No abstract

No abstract

No abstract

No abstract

A new formalism, termed the resolution space, is presented within which the theory of causal systems may be unified and extended. The resulting formalism, which is defined as a Hilbert space together with a resolution of the identity, readily includes the commonly encountered function and sequence space causality concepts yet is sufficiently straightforward to allow the various aspects of network and system theory which are dependent on the time parameter to be studied in an operator theoretic context without the detailed structure of a function space. Specific results include additive and multiplicative decomposition theorems for causal operators which naturally extend the "realizable part" and "spectral" decompositions of classical system theory and an integral representation theorem for linear operators on a resolution space. The general theory is illustrated with a number of examples concerning passive "networks", those including an operator theoretic approach to the passive synthesis problem over an arbitrary resolution space.In the various applications a number of different, but essentially equivalent, formulations of the causality concept have been given. These, of course, include the usual function and sequence space approaches (wherein operator [14], [34] and Hilbert space valued [3], [6], [12], [35], [37] functions are employed) as well as a number of invariant subspace formulations 7], 15], [21], [25]. Possibly *

No abstract

It is shown that a Hopfield neural network (with linear transfer functions) augmented by an additional feedforward layer can be used to compute the Moore-Penrose generalized inverse of a matrix. The resultant augmented linear Hopfield network can be used to solve an arbitrary set of linear equations or, alternatively, to solve a constrained least squares optimization problem. Applications in signal processing and robotics are considered. In the former case the augmented linear Hopfield network is used to estimate the "structured noise" component of a signal and adjust the parameters of an appropriate filter on-line, whereas in the latter case it is used to implement an on-line solution to the inverse kinematics problem.

No abstract