The root systems appearing in the theory of Lie superalgebras and Nichols algebras admit a large symmetry extending properly the one coming from the Weyl group. Based on this observation we set up a general framework in which the symmetry object is a groupoid. We prove that in our context the groupoid is generated by reflections and Coxeter relations. This answers a question of Serganova. Our w...

We show that the graded set of filter homotopy classes rel vertices of maps from the n-globe to a filtered space may be given the structure of (strict) globular ω-groupoid. The proofs use an analogous fundamental cubical ω-groupoid due to the author and Philip Higgins in 1981. This method also relates the construction to the fundamental crossed complex of a filtered space, and this relation all...

We construct an algebra of pseudodifferential operators on each groupoid in a class that generalizes differentiable groupoids to allow manifolds with corners. We show that this construction encompasses many examples. The subalgebra of regularizing operators is identified with the smooth algebra of the groupoid, in the sense of non-commutative geometry. Symbol calculus for our algebra lies in th...

We prove the existence of a strict deformation quantization for the canonical Poisson structure on the dual of an integrable Lie algebroid. It follows that any Lie groupoid C-algebra may be regarded as a result of a quantization procedure. The C-algebra of the tangent groupoid of a given Lie groupoid G (with Lie algebra G) is the C-algebra of a continuous field of C-algebras over R with fibers ...

Suppose G is a second countable, locally compact, Hausdorff groupoid with a fixed left Haar system. Let G/G denote the orbit space of G and C∗(G) denote the groupoid C∗-algebra. Suppose that the isotropy groups of G are amenable. We show that C∗(G) is CCR if and only if G/G is a T1 topological space and all of the isotropy groups are CCR. We also show that C∗(G) is GCR if and only if G/G is a T...

If one considers life on Earth since its very beginning, three levels of organization can be distinguished: the phylogenetic level concerns the temporal evolution of the genetic programs within individuals and species, the ontogenetic level concerns the developmental process of a single multicellular organism, and the epigenetic level concerns the learning processes during an individual organis...

The importance of Post-Occupancy Evaluation (POE) surveys is paramount in understanding building performance and evaluating building technologies. However, only a small percentage of completed buildings go through a POE. A reason is that clients and non-specialist designers often do not see the value POE can add, and consider it a specialist pursuit. This paper presents ways developed to addres...

This paper explains the new concept of buffered probability of exceedance (bPOE). Through averaging of data in the tail, bPOE compactly presents probability as well as loss value of the tail and has the capability to revolutionize the concept of risk-averse engineering. bPOE is demonstrated using tropical storm damage along the Atlantic and Gulf of Mexico coasts of the USA from 1900 to 2013 (av...