The exterior algebra ΛkRn and area minimization

The structure of the exterior algebra Λ^kRⁿ is studied in low dimensions, and consequences are drawn for k-dimensional area-minimizing surfaces in Rⁿ. For a general form ø∈Λ²R⁴, Section 3 gives explicit formulas for the comass 6ø6 and the face of the Grassmannian exposed by ø. Section 4 classifies the faces of the Grassmannian of the 3-planes in R⁶ and hence the associated geometries of area-minimizing surfaces (there are four types). Section 5 establishes an equality involving the comass norm in low dimension and draws implications on when the Cartesian product of area-minimizing surfaces is area-minimizing. New examples of area-minimizing integral currents with singularities follow.     

Stability of Regularized Bilevel Programming Problems

A bilevel programming problem S is considered. First, sufficient conditions of minimal character are given on the data of the problem in order to guarantee the lower semicontinuity of the marginal function of the upper level problem. Then, for >0, a regularized problem S() is considered for which continuity of the regularized marginal function and convergence of the approximate value, as goes to zero, are obtained. Moreover, under perturbations on the data, convergence results for the perturbed marginal functions and the solutions to the problem S _n() are given for any >0.     

A method to bypass the lack of solutions in MinSup problems under quasi-equilibrium constraints

MinSup problems with constraints described by quasi-equilibrium problems are considered in Banach spaces. The solutions set of such problems may be empty even in very good situations, so the aim of this paper is twofold. First, we determine appropriate regularizations (called inner regularizations) which allow to reach the value of the original problem. Then, among these regularizations we identify those which allow to bypass the lack of exact solutions to these problems by a suitable concept of “viscosity” solution whose existence is then proved under reasonable assumptions.     

Investigating changes in high-stakes mathematics examinations: a discursive approach

ABSTRACT

This article focuses on the theoretical-methodological question of how to identify reform-induced changes in school mathematics. The issue arose in our project The Evolution of the Discourse of School Mathematics (EDSM), in which we studied transformations in high-stakes examinations taken by students in England at the end of compulsory schooling. We have adopted a conceptualisation that draws on social semiotics and on a communicational approach, according to which school mathematics can be thought of as a discourse. Methods of comparing examinations of different years developed on the basis of this definition enable identification of subtle disparities that are nevertheless significant enough to make an important difference in students’ vision of mathematics, in their performance and, eventually, in their ability to cope with problems that can benefit from the use of mathematics. In this article, we present these methods and argue that they have wider application for comparative studies of school mathematics.     

Bilayer Networks within a Hydrogel Shell: A Robust Chassis for Artificial Cells and a Platform for Membrane Studies

The ability to make artificial lipid bilayers compatible with a wide range of environments, and with sufficient structural rigidity for manual handling, would open up a wealth of opportunities for their more routine use in real‐world applications. Although droplet interface bilayers (DIBs) have been demonstrated in a host of laboratory applications, from chemical logic to biosynthesis reaction vessels, their wider use is hampered by a lack of mechanical stability and the largely manual methods employed in their production. Multiphase microfluidics has enabled us to construct hierarchical triple emulsions with a semipermeable shell, in order to form robust, bilayer‐bound, droplet networks capable of communication with their external surroundings. These constructs are stable in air, water, and oil environments and overcome a critical obstacle of achieving structural rigidity without compromising environmental interaction. This paves the way for practical application of artificial membranes or droplet networks in diverse areas such as medical applications, drug testing, biophysical studies and their use as synthetic cells.     

Direct imaging of o-carborane molecules within single walled carbon nanotubes

Ortho-carborane molecules have been inserted into single walled carbon nanotubes (SWNTs) and imaged directly by high resolution transmission electron microscopy (HRTEM); both discrete molecules and 'zig-zag' 1D chains of o-carborane 'petit pois' were observed to pack into the tubule capillaries.     

A solid-phase route to N-cyanoamides

[reaction: see text] A new method for the solid-phase synthesis of cyanamides is described. The attachment of a secondary amine to solid support is accomplished using Merrifield resin. After functionalization, cleavage is readily achieved with cyanogen bromide to afford the desired cyanamide.