Mehler’s formula and functional calculus

Science China Mathematics - We show that Mehler’s formula can be used to handle several formulas involving the quantization of singular Hamiltonians. In particular, we diagonalize in the...     

Quantization of Compact Symplectic Manifolds

We develop the theory of Berezin–Toeplitz operators on any compact symplectic prequantizable manifold from scratch. Our main inspiration is the Boutet de Monvel–Guillemin theory, which we simplify in several ways to obtain a concise exposition. A comparison with the spin-c Dirac quantization is also included.     

FLOATING QUANTIZATION EFFECTS ON MULTIRATE SAMPLED-DATA NONLINEAR CONTROL SYSTEMS

In this article,floating quantization effects on multirate sampled-data control systems are studied.It shows that the solutions of multirate digital feedback control systems with nonlinear plant and with floating quantization in the controller are uniformly ultimately bounded if the associated linear systems consisting of linearization of the plant and controller with no quantization are Schur stable.Moreover,it also shows that the difference between the response of multirate digital controllers without quantizers and the same plant with floating quantization in the controllers can be made as small as desired by selecting proper quantization level.     

Poisson Equations Associated with Differential Second Quantization Operators in White Noise Analysis

In this paper we shall show the heredity of a differentiable one-parameter semigroup under the second quantization and then discuss the resolvent of the differential second quantization operator and the potentials of test white noise functionals. As an application, we shall investigate the existence of solutions of the Poisson-type equations associated with differential second quantization operators as well as operators similar to differential second quantization operators.     

Quantization of compressive samples with stable and robust recovery

In this paper we study the quantization stage that is implicit in any compressed sensing signal acquisition paradigm. We propose using Sigma–Delta (ΣΔ) quantization and a subsequent reconstruction scheme based on convex optimization. We prove that the reconstruction error due to quantization decays polynomially in the number of measurements. Our results apply to arbitrary signals, including compressible ones, and account for measurement noise. Additionally, they hold for sub-Gaussian (including Gaussian and Bernoulli) random compressed sensing measurements, as well as for both high bit-depth and coarse quantizers, and they extend to 1-bit quantization. In the noise-free case, when the signal is strictly sparse we prove that by optimizing the order of the quantization scheme one can obtain root-exponential decay in the reconstruction error due to quantization.     

Stability of quantization dimension and quantization for homogeneous Cantor measures

We effect a stabilization formalism for dimensions of measures and discuss the stability of upper and lower quantization dimension. For instance, we show for a Borel probability measure with compact support that its stabilized upper quantization dimension coincides with its packing dimension and that the upper quantization dimension is finitely stable but not countably stable. Also, under suitable conditions explicit dimension formulae for the quantization dimension of homogeneous Cantor measures are provided. This allows us to construct examples showing that the lower quantization dimension is not even finitely stable. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Quantization of classical mechanics: Shall we lie?

We propose a Lie-Noether-symmetry solution of two problems that arise with classical quantization: the quantization of higher-order (more than second) Euler-Lagrange ordinary differential equations of classical mechanics and the quantization of any second-order Euler-Lagrange ordinary differential equation that classically comes from a simple linear equation via nonlinear canonical transformations.     

A Class of Potentials for Which Exact Semiclassical Quantization Can Be Achieved

We consider a class of potentials for which the exact semiclassical quantization is achieved by a certain modification of the quantization condition. A list of potentials for which the new quantization condition is exact coincides with the list of potentials for which the spectrum is determined by the factorization method. We construct a one-parameter family of quantization conditions including the supersymmetric WKB condition as a special case. The new condition allows considering the interrelations between different modifications of the leading approximation and their validity ranges and also allows developing new approximate methods for calculating spectra.     

A note on the trade-off between sampling and quantization in signal processing

We discuss the trade-off between sampling and quantization in signal processing for the purpose of minimizing the error of the reconstructed signal subject to the constraint that the digitized signal fits in a given amount of memory. For signals with different regularities, we estimate the intrinsic errors from finite sampling and quantization, and determine the sampling and quantization resolutions.     

Optimal Quantization for Uniform Distributions on Cantor-Like Sets

In this paper, the problem of optimal quantization is solved for uniform distributions on some higher dimensional, not necessarily self-similar N-adic Cantor-like sets. The optimal codebooks are determined and the optimal quantization error is calculated. The existence of the quantization dimension is characterized and it is shown that the quantization coefficient does not exist. The special case of self-similarity is also discussed. The conditions imposed are a separation property of the distribution and strict monotonicity of the first N quantization error differences. Criteria for these conditions are proved and as special examples modified versions of classical fractal distributions are discussed. This work contains and generalizes some parts of the authors doctoral thesis (cf. 16).     

