Fuzzy Toric Geometries

We describe a construction of fuzzy spaces which approximate projective toric varieties. The construction uses the canonical embedding of such varieties into a complex projective space: The algebra of fuzzy functions on a toric variety is obtained by a restriction of the fuzzy algebra of functions on the complex projective space appearing in the embedding. We give several explicit examples for this construction; in particular, we present fuzzy weighted projective spaces as well as fuzzy Hirzebruch and del Pezzo surfaces. As our construction is actually suited for arbitrary subvarieties of complex projective spaces, one can easily obtain large classes of fuzzy Calabi-Yau manifolds and we comment on fuzzy K3 surfaces and fuzzy quintic three-folds. Besides enlarging the number of available fuzzy spaces significantly, we show that the fuzzification of a projective toric variety amounts to a quantization of its toric base.Comment: 1+25 pages, extended version, to appear in JHE

Branes, Quantization and Fuzzy Spheres

We propose generalized quantization axioms for Nambu-Poisson manifolds, which allow for a geometric interpretation of n-Lie algebras and their enveloping algebras. We illustrate these axioms by describing extensions of Berezin-Toeplitz quantization to produce various examples of quantum spaces of relevance to the dynamics of M-branes, such as fuzzy spheres in diverse dimensions. We briefly describe preliminary steps towards making the notion of quantized 2-plectic manifolds rigorous by extending the groupoid approach to quantization of symplectic manifolds.Comment: 18 pages; Based on Review Talk at the Workshop on "Noncommutative Field Theory and Gravity", Corfu Summer Institute on Elementary Particles and Physics, September 8-12, 2010, Corfu, Greece; to be published in Proceedings of Scienc

Towards a generalisation of formal concept analysis for data mining purposes

In this paper we justify the need for a generalisation of Formal Concept Analysis for the purpose of data mining and begin the synthesis of such theory. For that purpose, we first review semirings and semimodules over semirings as the appropriate objects to use in abstracting the Boolean algebra and the notion of extents and intents, respectively. We later bring to bear powerful theorems developed in the field of linear algebra over idempotent semimodules to try to build a Fundamental Theorem for K-Formal Concept Analysis, where K is a type of idempotent semiring. Finally, we try to put Formal Concept Analysis in new perspective by considering it as a concrete instance of the theory developed

Supersymmetric quantum theory and non-commutative geometry

Classical differential geometry can be encoded in spectral data, such as Connes' spectral triples, involving supersymmetry algebras. In this paper, we formulate non-commutative geometry in terms of supersymmetric spectral data. This leads to generalizations of Connes' non-commutative spin geometry encompassing non-commutative Riemannian, symplectic, complex-Hermitian and (Hyper-)Kaehler geometry. A general framework for non-commutative geometry is developed from the point of view of supersymmetry and illustrated in terms of examples. In particular, the non-commutative torus and the non-commutative 3-sphere are studied in some detail.Comment: 77 pages, PlainTeX, no figures; present paper is a significantly extended version of the second half of hep-th/9612205. Assumptions in Sect. 2.2.5 clarified; final version to appear in Commun.Math.Phy

The relationship between IR and multimedia databases

Modern extensible database systems support multimedia data through ADTs. However, because of the problems with multimedia query formulation, this support is not sufficient.\ud \ud Multimedia querying requires an iterative search process involving many different representations of the objects in the database. The support that is needed is very similar to the processes in information retrieval.\ud \ud Based on this observation, we develop the miRRor architecture for multimedia query processing. We design a layered framework based on information retrieval techniques, to provide a usable query interface to the multimedia database.\ud \ud First, we introduce a concept layer to enable reasoning over low-level concepts in the database.\ud \ud Second, we add an evidential reasoning layer as an intermediate between the user and the concept layer.\ud \ud Third, we add the functionality to process the users' relevance feedback.\ud \ud We then adapt the inference network model from text retrieval to an evidential reasoning model for multimedia query processing.\ud \ud We conclude with an outline for implementation of miRRor on top of the Monet extensible database system