Deformations of modules of maximal grade and the Hilbert scheme at determinantal schemes

Let R be a polynomial ring and M a finitely generated graded R-module of maximal grade (which means that the ideal I_t(\cA) generated by the maximal minors of a homogeneous presentation matrix, \cA, of M has maximal codimension in R). Suppose X:=Proj(R/I_t(\cA)) is smooth in a sufficiently large open subset and dim X > 0. Then we prove that the local graded deformation functor of M is isomorphic to the local Hilbert (scheme) functor at X \subset Proj(R) under a week assumption which holds if dim X > 1. Under this assumptions we get that the Hilbert scheme is smooth at (X), and we give an explicit formula for the dimension of its local ring. As a corollary we prove a conjecture of R. M. Mir\'o-Roig and the author that the closure of the locus of standard determinantal schemes with fixed degrees of the entries in a presentation matrix is a generically smooth component V of the Hilbert scheme. Also their conjecture on the dimension of V is proved for dim X > 0. The cohomology H^i_{*}({\cN}_X) of the normal sheaf of X in Proj(R) is shown to vanish for 0 < i < dim X-1. Finally the mentioned results, slightly adapted, remain true replacing R by any Cohen-Macaulay quotient of a polynomial ring.Comment: 24 page

On a graded q-differential algebra

Given a unital associatve graded algebra we construct the graded q-differential algebra by means of a graded q-commutator, where q is a primitive N-th root of unity. The N-th power (N>1) of the differential of this graded q-differential algebra is equal to zero. We use our approach to construct the graded q-differential algebra in the case of a reduced quantum plane which can be endowed with a structure of a graded algebra. We consider the differential d satisfying d to power N equals zero as an analog of an exterior differential and study the first order differential calculus induced by this differential.Comment: 6 pages, submitted to the Proceedings of the "International Conference on High Energy and Mathematical Physics", Morocco, Marrakech, April 200

On special quadratic birational transformations of a projective space into a hypersurface

We study transformations as in the title with emphasis on those having smooth connected base locus, called "special". In particular, we classify all special quadratic birational maps into a quadric hypersurface whose inverse is given by quadratic forms by showing that there are only four examples having general hyperplane sections of Severi varieties as base loci.Comment: Accepted for publication in Rendiconti del Circolo Matematico di Palerm

The application of Item Response Theory on a teaching strategy profile questionnaire

<p>Abstract</p> <p>Background</p> <p>In medical education research, various questionnaires are often used to study possible relationships between strategies and approaches to teaching and learning and the outcome of these. However, judging the applicability of such questionnaires or the interpretation of the results is not trivial.</p> <p>Methods</p> <p>As a way to develop teacher thinking, teaching strategy profiles were calculated for teachers in a research intensive department at Karolinska Institutet. This study compares the sum score, that was inherent in the questionnaire used, with an Item Response Theory (IRT) approach. Three teaching dimensions were investigated and the intended sum scores were investigated by IRT analysis.</p> <p>Results</p> <p>Agreements as well as important differences were found. The use of the sum score seemed to agree reasonably with an IRT approach for two of the dimensions, while the third dimension could not be identified neither by a the sum score, nor by an IRT approach, as the items included showed conflicting messages.</p> <p>Conclusions</p> <p>This study emphasizes the possibilities to gain better insight and more relevant interpretation of a questionnaire by use of IRT. A sum score approach should not be taken for granted. Its use has to be thoroughly evaluated.</p

Beyond academic development as institutional practice: advancing community-led approaches

We introduce the special issue entitled ‘Beyond academic development as institutional practice: advancing community-led approaches’ and offer critical commentary that advances the concept and practice of academic community development. Through repeated readings of the 10 articles from diverse contexts included in this issue, we draw out four main themes highlighting why and how academic communities beyond single institutions support academic development. First, we revise the initial conceptual framework of ‘academic community development (ACD)’, which focused the call for papers, presenting six dimensions rather than the previous three continua. Then, we consider the degree of ‘institutionalisation’ of this range of initiatives. From that analysis, we derive a typology of ACD initiatives. We then consider a) what common interests have been served by these communities, and whose they are; b) the focus of these communities in relation to the revised conceptual framework; and c) the essential ingredients for success. We emphasise that communities leverage assets in ways that traditional needs-based academic development often does not. We conclude with implications for typical academic development work