Search

Gravitational subsystems with boundaries carry the action of an infinite-dimensional symmetry algebra, with potentially profound implications for the quantum theory of gravity. We initiate an investigation into the quantization of this corner symmetry algebra for the phase space of gravity localized to a region bounded by a 2-dimensional sphere. Starting with the observation that the algebra sdiff (S2) of area-preserving diffeomorphisms of the 2-sphere admits a deformation to the finite-dimensional algebra su (N), we derive novel finite-N deformations for two important subalgebras of the gravitational corner symmetry algebra. Specifically, we find that the area-preserving hydrodynamical algebra sdiff S 2 ⊕ L ℝ S 2 arises as the large-N limit of sl N ℂ ⊕ ℝ and that the full area-preserving corner symmetry algebra sdiff S 2 ⊕ L sl 2 ℝ S 2 is the large-N limit of the pseudo-unitary group su (N, N). We find matching conditions for the Casimir elements of the deformed and continuum algebras and show how these determine the value of the deformation parameter N as well as the representation of the deformed algebra associated with a quantization of the local gravitational phase space. Additionally, we present a number of novel results related to the various algebras appearing, including a detailed analysis of the asymptotic expansion of the su (N) structure constants, as well as an explicit computation of the full diff (S2) structure constants in the spherical harmonic basis. A consequence of our work is the definition of an area operator which is compatible with the deformation of the area-preserving corner symmetry at finite N. [ABSTRACT FROM AUTHOR]

Abstract: We consider a quantum system of a Dirac particle interacting with the quantum radiation field, where the Dirac particle is in a -Hermitian matrix-valued potential V. Under the assumption that the total Hamiltonian is essentially self-adjoint (we denote its closure by ), we investigate properties of the Heisenberg operator () of the j-th position operator of the Dirac particle at time and its strong derivative (the j-th velocity operator), where is the multiplication operator by the j-th coordinate variable (the j-th position operator at time ). We prove that , the domain of the position operator , is invariant under the action of the unitary operator for all and establish a mathematically rigorous formula for . Moreover, we derive asymptotic expansions of Heisenberg operators in the coupling constant (the electric charge of the Dirac particle). [Copyright &y& Elsevier]

The present monograph studies the asymptotic behaviour of eigenvalues, products and functions of block Toeplitz matrices generated by the Fourier coefficients of a continuous matrix-valued function. This study is based on the concept of asymptotically equivalent sequences of non-square matrices. The asymptotic results on block Toeplitz matrices obtained are applied to vector asymptotically wide sense stationary processes. Therefore, this monograph is a generalization to block Toeplitz matrices of the Gray monograph entitled "Toeplitz and circulant matrices: A review", which was published in the second volume of Foundations and Trends in Communications and Information Theory, and which is the simplest and most famous introduction to the asymptotic theory on Toeplitz matrices. [ABSTRACT FROM AUTHOR]

Focuses on the stability of linear stochastic finite-difference systems with almost-periodic coefficients. Reduction of the stability problem of solutions of deterministic linear finite-difference equation; Establishment of a condition for exponential stability; Validity of the criterion of asymptotic mean-square stability.

Quantum robust control—which can employ fast leading order approximations, slower but more accurate asymptotic methods, or a combination thereof for quantification of robustness—enables control of moments of quantum observables and gates in the presence of Hamiltonian uncertainty or field noise. In this paper, we present a generalized quantum robust control theory that extends the previously described theory of quantum robust control in several important ways. We present robust control theory for control of any moment of arbitrary quantum control objectives, introducing moment-generating functions and transfer functions for quantum robust control that generalize the tools of frequency domain response theory to quantum systems, and extend the Pontryagin maximum principle for quantum control to control optimization in the presence of noise in the manipulated amplitudes or phases used to shape the control field. To provide guidelines as to the types of quantum control systems and control objectives for which asymptotic robustness analysis is important for accuracy, we introduce methods for assessing the Lie algebraic depth of quantum control systems, and illustrate through examples drawn from quantum information processing how such accurate methods for quantification of robustness to noise and uncertainty are more important for control strategies that exploit higher order quantum pathways. In addition, we define the relationship between leading order Taylor expansions and asymptotic estimates for quantum control moments in the presence of Hamiltonian uncertainty and field noise, and apply such leading order approximations to significant pathways analysis and dimensionality reduction of asymptotic quantum robust control calculations, describing numerical methods for implementation of these calculations. [ABSTRACT FROM AUTHOR]

