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.     

Study of kinetic,thermodynamic, and isotherm of Sr adsorption from aqueous solutions on graphene oxide (GO) and (aminomethyl)phosphonic acid–graphene oxide (AMPA–GO)

Journal of Radioanalytical and Nuclear Chemistry - In this work, graphene oxide (GO) and (aminomethyl)phosphonic acid–Graphene oxide (AMPA–GO) adsorbents were prepared by the modified...     

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.     

Weyl’s Law and Connes’ Trace Theorem for Noncommutative Two Tori

We prove the analogue of Weyl’s law for a noncommutative Riemannian manifold, namely the noncommutative two torus ${\mathbb{T}_{\theta}^{2}}$ equipped with a general translation invariant conformal structure and a Weyl conformal factor. This is achieved by studying the asymptotic distribution of the eigenvalues of the perturbed Laplacian on ${\mathbb{T}_{\theta}^{2}}$ . We also prove the analogue of Connes’ trace theorem by showing that the Dixmier trace and a noncommutative residue coincide on pseudodifferential operators of order ?2 on ${\mathbb{T}_{\theta}^{2}}$ .     

Urea enzymatic hydrolysis reaction: optimization by response surface methodology based on potentiometric measurements

The enzymatic hydrolysis reaction of urea by urease is optimized in this work by the chemometric response surface methodology (RSM), based on an initial rate potentiometric measurement using an NH(4)(+) ion-selective electrode (ISE). In this investigation, the ranges of critical variables determined by a preliminary "one at a time" (OVAT) procedure were used as input for the subsequent RSM chemometric analysis. The RSM quadratic response was found to be quite appropriate for modeling and optimization of the hydrolysis reaction as illustrated by the relatively high value of the determination coefficient (R(2)=90.1%), along with the satisfactory results obtained by the analysis of variance (ANOVA). All the evaluated analytical characteristics of the optimized method such as: the linear calibration curve, the upper and lower detection limits, the within-day precisions at low and at high levels, the assay recovery in pool serum media, along with the activation kinetic parameters, were also reported. Further, in order to check the quality of the optimization and the validity of the model, the assay of urea, both in aqueous laboratory and human serum samples, were performed. It has to be noted that the kinetic initial rate measurement method used in this work, permitted to overcome the general problem of NH(4)(+) ISE low selectivity against Na(+) and K(+) interfering ions in real samples.     

Isolation of beta-glucan from the cell wall of Saccharomyces cerevisiae

Beta-glucan, one of the major cell wall components of Saccharomyces cerevisiae (S. cerevisiae), has been found to enhance immune functions. At present study, we developed an optimal procedure to extract and purify beta-glucan. At first, yeast cells were grown in sabouraud dextrose agar and then cultured in yeast extract-peptone-glucose (YPG) broth. After incubation, cells were harvested, washed and disrupted by means of sonication method. The obtained cell walls were used to prepare alkali-soluble beta-glucan (glucan-S1). In this regard, 2% sodium hydroxide (NaOH) and 3% acetic acid were used in alkaline-acid extraction, respectively. This preparation contained 2.4% protein. In the next step, DEAE sephacel chromatography was used to remove remaining proteins (glucan-S2). Subsequently this preparation was applied into concanavalin-A sepharose column to remove manann. Finally, beta-glucan free of mannoprotein complexes was prepared (glucan-S3).