance of the developed schemes on a stochastic Nash–Cournot game with merely monotone and non-Lipschitzian maps.     

Relative T -Injective Modules and Relative T -Flat Modules

Let T be a Wakamatsu tilting module. A module M is called (n, T)-copure injective (resp. (n, T)-copure flat) if ɛ _T ¹ (N, M) = 0 (resp. Γ₁ ^T (N, M) = 0) for any module N with T-injective dimension at most n (see Definition 2.2). In this paper, it is shown that M is (n, T)-copure injective if and only if M is the kernel of an I _n (T)-precover f: A → B with A ∈ Prod T. Also, some results on Prod T-syzygies are presented. For instance, it is shown that every nth Prod T-syzygy of every module, generated by T, is (n, T)-copure injective.     

Roughness evolution of highly ordered nanoporous anodic aluminum oxide films

In this paper, we report on the surface roughness evolution of highly ordered anodic aluminum oxide (AAO) films based on an atomic force microscopy (AFM) study. Root mean square of the surface roughness was measured on AFM images taken from highly ordered AAO films produced by two-step anodization under different conditions including electrolyte type, anodization voltage, and anodization time. Surface roughness of highly ordered AAO films increases step by step through the two-step anodizing process including electropolishing, first-step anodization, dissolution, and second-step anodization. However, increase of the surface roughness is proportional to the anodization voltage and time. The surface roughness of AAO films changes as a function of length scale until it finally approaches a maximum termed the saturation roughness. The variation of roughness of the growth of AAO could be scaled with an anomalous dynamic behavior as it saturates over a critical length scale while the saturation roughness is dependent on the anodizing time and voltage.     

Solvent-free and room temperature synthesis of thiochromans in the presence of a catalytic amount of tungstophosphoric acid

Tungstophosphoric acid catalyzed rapid and good yielding reactions of α,β-unsaturated aldehydes with arenethiols to give the corresponding 4-thioaryl-1,2,3,4-tetrahydro-1-benzothiopyrans (thiochromans) under solvent-free and room temperature conditions.     

Vibrational assignment, structure and intramolecular hydrogen bond study of 3-amino-1-phenyl-2-buten-1-one

Fourier transform infrared and Fourier transform Raman spectra of 3-amino-1-phenyl-2-buten-1-one and its deuterated analogue were recorded in the regions 400-4,000 and 150-4,000 cm(-1), respectively. Furthermore, the molecular structure and vibrational frequencies of title compound were investigated by a series of density functional theoretical, DFT, and ab initio calculations at the post-Hartree-Fock (MP2) level. Although, the calculated frequencies are generally in agreement with the observed spectra but the DFT results are in much better quantitative agreement with the observed spectra than the MP2 results. The observed wavenumbers were analyzed and assigned to different normal modes of vibration of the molecule. The calculated geometrical parameters show a strong intramolecular hydrogen bond with a N...O distance of 2.621-2.668 A. This bond length is shorter than that of its parent, 4-amino-3-penten-2-one (with two methyl groups in the beta-position), which is in agreement with spectroscopic results. The topological properties of the electron density contributions for intramolecular hydrogen bond in 3-amino-1-phenyl-2-buten-1-one and 4-amino-3-penten-2-one have been analyzed in term of the Bader theory of atoms in molecules (AIM). These results also support the stronger hydrogen bond in the title compound with respect to the parent molecule.     

A distance function for computing on finite subsets of Euclidean spaces

In practical purposes for some geometrical problems, specially the fields in common with computer science, we deal with information of some finite number of points. The problem often arises here is: “How are we able to define a plausible distance function on a finite three dimensional space?” In this paper, we define such a distance function in order to apply it to further purposes, e.g. in the field settings of transportation theory and geometry. More precisely, we present a new model for traveling salesman problem and vehicle routing problem for two dimensional manifolds in three dimensional Euclidean space, the second problem on which we focus on this line is, three dimensional triangulation.     

High Pressure High Temperature Dynamic Rheometry of Sulfonated Polyacrylamide Solutions in Electrolyte Media as Used for Oil Recovery Applications

Sulfonated polyacrylamide (SPAA) solutions were prepared and the effects of pressure, polymer concentration, and water temperature, pH and salinity on their rheological behavior were investigated using a concentric cylinder dynamic rheometer equipped with a high pressure cell. According to the rheological flow curves the shear stress of SPAA solutions increased less than in proportion to their shear rates; that is, a shear thinning effect occurred. For polymer solutions containing 15,000 ppm of SPAA, shear viscosity, and stress were nearly insensitive to pressure. However, the shear viscosity and stress of SPAA solutions were affected by temperature and this effect was more evident at lower pressure. The flow curves indicated the shear viscosity and stress of the samples increased with increasing SPAA concentration and pH of the water, but were decreased with increasing water salinity and temperature.     

Rheological and Mechanical Characterization of Multi-Walled Carbon Nanotubes/Polypropylene Nanocomposites

The rheology and morphology of multi-walled carbon nanotube (MWNT)/polypropylene (PP) nanocomposites prepared via melt blending was investigated. The minor phase content of MWNT varied between 0.25 and 8 wt%. From morphological studies using a scanning electron microscopy technique a good dispersion of carbon nanotubes in the PP matrix was observed. The rheological studies were performed by a capillary rheometer, and mechanical properties of the nanocomposites were studied using a tensile and flexural tester. Both PP and its nanocomposites showed non-Newtonian behavior. At low shear rates the addition of MWNT content causes an increase in viscosity; however, viscosity is less sensitive to addition of MWNT content at higher shear rates. Flow activation energy for the nanocomposites was calculated using an Arrhenius type equation. From this calculation it was concluded that the temperature sensitivity of nanocomposites was increased by increasing of nanotube content. An increase in tensile and flexural moduli and Izod impact strength was also observed by increasing the MWNT content. From rheological and mechanical tests it was concluded that the mechanical and rheological percolation threshold is at 1.5 wt%.