Locally most powerful invariant test (LMPIT) for sphericity of Gaussian vectors has been derived by Ramírez et al. Nevertheless, the decision threshold of the LMPIT is not accurate and its detection performance has not yet been addressed. In this paper, the LMPIT is performed for target detection in multiple-input multiple-output (MIMO) radar, and its theoretical decision threshold as well as detection probability are accurately determined. Utilizing asymptotic expansion approach, we calculate the asymptotic null distribution as a function of central chi-square distributions, resulting in precise closed-form formula for thresholding. On the other hand, the nonnull distribution is approximated by weighted sum of noncentral chi-square distributions and Gamma distribution for close and far hypotheses, respectively. This enables us to derive a closed-form formula to precisely evaluate the detection power of the LMPIT. Numerical results demonstrate that our theoretical computations are very accurate in determining the decision threshold and predicting the behaviors of the LMPIT. Moreover, the superiority of the LMPIT for MIMO radar target detection over state-of-the-art methods is demonstrated for spatially colored but temporarily white noise. [ABSTRACT FROM AUTHOR]

The Schrödinger operator in a thin infinite strip with PT -symmetric boundary conditions and a localized potential is studied. The case of a virtual level on the threshold of the essential spectrum of an efficient one-dimensional operator is considered. Sufficient conditions for the transformation of this level into an isolated eigenvalue are obtained and the first terms of the asymptotic expansion are calculated for this eigenvalue. Sufficient conditions for the absence of such an eigenvalue are also obtained. [ABSTRACT FROM AUTHOR]

In this paper, the quasi-convexity of a sum of quadratic fractions in the form \sumi=1^n {1+c_i x^2\over (1+d_ix)^2} is demonstrated where c_i and d_i are strictly positive scalars, when defined on the positive real axis \BBR^+. It will be shown that this quasi-convexity guarantees it has a unique local (and hence global) minimum. Indeed, this problem arises when considering the optimization of the weighting coefficient in regularized semi-blind channel identification problem, and more generally, is of interest in other contexts where we combine two different estimation criteria. Note that V. Buchoux have noticed by simulations that the considered function has no local minima except its unique global minimum but this is the first time this result, as well as the quasi-convexity of the function is proved theoretically. [ABSTRACT FROM AUTHOR]

We study the performance of systems for signal parameters estimation, which are based on the linear minimum mean square error (LMMSE) filtering. Such an estimation technique is widely used in practical scenarios, specifically wireless sensor networks and MIMO communications. We model the estimation system through sums and products of random matrices, involving a d-fold Vandermonde matrix (d\geq 1) with entries uniformly distributed on the complex unit circle. Then, we derive the mean square error (MSE) of the estimated signal by resorting to asymptotic analysis and by applying the \eta-transform operator. We describe how our results can be exploited for the study of practical systems, and we show the agreement existing between the asymptotic results derived through our analysis and the simulation results obtained for small values of d. [ABSTRACT FROM AUTHOR]

The present monograph studies the asymptotic behaviour of eigenvalues, products and functions of block Toeplitz matrices generated by the Fourier coefficients of a continuous matrix-valued function. This study is based on the concept of asymptotically equivalent sequences of non-square matrices. The asymptotic results on block Toeplitz matrices obtained are applied to vector asymptotically wide sense stationary processes. Therefore, this monograph is a generalization to block Toeplitz matrices of the Gray monograph entitled "Toeplitz and circulant matrices: A review", which was published in the second volume of Foundations and Trends in Communications and Information Theory, and which is the simplest and most famous introduction to the asymptotic theory on Toeplitz matrices. [ABSTRACT FROM AUTHOR]