Error bounds for high-resolution quantization with Rényi-α-entropy constraints

We consider the problem of optimal quantization with norm exponent r > 0 for Borel probability measures on ? ^d under constrained Rényi-α-entropy of the quantizers. If the bound on the entropy becomes large, then sharp asymptotics for the optimal quantization error are well-known in the special cases α = 0 (memory-constrained quantization) and α = 1 (Shannon-entropy-constrained quantization). In this paper we determine sharp asymptotics for the optimal quantization error under large entropy bound with entropy parameter α ∈ [1+r/d,∞]. For α ∈ [0,1 + r/d] we specify the asymptotical order of the optimal quantization error under large entropy bound. The optimal quantization error is decreasing exponentially fast with the entropy bound and the exact rate is determined for all α ∈ [0, ∞].     

Symmetry groups in the extended quantization scheme

An extended quantization scheme that in essence is similar to geometric quantization is considered. The phase space is extended, and the methods of quantizing constrained systems are used. A condition for choosing coordinates in which the quantization preserves the symmetry group is obtained. A mechanism for determining the scalar product in Dirac's quantization method is proposed.In memory of Mikhail Konstantinovich PolivanovA. M. Razmadze Tbilisi Mathematics Institute, Georgian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 93, No. 2, pp. 231–248, November, 1992.     

Quantization of Lie bialgebras,I

In the paper [Dr3] V. Drinfeld formulated a number of problems in quantum group theory. In particular, he raised the question about the existence of a quantization for Lie bialgebras, which arose from the problem of quantization of Poisson Lie groups. When the paper [KL] appeared Drinfeld asked whether the methods of [KL] could be useful for the problem of quantization of Lie bialgebras. This paper gives a positive answer to a number of Drinfeld's questions, using the methods and ideas of [KL]. In particular, we show the existence of a quantization for Lie bialgebras. The universality and functoriality properties of this quantization will be discussed in the second paper of this series. We plan to provide positive answers to most of the remaining questions in [Dr3] in the following papers of this series.     

Stochastic integral representations of second quantization operators

We give a necessary and sufficient condition for the second quantization operator Γ(h) of a bounded operator h on , or for its differential second quantization operator λ(h), to have a representation as a quantum stochastic integral. This condition is exactly that h writes as the sum of a Hilbert-Schmidt operator and a multiplication operator. We then explore several extensions of this result. We also examine the famous counterexample due to Journé and Meyer and explain its representability defect.     

Optimal vector quantization in terms of Wasserstein distance

The optimal quantizer in memory-size constrained vector quantization induces a quantization error which is equal to a Wasserstein distortion. However, for the optimal (Shannon-)entropy constrained quantization error a proof for a similar identity is still missing. Relying on principal results of the optimal mass transportation theory, we will prove that the optimal quantization error is equal to a Wasserstein distance. Since we will state the quantization problem in a very general setting, our approach includes the Rényi-α-entropy as a complexity constraint, which includes the special case of (Shannon-)entropy constrained (α=1) and memory-size constrained (α=0) quantization. Additionally, we will derive for certain distance functions codecell convexity for quantizers with a finite codebook. Using other methods, this regularity in codecell geometry has already been proved earlier by György and Linder (2002, 2003) [11] and [12].     

基于自适应小波变换和矢量量化的图像压缩编码

运用小波变换进行图像压缩的算法其核心都是小波变换的多分辨率分析以及对不同尺度的小波系数的量化和编码 .本文提出了一种基于能量的自适应小波变换和矢量量化相结合的压缩算法 .即在一定的能量准则下 ,根据子图像的能量大小决定是否进行小波分解 ,然后给出恰当的小波系数量化 .在量化过程中 ,采用一种改进的LBG算法进行码书的训练 .实验表明 ,本算法广泛适用于不同特征的数字图像 ,在取得较高峰值信噪比的同时可以获得较高的重建图像质量 .     

Some reflections on mathematicians’ views of quantization

We start with a short presentation of the difference in attitude between mathematicians and physicists even in their treatment of physical reality, and look at the paradigm of quantization as an illustration. In particular, we stress the differences in motivation and realization between the Berezin and deformation quantization approaches, exposing briefly Berezin’s view of quantization as a functor. We continue with a schematic overview of deformation quantization and of its developments in contrast with the latter and discuss related issues, in particular, the spectrality question. We end by a very short survey of two main avatars of deformation quantization, quantum groups and quantum spaces (especially noncommutative geometry) presented in that perspective. Bibliography: 74 titles. This paper is dedicated to the memory of “Alik” Berezin Published in Zapiski Nauchnykh Seminarov POMI, Vol. 331, 2006, pp. 199–220.     

On Martin Bordemann's proof of the existence of projectively equivariant quantizations

The paper explains the notion of projectively equivariant quantization. It gives a sketch of Martin Bordemann's proof of the existence of projectively equivariant quantization on arbitrary manifolds.