Calculus students’ understanding of the vertex of the quadratic function in relation to the concept of derivative

The purpose of this study was to gain insight into 30, first year calculus students’ understanding of the relationship between the concept of vertex of a quadratic function and the concept of the derivative. APOS (action-process-object-schema) theory was applied as a guiding framework of analysis on student written work, think-aloud and follow up group interviews. Students’ personal meanings of the vertex, including misconceptions, were explored, along with students’ understanding to solve problems pertaining to the derivative of a quadratic function. Results give evidence of students’ weak schema of the vertex, lack of connection between different problem types and the importance of linguistics in relation to levels of APOS theory. A preliminary genetic decomposition was developed based on the results. Future research is suggested as a continuation to improve student understanding of the relationship between the vertex of quadratic functions and the derivative.     

Rational approximation of vertical segments

In many applications, observations are prone to imprecise measurements. When constructing a model based on such data, an approximation rather than an interpolation approach is needed. Very often a least squares approximation is used. Here we follow a different approach. A natural way for dealing with uncertainty in the data is by means of an uncertainty interval. We assume that the uncertainty in the independent variables is negligible and that for each observation an uncertainty interval can be given which contains the (unknown) exact value. To approximate such data we look for functions which intersect all uncertainty intervals. In the past this problem has been studied for polynomials, or more generally for functions which are linear in the unknown coefficients. Here we study the problem for a particular class of functions which are nonlinear in the unknown coefficients, namely rational functions. We show how to reduce the problem to a quadratic programming problem with a strictly convex objective function, yielding a unique rational function which intersects all uncertainty intervals and satisfies some additional properties. Compared to rational least squares approximation which reduces to a nonlinear optimization problem where the objective function may have many local minima, this makes the new approach attractive.     

On the Froissart phenomenon in multivariate homogeneous Padé approximation

In univariate Padé approximation we learn from the Froissart phenomenon that Padé approximants to perturbed Taylor series exhibit almost cancelling pole–zero combinations that are unwanted. The location of these pole–zero doublets was recently characterized for rational functions by the so‐called Froissart polynomial. In this paper the occurrence of the Froissart phenomenon is explored for the first time in a multivariate setting. Several obvious questions arise. Which definition of Padé approximant is to be used? Which multivariate rational functions should be investigated? When considering univariate projections of these functions, our analysis confirms the univariate results obtained so far in [13], under the condition that the noise is added after projection. At the same time, it is apparent from section 4 that for the unprojected multivariate Froissart polynomial no conjecture can be formulated yet.

     

On the continuity of the multivariate Padé operator

Continuity of the univariate Padé operator was proved in [5,6]. We discuss the limitations of a multivariate generalization and prove a multivariate analogon of the continuity property.     

A new method for the stereoselective synthesis of alpha- and beta-glycosylamines using the Burgess reagent

Although glycosylamines constitute an important group of carbohydrates from the standpoint of biology and medicine, methods for their synthesis typically lack substrate generality and/or result in variable stereoselectivity, especially in complex contexts. In this communication, we report an operationally simple method for the synthesis of both alpha- and beta-glycosylamines using the Burgess reagent that overcomes many of these limitations in a bare minimum of synthetic steps.     

Fluorescent polymer cubosomes and hexosomes with aggregation-induced emission

Fluorescent polymer cubosomes and hexosomes with aggregation-induced emission (AIE) were prepared from amphiphilic block copolymers PEG-b-PTPEMA where the hydrophobic block PTPEMA was a polymethacrylate with tetraphenylethene (TPE) as the AIE side group. Four highly asymmetric block copolymers with hydrophilic block weight ratio f_PEG ≤ 20% were synthesized. Cubosomes and hexosomes with strong fluorescence emission were obtained by nanoprecipitation of polymers with f_PEG < 9% in dioxane/water and THF/water systems. Their ordered internal structures were studied by electron microscopy (cryo-EM, SEM and TEM) and the X-ray scattering technique (SAXS). To elucidate the formation mechanisms of these inverted colloids, other parameters influencing the morphologies, like the water content during self-assembly and the organic solvent composition, were also investigated. This study not only inspires people to design novel building blocks for the preparation of functional cubosomes and hexosomes, but also presents the first AIE fluorescent polymer cubosome and hexosome with potential applications in bio-related fields.

Fluorescent Im$\bar{3}$m cubosome and P6mm hexosome with aggregation-induced emission (AIE) were reported, which were formed by amphiphilic block copolymers PEG-b-PTPEMA. The length of hydrophobic block PTPEMA was adjusted to control morphology formation.