Characterization of vector diffraction-free beams
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Abstract:
It is observed that a constant unit vector denoted by I is needed to characterize a complete orthonormal set of vector diffraction-free beams. The previously found diffraction-free beams are shown to be included as special cases. The I-dependence of the longitudinal component of diffraction-free beams is also discussed.Keywords:
Characterization
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Subspace tracking is important for many communications and signal processing tasks. Many of the simplest subspace tracking methods, however, only approximately maintain the orthonormality of the subspace matrix estimate. In this paper, we describe a generalized procedure for designing principal subspace tracking algorithms that maintains the orthonormality of the subspace matrix estimate in a numerically-robust fashion. Our generalized algorithm families include two orthonormal update principal subspace tracking algorithms as special cases, and all but one of the new algorithms are computationally-simpler than these existing approaches. Moreover, we show how to modify these algorithms to perform minor subspace tracking in a numerically-stable fashion. Simulations verify the numerically-robust performances of the algorithms in principal and minor subspace tracking tasks, respectively.
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Reinforcement Learning with Orthonormal Basis Adaptation Based on Activity-Oriented Index Allocation
An orthonormal basis adaptation method for function approximation was developed and applied to reinforcement learning with multi-dimensional continuous state space. First, a basis used for linear function approximation of a control function is set to an orthonormal basis. Next, basis elements with small activities are replaced with other candidate elements as learning progresses. As this replacement is repeated, the number of basis elements with large activities increases. Example chaos control problems for multiple logistic maps were solved, demonstrating that the method for adapting an orthonormal basis can modify a basis while holding the orthonormality in accordance with changes in the environment to improve the performance of reinforcement learning and to eliminate the adverse effects of redundant noisy states.
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Basis function
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Orthonormality
Basis (linear algebra)
Truncation (statistics)
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Truncation error
Constant (computer programming)
Orthogonal functions
Function Approximation
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This paper introduces an orthonormal version of the sliding-window projection approximation subspace tracker (PAST). The new algorithm guarantees the orthonormality of the signal subspace basis at each iteration. Moreover, it has the same complexity as the original PAST algorithm, and like the more computationally demanding natural power (NP) method, it satisfies a global convergence property, and reaches an excellent tracking performance.
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Abstract We show how to easily create a right-handed orthonormal basis, given a unit vector, in 2−, 3−, and 4-space.
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Interacting boson model
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In this article we consider the Data Projection Method (DPM), which constitutes a simple and reliable means for adaptively estimating and tracking subspaces. Specifically we propose a fast and numerically robust implementation of DPM. Existing schemes can track subspaces corresponding either to the largest or the smallest singular values. DPM, on the other hand, with a simple change of sign in its step size, can switch from one subspace type to the other. Our fast implementation of DPM preserves the simple structure of the original DPM having also a considerably lower computational complexity. The proposed version provides orthonormal vector estimates of the subspace basis which are numerically stable. In other words, our scheme does not accumulate roundoff errors and therefore preserves orthonormality in its estimates. In fact, our scheme constitutes the only numerically stable, low complexity, algorithm for tracking subspaces corresponding to the smallest singular values. In the case of tracking subspaces corresponding to the largest singular values, our scheme exhibits the fastest convergence-towards-orthonormality among all other subspace tracking algorithms of similar complexity.
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Orthonormal basis (plural orthonormal bases): a set B of vectors in Euclidean or Hilbert space such that every vector can be written as a (finite or infinite) linear combination of vectors from B, while all vectors from B have length 1 and any two of them are orthogonal. The number of vectors in B then equals the dimension of the space, which can be finite or infinite.
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Abstract The method of orthonormality‐constrained variation is extended using a dual‐basis set instead of a single orthonormal basis. The complete and the partial variation methods are discussed and applied to electronic systems. It is found that the present formulation leads to the most general equation in the coupling operator method.
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Abstract The space‐vector theory, which allows to describe three linearly dependent quantities (e. g. three voltages or three currents) using only two linearly independent quantities represented in an orthonormal coordinate system, is widely used in the fields of AC machine theory, power definitions and active filtering. It can be used very effectively in case of circuits with only three terminals. Problems occur in case of four terminals, when four linearly dependent quantities (e. g. four currents, the sum of which is always zero) exist. In this case a zero‐sequence quantity can be additionally introduced, which has to be treated separately with special equations. Vector operations like cross product or dot product, which are very useful to calculate e. g. power quantities, can no longer be used in the general case. This paper presents a method transforming any system represented by four linearly dependent quantities or by three linearly independent quantities into a three‐dimensional orthonormal coordinate system. All four members of a set of linearly dependent quantities are treated equally. We suggest to call this representation Hyper Space Vector (HSV). A lot of vector operations can be applied without problems to HSV, making calculations much easier and more graphic.
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