Search results for “group schemes”

A proof of Raynaud's theorem: every finite flat commutative group scheme over a locally noetherian base embeds, Zariski-locally, as a closed subgroup of an abelian scheme.

Every finite flat commutative group scheme over a noetherian local ring is the torsion component of the Picard scheme of a smooth projective scheme with 3-dimensional fibers, built as a quotient of a complete intersection. An application: Hodge numbers that jump in a smooth projective family.

A rigidity condition on schemes, strictly stronger than smoothness, introduced with examples and a look at where it fails to be geometric — no known application, just a curiosity.

An example, due to Serre, of a smooth projective surface in positive characteristic whose Picard scheme fails to be reduced, worked out via the two governing dimension inequalities.

With an appendix by Sean Cotner, Joshua M. Lansky, and Loren Spice.

The Millennium Meeting, Paris, 24 May 2000, at which the seven problems were announced.

48th International Mathematical Olympiad, 2007.

55th International Mathematical Olympiad. Cape Town, South Africa, 2014.

Not intended for publication.

With the increasing popularity of cashless payment methods for everyday, seasonal and special expenses popular banks accumulate huge amount of data about customer operations. In the article, we report a successful application of topic modelling to extract behaviour patterns from the data. The resulting models are built with BigARTM framework: flexible and efficient tool for topic modelling. The framework allows us to experiment with various models including PLSA, LDA and beyond. Results demonstrate ability of the approach to aggregate information about behaviour patterns of different customer groups. The results analysis allows to see the topics of such people clusters varying from travellers to mortgage holders. Moreover, low-dementional embeddings of the customers, which was given with topic model, were studied. We display that the client vector representations store demographic information as well as source data. We also test for a best way of preparing data for the model with metric above in mind.