We review the notion of a vector space, basis and dimension, linear transformations between vector spaces, dual vector spaces and transformations, spectral decomposition for normal operators (which includes symmetric, Hermitian, orthogonal, and unitary operators), and determinants. Along the way we review direct-sum decompositions, bilinear forms and inner product spaces, adjoints, characterist...

Analytic-bilinear approach for construction and study of integrable hierarchies, in particular, the KP hierarchy is discussed. It is based on the generalized Hirota identity. This approach allows to represent generalized hierarchies of integrable equations in a condensed form of finite functional equations. Resolution of these functional equations leads to the τ-function and addition formulae t...

Based on an extension of the bilinear transformation, a rational implementation for distributed delay in linear control laws is proposed. This implementation converges much faster than the rational implementation inspired from the δ-operator. The implementation has an elegant structure of chained bi-proper nodes cascaded with a strictly proper node. The stability of each node is determined by t...

It is helpful to have available systematic ways to find a variety useful conformal mappings. Our text treats one such method: bilinear transformations. Here we study a second: Schwarz-Christoffel transformations. These are discussed in many references; I recommend particularly [1], a lovely book on complex variable theory which has a decided applied slant. The Schwarz-Christoffel transformation...

A noncommutative version of the KP equation and two families of its solutions expressed as quasideterminants are discussed. The origin of these solutions is explained by means of Darboux and binary Darboux transformations. Additionally, it is shown that these solutions may also be verified directly. This approach is reminiscent of the wronskian technique used for the Hirota bilinear form of the...

Two different approaches for describing discrete-time fractional linear systems are presented. The first is based on the nabla and delta derivatives. In this case, suitable exponentials introduced used to define discrete Laplace transforms. second approach bilinear (Tustin) transformations. For both cases, appropriate algorithms obtaining impulse, step, frequency responses state-variable repres...

For a principal SU(n)-bundle over a compact manifold of dimension 2, 3, 4, we determine the orbit types of the action of the gauge group on the space of connections modulo pointed local gauge transformations. We find that they are given by Howe subgroups of SU(n) for which a certain characteristic equation is solvable. Depending on the base manifold, this equation leads to a linear, bilinear, o...

The paper discusses the synthesis of switched-current filters from continuous-time prototypes by the simulation of components, particularly capacitors and transconductors in OTA-C filters. The obtained structures implement discretetime versions of the nodal equations of the prototype circuit using Euler or bilinear transformations over the circuit elements. All the proposed structures work with...