We prove that the Jordan Normal Form of a rational matrix can be computed exactly in polynomial time. We obtain the transformation matrix and its inverse exactly, and we show how to apply the basis transformation to any commuting matrices.

Besides Karatsuba algorithm, optimal Toeplitz matrix-vector product (TMVP) formulae is another approach to design GF (2) subquadratic multipliers. However, when GF (2) elements are represented using a shifted polynomial basis, this approach is currently appliable only to GF (2)s generated by all irreducible trinomials and a special type of irreducible pentanomials, not all general irreducible p...

An optical neural network is proposed and demonstrated with programmable matrix transformation nonlinear activation function of photodetection (square-law detection). Based on discrete phase-coherent spatial modes, the dimensionality operations 30~37, which implemented by light modulators. With this architecture, all-optical classification tasks handwritten digits, objects depth images are perf...

Presented to the Society, December, 1955 under the title ^4 necessary condition for nonoscillation of a system of second order differential equations, and August, 1956; received by the editors August 2, 1956. 1 These results were obtained while the author held a National Science Foundation grant, NSF-G1825 and was at Yale University. Now at the University of Utah. 2 This paper contains a biblio...

In the Matrix approach to graph transformation we represent simple digraphs and rules with Boolean matrices and vectors, and the rewriting is expressed using Boolean operations only. In previous works, we developed analysis techniques enabling the study of the applicability of rule sequences, their independence, stated reachability and the minimal digraph able to fire a sequence. See [19] for a...

For the Hamiltonian integrable systems governed by SL(2)-invariant r-matrix (such as Heisenberg magnet, Toda lattice, nonlinear Schrödinger equation) a general procedure for constructing Bäcklund transformation is proposed. The corresponding BT is shown to preserve the Poisson bracket. The proof is given by a direct calculation using the r-matrix expression for the Poisson bracket.