This is a slight extension of an expository paper I wrote a while ago as a supplement to my joint work with Declan Quinn on Burnside’s theorem for Hopf algebras. It was never published, but may still be of interest to students and beginning researchers. Let K be a field and let A be an algebra over K. Then the tensor product A⊗A = A⊗K A is also a K-algebra, and it is quite possible that there e...

The geometry of Rn is controlled algebraically by the dot product. We will abstract the dot product on Rn to a bilinear form on a vector space and study algebraic and geometric notions related to bilinear forms (especially the concept of orthogonality in all its manifestations: orthogonal vectors, orthogonal subspaces, and orthogonal bases). Section 1 defines a bilinear form on a vector space a...

A possible notion of superspace associated to a given 3-dimensional Lie algebra g0 might be a Lie superalgebra g g0 ⊕ g1. If one is to introduce a minimal set of assumptions, it seems quite natural to consider those superalgebras for which g1 g0 and for which the g0 action on g1 is the adjoint representation. A complete classification of the real and complex 3, 3 -dimensional Lie superalgebras ...

We describe the classical o(3, 2) r-matrices as generating the quantum deformations of either D=3 conformal algebra with mass-like deformation parameters or D=4 AdS algebra with dimensionless deformation parameters. We describe the quantization of classical o(3, 2) r-matrices via Drinfeld twist method which locates the deformation in the coalgebra sector. Further we obtain the quantum o(3, 2) a...

We describe the classical o(3, 2) r-matrices as generating the quantum deformations of either D=3 conformal algebra with mass-like deformation parameters or D=4 AdS algebra with dimensionless deformation parameters. We describe the quantization of classical o(3, 2) r-matrices via Drinfeld twist method which locates the deformation in the coalgebra sector. Further we obtain the quantum o(3, 2) a...

The authors prove that bilinear operators given by finite sums of products of Calderón-Zygmund operators on Rn are bounded from HK̇11 q1 × HK̇ α2,p2 q2 into HK̇ q if and only if they have vanishing moments up to a certain order dictated by the target space. Here HK̇ q are homogeneous Herz-type Hardy spaces with 1/p = 1/p1 +1/p2, 0 < pi ≤ ∞, 1/q = 1/q1 +1/q2, 1 < q1, q2 < ∞, 1 ≤ q < ∞, α = α1 + α2 a...

A difference analogue of the logistic equation, which preserves integrability, is derived from Hirota’s bilinear difference equation. The integrability of the map is shown to result from the large symmetry associated with the Bäcklund transformation of the KP hierarchy. We introduce a scheme which interpolates between this map and the standard logistic map and enables us to study integrable and...

PURPOSE Dual-energy CT (DECT) is arguably the most accurate energy mapping technique in CT-based attenuation correction (CTAC) implemented on hybrid PET/CT systems. However, this approach is not attractive for clinical use owing to increased patient dose. The authors propose a novel energy mapping approach referred to as virtual DECT (VDECT) taking advantage of the DECT formulation but using CT...

Filter-bank outputs are extended into tensors to yield precise acoustic features for speech recognition using deep neural networks (DNNs). The filter-bank outputs with temporal contexts form a time-frequency pattern of speech and have been shown to be effective as a feature parameter for DNN-based acoustic models. We attempt to project the filter-bank outputs onto a tensor product space using d...

There are many good references for this material including [EKM], [L], [Pf] and [S]. 1 Quadratic forms Let k be a field with char k = 2. Definition 1.1. A quadratic form q : V → k on a finite-dimensional vector space V over k is a map satisfying: 1. q(λv) = λ 2 q(v) for v ∈ V , λ ∈ k. 2. The map b q : V × V → k, defined by b q (v, w) = 1 2 [q(v + w) − q(v) − q(w)] is bilinear. We denote a quadr...