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In his famous paper (Gersho, IEEE Trans. Inf. Theory 25(4):373-380, 1979), Gersho stressed that the codecells of optimal quantizers asymptotically make an equal contribution to the distortion of the quantizer. Motivated by this fa...
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In his famous paper (Gersho, IEEE Trans. Inf. Theory 25(4):373-380, 1979), Gersho stressed that the codecells of optimal quantizers asymptotically make an equal contribution to the distortion of the quantizer. Motivated by this fact, we investigate in this paper quantizers in the scalar case, where each codecell contributes with exactly the same portion to the quantization error. We show that such quantizers of Gersho type-or Gersho quantizers for short-exist for nonatomic scalar distributions. As a main result, we prove that Gersho quantizers are asymptotically optimal.
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This is a survey on our recent works Chan et al. [5-7], which reveal new relationships among deformation quantization, geometric quantization, Berezin-Toeplitz quantization and BV quantization on Kahler manifolds. (C) 2021 Elsevie...
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This is a survey on our recent works Chan et al. [5-7], which reveal new relationships among deformation quantization, geometric quantization, Berezin-Toeplitz quantization and BV quantization on Kahler manifolds. (C) 2021 Elsevier B.V. All rights reserved.
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In this paper, first we have defined a uniform distribution on the boundary of a regular hexagon and then investigate the optimal sets of n-means and the nth quantization errors for all positive integers n. We give an exact formul...
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In this paper, first we have defined a uniform distribution on the boundary of a regular hexagon and then investigate the optimal sets of n-means and the nth quantization errors for all positive integers n. We give an exact formula to determine them, if n is of the form n=6k for some positive integer k. We further calculate the quantization dimension, the quantization coefficient, and show that the quantization dimension is equal to the dimension of the object, and the quantization coefficient exists as a finite positive number. Then, we define a mixture of two uniform distributions on the boundary of a semicircular disc and obtain a sequence and an algorithm, with the help of which we determine the optimal sets of n-means and the nth quantization errors for all positive integers n with respect to the mixed distribution. Finally, for a uniform distribution defined on an elliptical curve, we investigate the optimal sets of n-means and the nth quantization errors for all positive integers n.
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For a large class of dyadic homogeneous Cantor distributions in R, which are not necessarily self-similar, we determine the optimal quantizers, give a characterization for the existence of the quantization dimension, and show the ...
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For a large class of dyadic homogeneous Cantor distributions in R, which are not necessarily self-similar, we determine the optimal quantizers, give a characterization for the existence of the quantization dimension, and show the non-existence of the quantization coefficient. The class contains all self-similar dyadic Cantor distributions, with contraction factor less than or equal to 1/3. For these distributions we calculate the quantization errors explicitly.
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Consider a physical system for which a mathematically rigorous geometric quantization procedure exists. Now subject the system to a finite set of irreducible first class (bosonic) constraints. It is shown that there is a mathemati...
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Consider a physical system for which a mathematically rigorous geometric quantization procedure exists. Now subject the system to a finite set of irreducible first class (bosonic) constraints. It is shown that there is a mathematically rigorous BRST quantization of the constrained system whose cohomology at ghost number zero recovers the constrained quantum states. Moreover this space of constrained states has a well-defined Hilbert space structure inherited from that of the original system. Treatments of these ideas in the physics literature are more general but suffer from having states with infinite or zero "norms" and thus are not admissible as states. Also BRST operators for many systems require regularization to be well-defined. In our more restricted context, we show that our treatment does not suffer from any of these difficulties.
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摘要 :
Consider a physical system for which a mathematically rigorous geometric quantization procedure exists. Now subject the system to a finite set of irreducible first class (bosonic) constraints. It is shown that there is a mathemati...
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Consider a physical system for which a mathematically rigorous geometric quantization procedure exists. Now subject the system to a finite set of irreducible first class (bosonic) constraints. It is shown that there is a mathematically rigorous BRST quantization of the constrained system whose cohomology at ghost number zero recovers the constrained quantum states. Moreover this space of constrained states has a well-defined Hilbert space structure inherited from that of the original system. Treatments of these ideas in the physics literature are more general but suffer from having states with infinite or zero "norms" and thus are not admissible as states. Also BRST operators for many systems require regularization to be well-defined. In our more restricted context, we show that our treatment does not suffer from any of these difficulties.
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Polymer Quantum Mechanics is based on some of the techniques used in the loop quantization of gravity that are adapted to describe systems possessing a finite number of degrees of freedom. It has been used in two ways: on one hand...
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Polymer Quantum Mechanics is based on some of the techniques used in the loop quantization of gravity that are adapted to describe systems possessing a finite number of degrees of freedom. It has been used in two ways: on one hand it has been used to represent some aspects of the loop quantization in a simpler context, and, on the other, it has been applied to each of the infinite mechanical modes of other systems. Indeed, this polymer approach was recently implemented for the free scalar field propagator. In this work we compute the polymer propagators of the free particle and a particle in a box; amusingly, just as in the non polymeric case, the one of the particle in a box may be computed also from that of the free particle using the method of images. We verify the propagators hereby obtained satisfy standard properties such as: consistency with initial conditions, composition and Green's function character. Furthermore they are also shown to reduce to the usual Schr?dinger propagators in the limit of small parameter _(μ0), the length scale introduced in the polymer dynamics and which plays a role analog of that of Planck length in Quantum Gravity.
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We show that continuous and differential deformation theories of the algebra of smooth functions on R(N) are the same, and that the same result holds for the algebra of formal series. We show that preferred quantizations of formal...
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We show that continuous and differential deformation theories of the algebra of smooth functions on R(N) are the same, and that the same result holds for the algebra of formal series. We show that preferred quantizations of formal groups are always differential. [References: 11]
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In this paper, we have considered a uniform probability distribution supported by a stretched Sierpinski triangle. For this probability measure, the optimal sets of n-means and the nth quantization errors are determined for all n ...
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In this paper, we have considered a uniform probability distribution supported by a stretched Sierpinski triangle. For this probability measure, the optimal sets of n-means and the nth quantization errors are determined for all n >= 2. In addition, it is shown that the quantization coefficient for such a measure does not exist though the quantization dimension exists.
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The estimation of signal parameters using quantized data is a recurrent problem in electrical engineering. As an example, this includes the estimation of a noisy constant value and of the parameters of a sinewave, that is, its amp...
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The estimation of signal parameters using quantized data is a recurrent problem in electrical engineering. As an example, this includes the estimation of a noisy constant value and of the parameters of a sinewave, that is, its amplitude, initial record phase, and offset. Conventional algorithms, such as the arithmetic mean, in the case of the estimation of a constant, are known not to be optimal in the presence of quantization errors. They provide biased estimates if particular conditions regarding the quantization process are not met, as it usually happens in practice. In this paper, a quantile-based estimator is presented, which is based on the Gauss–Markov theorem. The general theory is first described and the estimator is then applied to both direct current and alternate current input signals with unknown characteristics. Using simulations and experimental results, it is shown that the new estimator outperforms conventional estimators in both problems, by removing the estimation bias.
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