Boundedness, rigidity and global rigidity of direction�Clength frameworks

We describe characterizations of boundedness and rigidity of generic 2-dimensional direction?Clength frameworks, and give partial results for the open problem of characterizing the global rigidity of such frameworks.     

STUDENT TEACHERS’ UNDERSTANDING OF RATIONAL NUMBER: PART-WHOLE AND NUMERICAL CONSTRUCTS

Previous research has shown that secondary school students’ understanding of fractions is dominated by the part-whole concept to the possible detriment of their understanding of a fraction as a number in its own right. The present paper reports on an investigation into the understanding of intending primary teachers in this area. Four representatives of a cohort of sixty students on a PGCE course specialising in the lower primary age range were asked detailed questions probing their knowledge of fractions. The conclusion was that the part-whole concept dominates. All of the students were familiar with the numerical concept from their work on the PGCE course, but they reverted to the more familiar part-whole ideas in attempting to solve problems.     

A transformation theorem for one-dimensional F-expansions

If f is a monotone function subject to certain restrictions, then one can associate with any real number x between zero and one a sequence {a_n(x)} of integers such that

$\text{x=f(a}{}_1\text{(x) + f(a}{}_2\text{(x) +f(a}{}_3\text{(x) +…)))}$

. In this paper properties of the function F defined by

$\text{Fx=g(a}{}_1\text{(x) + g(a}{}_2\text{(x) +g(a}{}_3\text{(x) +…)))}$

, where g is any function satisfying the same restrictions as f, are discussed. Principally, F is found to be useful in finding stationary measures on the sequences {a_n(x)}.     

Electronic inhomogeneities,electron-lattice and pairing interactions in high-T c superconductors

The presence of electronic inhomogeneities strongly reduces the screening of the electron-ion interaction in high-temperature superconductors. This implies the existence of an non-totally screened long-range contribution to the electron-lattice coupling and opens an additional channel for the formation of copper pairs. We calculate the superconducting order parameter taking into account a) the longrange and the short-range parts of the electron-lattice interaction and b) the Coulomb repulsion between charge-carriers. We show that whereas the long-range electron-lattice coupling determines the anisotropy of the order parameter, the Coulomb repulsion and the short-range interactions determine its symmetry. Thus, different high-T _c superconductors may have s- or d-wave symmetry, depending on the relative strength of the interactions involved in the pairing.     

Bounded Direction–Length Frameworks

A direction–length framework is a pair (G,p) where G=(V;D,L) is a ‘mixed’ graph whose edges are labelled as ‘direction’ or ‘length’ edges and p is a map from V to ℝ ^d for some d. The label of an edge uv represents a direction or length constraint between p(u) and p(v). Let G ⁺ be obtained from G by adding, for each length edge e of G, a direction edge with the same end vertices as e. We show that (G,p) is bounded if and only if (G ⁺,p) is infinitesimally rigid. This gives a characterization of when (G,p) is bounded in terms of the rank of the rigidity matrix of (G ⁺,p). We use this to characterize when a mixed graph is generically bounded in ℝ ^d . As an application we deduce that if (G,p) is a globally rigid generic framework with at least two length edges and e is a length edge of G then (G∖e,p) is bounded.