Transforming the canonical piecewise-linear model into a smooth-piecewise representation

Transforming the canonical piecewise-linear model into a smooth-piecewise representation

A smoothed representation (based on natural exponential and logarithmic functions) for the canonical piecewise-linear model, is presented. The result is a completely differentiable formulation that exhibits interesting properties, like preserving the parameters of the original piecewise-linear model in such a way that they can be directly inherited to the smooth model in order to determine thei...

A Complete 3-D Canonical Piecewise-Linear Representation

A complete 3-D Canonical Piecewise-Linear (CPWL) representation is developed constructively in this paper. The key to the representation is the establishment of the explicit functional formulation of basis function. It is proved that basis function is the most elementary generating function from which a fully general 3-D PWL function can be formulated. This CPWL representation laid a solid theo...

Canonical Bases and Piecewise-linear Combinatorics

Let Uq be the quantum group associated to a Lie algebra g of rank n. The negative part U q of Uq has a canonical basis B with favourable properties (see Kashiwara [3] and Lusztig [6, §14.4.6]). The approaches of Lusztig and Kashiwara lead to a set of alternative parametrizations of the canonical basis, one for each reduced expression for the longest word in the Weyl group of g. We describe the ...

8 Piecewise Linear Parametrization of Canonical Bases

In [L1] the author introduced the canonical basis for the plus part of a quantized enveloping algebra of type A,D or E. (The same method applies for nonsimplylaced types, see [L3, 12.1].) Another approach to the canonical basis was later found in [Ka]. In [L1] we have also found that the set parametrizing the canonical basis has a natural piecewise linear structure that is, a collection of bije...

Piecewise Linear Approximation of Smooth Compact Fibers