54,441 research outputs found
Generalized Triangular Decomposition in Transform Coding
A general family of optimal transform coders (TCs) is introduced here based on the generalized triangular decomposition (GTD) developed by Jiang This family includes the Karhunen-Loeve transform (KLT) and the generalized version of the prediction-based lower triangular transform (PLT) introduced by Phoong and Lin as special cases. The coding gain of the entire family, with optimal bit allocation, is equal to that of the KLT and the PLT. Even though the original PLT introduced by Phoong is not applicable for vectors that are not blocked versions of scalar wide sense stationary processes, the GTD-based family includes members that are natural extensions of the PLT, and therefore also enjoy the so-called MINLAB structure of the PLT, which has the unit noise-gain property. Other special cases of the GTD-TC are the geometric mean decomposition (GMD) and the bidiagonal decomposition (BID) transform coders. The GMD-TC in particular has the property that the optimum bit allocation is a uniform allocation; this is because all its transform domain coefficients have the same variance, implying thereby that the dynamic ranges of the coefficients to be quantized are identical
Quantum dynamical Yang-Baxter equation over a nonabelian base
In this paper we consider dynamical r-matrices over a nonabelian base. There
are two main results. First, corresponding to a fat reductive decomposition of
a Lie algebra \frakg =\frakh \oplus \frakm, we construct geometrically a
non-degenerate triangular dynamical r-matrix using symplectic fibrations.
Second, we prove that a triangular dynamical r-matrix r: \frakh^* \lon
\wedge^2 \frakg corresponds to a Poisson manifold \frakh^* \times G. A
special type of quantizations of this Poisson manifold, called compatible star
products in this paper, yields a generalized version of the quantum dynamical
Yang-Baxter equation
(or Gervais-Neveu-Felder equation). As a result, the quantization problem of
a general dynamical r-matrix is proposed.Comment: 23 pages, minor changes made, final version to appear in Comm. Math.
Phy
Highest-Weight Theory for Truncated Current Lie Algebras
Let g denote a Lie algebra over a field of characteristic zero, and let T(g)
denote the tensor product of g with a ring of truncated polynomials. The Lie
algebra T(g) is called a truncated current Lie algebra, or in the special case
when g is finite-dimensional and semisimple, a generalized Takiff algebra. In
this paper a highest-weight theory for T(g) is developed when the underlying
Lie algebra g possesses a triangular decomposition. The principal result is the
reducibility criterion for the Verma modules of T(g) for a wide class of Lie
algebras g, including the symmetrizable Kac-Moody Lie algebras, the Heisenberg
algebra, and the Virasoro algebra. This is achieved through a study of the
Shapovalov form.Comment: 42 pages. An extract from the author's PhD thesis. See also:
http://www.maths.usyd.edu.au/u/benw
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