This paper examines the relationship between certain non-commutative analogues of projective 3space, P 3 , and the quantized enveloping algebras Uq(sl2). The relationship is mediated by certain non-commutative graded algebras S, one for each q ∈ C × , having a degree-two central element c such that S[c −1 ]0 ∼ = Uq(sl2). The non-commutative analogues of P 3 are the spaces Proj nc (S). We show how the points, fat points, lines, and quadrics, in Proj nc (S), and their incidence relations, correspond to finite dimensional irreducible representations of Uq(sl2), Verma modules, annihilators of Verma modules, and homomorphisms between them.

Sensor networks are increasingly being used to provide timely information about the physical, urban and human environment. Algorithms that depend on sensor data often assume that the readings are complete. However, node failures or communication breakdowns result in missing data entries, preventing the use of such algorithms. To impute these missing values, we propose a method of exploiting spatial correlations which is based on the sparse autoencoder and inspired by the conditional Restricted Boltzmann Machine that contested for the Netflix Prize. We modify the autoencoder to cope with missing data, and test it on data from a sensor testbed in Santander, Spain. We show that our algorithm extracts features from datasets with high proportions of missing data and uses these features to accurately and efficiently impute missing entries.

We define and study opfibrations of V-enriched categories when V is an extensive monoidal category whose unit is terminal and connected. This includes sets, simplicial sets, categories, or any locally cartesian closed category with disjoint coproducts and connected unit. We show that for an ordinary category B, there is an equivalence of 2categories between V-enriched opfibrations over the free V-category on B, and pseudofunctors from B to the 2-category of V-categories. This generalizes the classical (Set-enriched) Grothendieck correspondence.

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