Diptych varieties. I

We present a new class of affine Gorenstein 6-folds obtained by smoothing the 1-dimensional singular locus of a reducible affine toric surface; their existence is established using explicit methods in toric geometry and serial use of Kustin-Miller Gorenstein unprojection. These varieties have applications as key varieties in constructing other varieties, including local models of Mori flips of Type A.Comment: 50 pages. The webpage at www-staff.lboro.ac.uk/~magdb/aflip.html contains links to auxiliary materia

Gopakumar-Vafa invariants do not determine flops

Two 3-fold flops are exhibited, both of which have precisely one flopping curve. One of the two flops is new, and is distinct from all known algebraic D4-flops. It is shown that the two flops are neither algebraically nor analytically isomorphic, yet their curve-counting Gopakumar-Vafa invariants are the same. We further show that the contraction algebras associated to both are not isomorphic, so the flops are distinguished at this level. This shows that the contraction algebra is a finer invariant than various curve-counting theories, and it also provides more evidence for the proposed analytic classification of 3-fold flops via contraction algebras.Comment: 10 pages, final versio

Implementation of computer assisted assessment: lessons from the literature

This paper draws attention to literature surrounding the subject of computer-assisted assessment (CAA). A brief overview of traditional methods of assessment is presented, highlighting areas of concern in existing techniques. CAA is then defined, and instances of its introduction in various educational spheres are identified, with the main focus of the paper concerning the implementation of CAA. Through referenced articles, evidence is offered to inform practitioners, and direct further research into CAA from a technological and pedagogical perspective. This includes issues relating to interoperability of questions, security, test construction and testing higher cognitive skills. The paper concludes by suggesting that an institutional strategy for CAA coupled with staff development in test construction for a CAA environment can increase the chances of successful implementation

Modular Autoencoders for Ensemble Feature Extraction

We introduce the concept of a Modular Autoencoder (MAE), capable of learning a set of diverse but complementary representations from unlabelled data, that can later be used for supervised tasks. The learning of the representations is controlled by a trade off parameter, and we show on six benchmark datasets the optimum lies between two extremes: a set of smaller, independent autoencoders each with low capacity, versus a single monolithic encoding, outperforming an appropriate baseline. In the present paper we explore the special case of linear MAE, and derive an SVD-based algorithm which converges several orders of magnitude faster than gradient descent.Comment: 18 pages, 8 figures, to appear in a special issue of The Journal Of Machine Learning Research (vol.44, Dec 2015

Gorenstein formats, canonical and Calabi-Yau threefolds

We extend the known classification of threefolds of general type that are complete intersections to various classes of non-complete intersections, and find other classes of polarised varieties, including Calabi-Yau threefolds with canonical singularities, that are not complete intersections. Our methods apply more generally to construct orbifolds described by equations in given Gorenstein formats.Comment: 25 page