Daniel Alpay

Daniel Alpay, Izchak Lewkowicz

We here extend the well known Positive Real Lemma (also known as the Kalman-Yakubovich-Popov Lemma) to complex matrix-valued generalized positive rational function, when non-minimal realizations are considered. We then exploit this result to provide an easy construction procedure of all (not necessarily minimal) state space realizations of generalized positive functions. As a by-product, we partition all state space realizations into subsets: Each is identified with a set of matrices satisfying the same Lyapunov inclusion and thus form a convex invertible cone, cic in short. Moreover, this approach enables us to characterize systems which may be brought to be generalized positive through static output feedback. The formulation through Lyapunov inclusions suggests the introduction of an equivalence class of rational functions of various dimensions associated with the same system matrix.

Daniel Alpay, Palle Jorgensen, Izchak Lewkowicz

We here use notions from the theory linear shift-invariant dynamical systems to provide an easy-to-compute characterization of all rational wavelet filters. For a given N bigger or equql to 2, the number of inputs, the construction is based on a factorization to an elementary wavelet filter along with of m elementary unitary matrices. We shall call this m the index of the filter. It turns out that the resulting wavelet filter is of McMillan degree $N((N-1)/2+m). Rational wavelet filters bounded at infinity, admit state space realization. The above input-output parameterization is exploited for a step-by-step construction (where in each the index m is increased by one) of state space model of wavelet filters.

Daniel Alpay, Kamal Diki, Mihaela Vajiac

We study and introduce new gradient operators in the complex and bicomplex settings, inspired from the well-known Least Mean Square (LMS) algorithm invented in 1960 by Widrow and Hoff for Adaptive Linear Neuron (ADALINE). These gradient operators will be used to formulate new learning rules for the Bicomplex Least Mean Square (BLMS) algorithms and we will also formulate these learning rules will for the case of multicomplex LMS algorithms (MLMS). This approach extends both the classical real and complex LMS algorithms.

Daniel Alpay, Kamal Diki, Mihaela Vajiac

In this paper we first write a proof of the perceptron convergence algorithm for the complex multivalued neural networks (CMVNNs). Our primary goal is to formulate and prove the perceptron convergence algorithm for the bicomplex multivalued neural networks (BMVNNs) and other important results in the theory of neural networks based on a bicomplex algebra.