The effect of anhydrous ammonia on the crystalline-state deformation of nylons 6 and 6,6

The aliphatic polyamides nylon 6 and 6,6 have been cold drawn by using the new concept of reversible plasticization with NH₃. Two aspects were investigated: the effect of NH₃ plasticization on the stress-strain behavior of nylon 6,6, and the extent of NH₃ absorption as determined by x-ray analysis for the structurally simpler nylon 6. X-ray analysis shows that ammonia is absorbed preferentially by the amorphous phase and that the plasticization has a significant effect on the unit-cell orientation, the small-angle void scattering, and the interplanar spacings between sheets of hydrogen-bonded chains. The effect of plasticization was seen in the lower yield stress, the higher draw ratio (up to 8X) and tensile modulus (10 GPa) not attainable with unplasticized specimens.     

On Clarkson's inequality,type and cotype for the Edmunds-Triebel logarithmic spaces

We show that the (p, p') Clarkson's inequality holds in the Edmunds-Triebel logarithmic spaces A_q(logA)_b,q A_{\theta}({\log}A)_{b,q} and in the Zygmund spaces L_p(logL)_b(W) L_p({\log}L)_b(\Omega) , for b ? \mathbbR b \in \mathbb{R} and for suitable 1 ￡ p ￡ 2 1 \leq p \leq 2 . As a consequence of these results we also obtain some new information about the types and the cotypes of these spaces.     

Studying teachers’ mathematical argumentation in the context of refuting students’ invalid claims

This study investigates teachers’ argumentation aiming to convince students about the invalidity of their mathematical claims in the context of calculus. 18 secondary school mathematics teachers were given three hypothetical scenarios of a student's proof that included an invalid algebraic claim. The teachers were asked to identify possible mistakes and explain how they would refute the student's invalid claims. Two of them were also interviewed. The data were analysed in terms of the content and structure of argumentation and the types of counterexamples the teachers generated. The findings show that teachers used two main approaches to refute students’ invalid claims, the use of theory and the use of counterexamples. The role of these approaches in the argumentation process was analysed by Toulmin's model and three types of reasoning emerged that indicate the structure of argumentation in the case of refutation. Concerning the counterexamples, the study shows that few teachers use them in their argumentation and in general they underestimate their value as a proof method.     

The melt anisotropy of ultrahigh-molecular-weight polyethylene

In the course of melt-flow crystallization studies with ultrahigh-molecular-weight polyethylene (UHMWPE), we observed that the melt of UHMWPE is highly anisotropic above its equilibrium melting point and has a tendency to fibrillate. An examination of the melt anisotropy of UHMWPE by optical, Thermal, and x-ray analysis indicates that the melt anisotropy persists at 345°C, i.e., the temperature at which the polymer degrades under nitrogen, and appears similar to a smectic liquid-crystalline phase.     

Solid-state coextrusion of high-density polyethylene. I. Effects of geometric factors

The solid (crystalline) state coextrusion of two high-density polyethylenes (HDPE) having weight-average molecular weights (M_w) of 59,000 and 200,000 have been studied as a function of the geometrical arrangement and the volume fraction of the components. The extrusion rate increased nonlinearly with the volume fraction of the low-M_w, component. The rate was faster when the low-M_w, component was the core rather than the sheath in the initial cylindrical concentric billet. Thus the slow extrusion rate of high-M_w HDPE alone was increased up to ten times by coextrusion with a small fraction of the low-M_w, HDPE component in its center. Generally, the deformation flow profile changed gradually from a parabolic to a W-shaped pattern as the volume fraction of the high-M_w, component increased. However, the geometric arrangement of the two different M_w components also had a pronounced effect on the deformation. The deformation patterns showed that upon coextrusion the low- and high-M_w HDPE's were extruded at the same rate and extrusion draw ratio. The geometrical arrangement had no substantial effects on the tensile modulus and strength of the extrudates; i.e., they increased linearly with volume fraction of the high-M_w HDPE.     

