Hamilton and Jacobi come full circle: Jacobi algorithms for structured Hamiltonian eigenproblems

Computer vision needs suitable methods of shape representation and contour reconstruction. One of them called method of Hurwitz-Radon Matrices (MHR) can be used in representation and reconstruction of shapes of the objects in the plane. Another problem is connected with shape coefficients. This paper contains the way of length estimation and area estimation via MHR method. Proposed method is based on a family of Hurwitz-Radon (HR) matrices. The matrices are skew-symmetric and possess columns composed of orthogonal vectors. The operator of Hurwitz-Radon (OHR), built from these matrices, is described. The shape is represented by the set of nodes. It is shown how to create the orthogonal and discrete OHR and how to use it in a process of shape representation and reconstruction. MHR method is interpolating the curve point by point without using any formula or function.

Shape representation and shape coefficients via method of Hurwitz-Radon matrices

This paper presents how to find an architecture for very large scale lossless neural nets, which can be used as Haar-Walsh spectrum analyzers. This analysis relies on the orthogonality of weight matrices W, where W could be Hurwitz-Radon matrices. The unique feature of these nets is the possibility to treat them either as algorithms or as Hamiltonian physical objects (Haar-Walsh Signal Processors).

http://ieeexplore.ieee.org/iel5/8430/26552/01182910.pdf

Hamiltonian neural nets as a universal signal processor

Shape Coefficients via Method of Hurwitz-Radon Matrices

Computer vision needs suitable methods of shape representation and contour reconstruction One of them, invented by the author and called method of Hurwitz-Radon Matrices (MHR), can be used in representation and reconstruction of shapes of the objects in the plane Proposed method is based on a family of Hurwitz-Radon (HR) matrices The matrices are skew-symmetric and possess columns composed of orthogonal vectors Shape is represented by the set of nodes It is shown how to create the orthogonal and discrete OHR operator and how to use it in a process of shape representation and reconstruction MHR method is interpolating the curve point by point without using any formula or function

Application of Hurwitz – Radon matrices in shape representation

T his is largely, but not entirely, a historical survey. It puts various matters together that are usually considered in separate contexts. Moreover, it leads to open, probably quite difficult problems and has analogues in contemporary mathematics. There are three parts. The first part is about a certain range of classical results in algebraic topology concerning continuous real functions and maps, vector fields, etc., that can be stated in a very simple way: just replace \" continuous \" by \" linear \". They thus seem to be reduced to problems of algebra, essentially linear algebra, where the solution is relatively easy. The proofs of all these statements, however, do not use such a reduction principle. They are beautiful and in general quite difficult, using elaborate ideas and techniques of topology such as cohomology operations, spectral sequences, K-theory, and so on. In these cases the absence of a reduction principle from continuity to linearity is thus a missing link between two areas. Of course, such a link is not necessary since the results are proved. Still, it might be interesting to have, at least in special cases, a direct reduction to linearity which could throw new light onto old and new mathematics. We begin, in that first part, with a very elementary example. It contains in a nutshell the problem to be discussed later for analogous but more complex phenomena. The second part is about the homotopy of the unitary groups. It leads to a different linearization phenomenon stating that maps of a sphere into a unitary group are homotopic in the infinite unitary group to a linear map (in the situation of the first part it is, in general, not true that any continuous solution is homotopic to a linear one). Again, no direct proof is known—by differential geometry, by approximation, or by optimization. This lin-earization result is due to the close relation between the classical Hurwitz-Radon matrices and Bott periodicity. Since the latter is the source of topological K-theory and general homology theories , a direct approach would establish a link between an important old matrix problem and refined methods of algebraic topology. The third part is a short outlook concerning topics of contemporary interest. It is about recent trends in homology theories for spaces with infinite fundamental group G using Hilbert space methods (Hilbert-G-modules, 2-homology, and 2-Betti numbers). The presentation, necessarily a little technical, turns around the von …

Topology, algebra, analysis: relations and missing links

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Topology, algebra, analysis: relations and missing links