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Supersymmetric Quantum Stochastic Filtering Theory
Published in Harish Parthasarathy, Supersymmetry and Superstring Theory with Engineering Applications, 2023
and also anticommute with the Fermionic operators. Physical meaning to the filtering results in such coherent states can be obtained only after averaging the results by means of a weight function dependent upon the Grassmannian parameters using the Berezin integral. This amounts to choosing our state as a weighted linear combination of Fermionic coherent states. This idea also plays a fundamental role in the theory of quantum antennas wherein the electromagnetic field produced by the current density generated by electrons and positrons in the Dirac picture is a quadratic combination of the Fermionic creation and annihilation operators and hence by applying the retarded potential formula, the electromagnetic field radiated out by such a quantum antenna is also a quadratic functional of the Fermionic creation and annihilation operators. Computing the quantum statistical moments of the radiation field in a Fermionic coherent state results in a function of the Grassmannian parameters which after a weighted Berezin integral approximation gives real and complex numbers which can be interpreted physically in terms of correlations, energy, power and phase.
Operators in the Cowen-Douglas Class and Related Topics
Published in Kehe Zhu, Handbook of Analytic Operator Theory, 2019
defines a holomorphic map into the Grassmannian G(ℋ, n) on the open set U*. The pull-back E0 of the canonical bundle on G(ℋ, n) under this map then defines a holomorphic Hermitian bundle on U*. Clearly, the decomposition of K given in our Theorem is not canonical in any way. So, we can’t expect the corresponding vector bundle E0 to reflect the properties of the Hilbert module ℳ. However, it is possible to obtain a canonical decomposition following the construction in [21]. It then turns out that the equivalence class of the corresponding vector bundle E0 obtained from this canonical decomposition is an invariant for the isomorphism class of the Hilbert module ℳ, These invariants are by no means easy to compute. At the end of this subsection, we indicate, how to construct invariants which are more easily computable. For now, the following corollary to the decomposition theorem is immediate.
Feedback Techniques for MIMO Channels
Published in George Tsoulos, MIMO System Technology for Wireless Communications, 2018
David J. Love, Robert W. Heath
The challenge of direct channel quantization [37–39, 46] is the large number of parameters. To illustrate, Table 5.1 provides the number of parameters for different kinds of transmit processing. Notice that more sophisticated forms of precoding require more feedback. Quantization of instantaneous channel state information is the alternative to direct channel quantization. Initial work in this area stems from the channel quantization work of Narula et al. [69] and the work by Heath and Paulraj on beamforming phase quantization [26] and antenna selection [22,29]. This kind of technique was then extended by Love et al. [53,63], Mukkavilli et al. [66,67] Santipach et al. [79,80], and Zhou et al. in [108]. Love et al. and Mukkavilli et al. independently found that the problem of feedback design for beamforming relates to the applied mathematics problem of Grassmannian line packing [10,87]. The codebook vectors are thought of as lines and are designed using subspace coding techniques. An overview of work in limited feedback MIMO can be found in [63b].
Continuity of the solution set to a linear PDE with constant coefficients
Published in International Journal of Control, 2022
We remind that the Grassmannian of a (finite-dimensional) linear space V is the set of all linear subspaces of V. One writes to denote the Grassmannian of k-dimensional linear subspaces of V. The set has the structure of a topological space (even the structure of a differentiable manifold), and the subsets () are connected components of . For a few facts about Grassmanians that we need the reader is referred to Appendix A in Lomadze (2020).
Solvable stochastic differential games in rank one compact symmetric spaces1
Published in International Journal of Control, 2018
Tyrone E. Duncan, Bozenna Pasik-Duncan
Some natural generalisations exist for this family of differential games. An infinite time horizon differential game with the stochastic equation (12) could be considered with a discounted payoff or a long-range average payoff. Compact symmetric spaces of higher rank could be considered as the space for the stochastic differential games. These latter spaces include many well-known Grassmannians.