Calculus in European classrooms: curriculum and teaching in different educational and cultural contexts

In this paper we provide a first overview of the landscape with respect to calculus teaching in European classrooms, an area where research is very limited. In particular through a small expert-based survey and a literature review, we trace the development of calculus teaching at schools in a number of European countries and identify commonalities and differences. In the current curriculum developments, we notice a reduction in the content of calculus and a more informal approach. The use of digital tools has started to be integrated in calculus teaching in most countries. However in some nations, teaching of calculus in the classroom is rather traditional, focusing on procedural aspects of knowledge. Moreover, in cases where more informal and conceptual teaching approaches are used in the classroom, often contradictions seem to exist with other contextual matters such as examination requirements. Finally, we discuss the future of calculus teaching in Europe.     

Weakly stable Banach spaces

The class of stable Banach spaces, inspired by the stability theory in mathematical logic, was introduced by Krivine and Maurey and provided the proper context for the abstract formulation of Aldous’ result of subspaces ofL ¹. In this paper we study the wider class of weakly stable Banach spaces, where the exchangeability of the iterated limits occurs only for sequences belonging to weakly compact subsets, introduced independently by Garling (in an earlier unpublished version of his expository paper on stable Banach spaces brought recently to our attention) and by the authors. Taking into account Rosenthal’s application of the study of pointwise compact sets of Baire-1 functions (Rosenthal compact spaces) in the study of Banach spaces (for whichl ¹ does not embed isomorphically) and of the study of Rosenthal compact sets by Rosenthal and Bourgain-Fremlin-Talagrand, we prove the following analogue of the Krivine-Maurey theorem for weakly stable spaces:If X is infinite dimensional and weakly stable then either l ^p for some p≧1or c_o embeds isomorphically in X (§1). Garling (in the above reference) proved this result under the additional assumption thatX* is separable. We also construct an example of a Banach spaceX which is weakly stable, without an equivalent stable norm, and such thatl ² embeds isomorphically in every infinite dimensional subspace ofX (§3).     

Solid-State coextrusion of high-density polyethylene. II. Effect of molecular weight and molecular weight distribution

The recently developed technique of solid-state coextrusion for ultradrawing semicrystalline thermoplastics has been applied in the preparation of self-reinforced high-density polyethylene extrudates. The extrudates consist of definite core and sheath phases composed of different molecular weights (M_w) in the range of 60,000–250,000 and different molecular weight distributions (M_w/M_n = 3.0–20). Concentric billets of two different phases were prepared for extrusion by in serting a polyethylene rod within a tubular billet of a different high-density polyethylene followed by melting the two phases to obtain bonding between them. The billet was then split longitudinally to increase extrusion speed and extruded at 120°C, 0.23 GPa through a conical die of extrusion draw ratio 25. Extrudates of high tensile modulus (38 GPa) and strength (0.50 GPa) could be produced in a steady state process at a rate near 0.25 cm/min. The tensile properties of the extrudates from either the single or concentric billets increased with average molecular weight and were insensitive to the molecular weight distribution of the constituent phases. Thermal analysis indicated a high deformation efficiency for the sheath and core phases of the extrudates by the coextrusion technique.     

First year mathematics undergraduates’ settled images of tangent line

This study concerns 182 first year mathematics undergraduates’ perspectives on the tangent line of function graph in the light of a previous study on Year 12 pupils’ perspectives. The aim was the investigation of tangency images that settle after undergraduates’ distancing from the notion for a few months and after their participation in university admission examination. To this end we related the performances of the undergraduates and the pupils in the same questions of a questionnaire; we classified the undergraduates in distinct groups through Latent Class Analysis; and, we examined this classification according to the Analytical Local, Geometrical Global and Intermediate Local perspectives on tangency we had identified among pupils. The findings suggest that more undergraduates than pupils demonstrated intermediate perspectives on tangency. Also, the undergraduates’ settled images were influenced by persistent images about tangency and their prior experience in the context of preparation for and participation in the examination.