Since the work of Crown (J. Natur. Sci. Math. 15(1-2), 11-25 1975) in the 1970's, it has been known that the projections of a finite-dimensional vector bundle E form an orthomodular poset (OMP) P(E). This result lies in the intersection of a number of current topics, including the categorical quantum mechanics of Abramsky and Coecke (2004), and the approach via decompositions of Harding (Trans. Amer. Math. Soc. 348(5), 1839-1862 1996). Moreover, it provides a source of OMPs for the quantum logic program close to the Hilbert space setting, and admitting a version of tensor products, yet having important differences from the standard logics of Hilbert spaces. It is our purpose here to initiate a basic investigation of the quantum logic program in the vector bundle setting. This includes observations on the structure of the OMPs obtained as P(E) for a vector bundle E, methods to obtain states on these OMPs, and automorphisms of these OMPs. Key theorems of quantum logic in the Hilbert setting, such as Gleason's theorem and Wigner's theorem, provide natural and quite challenging problems in the vector bundle setting.
There is a family of constructions to produce orthomodular structures from modular lattices, lattices that are M and M * -symmetric, relation algebras, the idempotents of a ring, the direct product decompositions of a set or group or topological space, and from the binary direct product decompositions of an object in a suitable type of category. We show that an interval [0, a] of such an orthomodular structure constructed from A is again an orthomodular structure constructed from some B built from A. When A is a modular lattice, this B is an interval of A, and when A is an object in a category, this B is a factor of A.
A recursive estimation algorithm for FIR systems is proposed using the 3rd and 4th order cumulants. From the 3rd and 4th order cumulants relationship, we construct a certain matrix form whose entry is consists of the system output sequence. Using this matrix form, the proposed recursive algorithm is developed by Overdetermined Recursive Instrumental Variable(0RIV) method. The proposed algorithm provides improved estimation accuracy when additive Gaussian noise is present and can be applied to a time varying system as well. Simulation results are presented to compare the performance with other HOS-based algorithms.
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