The present monograph studies the asymptotic behaviour of eigenvalues, products and functions of block Toeplitz matrices generated by the Fourier coefficients of a continuous matrix-valued function. This study is based on the concept of asymptotically equivalent sequences of non-square matrices. The asymptotic results on block Toeplitz matrices obtained are applied to vector asymptotically wide sense stationary processes. Therefore, this monograph is a generalization to block Toeplitz matrices of the Gray monograph entitled "Toeplitz and circulant matrices: A review", which was published in the second volume of Foundations and Trends in Communications and Information Theory, and which is the simplest and most famous introduction to the asymptotic theory on Toeplitz matrices. [ABSTRACT FROM AUTHOR]

The present monograph studies the asymptotic behaviour of eigenvalues, products and functions of block Toeplitz matrices generated by the Fourier coefficients of a continuous matrix-valued function. This study is based on the concept of asymptotically equivalent sequences of non-square matrices. The asymptotic results on block Toeplitz matrices obtained are applied to vector asymptotically wide sense stationary processes. Therefore, this monograph is a generalization to block Toeplitz matrices of the Gray monograph entitled "Toeplitz and circulant matrices: A review", which was published in the second volume of Foundations and Trends in Communications and Information Theory, and which is the simplest and most famous introduction to the asymptotic theory on Toeplitz matrices. [ABSTRACT FROM AUTHOR]

The present monograph studies the asymptotic behaviour of eigenvalues, products and functions of block Toeplitz matrices generated by the Fourier coefficients of a continuous matrix-valued function. This study is based on the concept of asymptotically equivalent sequences of non-square matrices. The asymptotic results on block Toeplitz matrices obtained are applied to vector asymptotically wide sense stationary processes. Therefore, this monograph is a generalization to block Toeplitz matrices of the Gray monograph entitled "Toeplitz and circulant matrices: A review", which was published in the second volume of Foundations and Trends in Communications and Information Theory, and which is the simplest and most famous introduction to the asymptotic theory on Toeplitz matrices. [ABSTRACT FROM AUTHOR]

We consider the sum capacity of a multi-input multi-output (MIMO) multiple access channel (MAC) where the receiver has the perfect channel state information (CS!), while the transmitters have either no or partial CS!. When the transmitters have partial CS!, it is in the form of either the covariance matrix of the channel or the mean matrix of the channel. For the covariance feedback case, we mainly consider physical models that result in single-sided correlation structures. For the mean feedback case, we consider physical models that result in in-phase received signals. Under these assumptions, we analyze the MIMO-MAC from three different viewpoints. First, we consider a finite-sized system. We show that the optimum transmit directions of each user are the eigenvectors of its own channel covariance and mean feedback matrices, in the covariance and mean feedback models, respectively. Also, we find the conditions under which beamforming is optimal for all users. Second, in the covariance feedback case, we prove that the region where beamforming is optimal for all users gets larger with the addition of new users into the system. In the mean feedback case, we show through simulations that this is not necessarily true. Third, we consider the asymptotic case where the number of users is large. We show that in both no and partial CS! cases, beamforming is asymptotically optimal. In particular, in the case of no CS!, we show that a simple form of beamforming, which may be characterized as an arbitrary antenna selection scheme, achieves the sum capacity. In the case of partial CSI, we show that beamforming in the direction of the strongest eigenvector of the channel feedback matrix achieves the sum capacity. Finally, we generalize our covariance feedback results to double-sided correlation structures in the Appendix. [ABSTRACT FROM AUTHOR]

In many detection and estimation problems associated with processing of second-order stationary random processes, the observation data are the sum of two zero-mean second-order stationary processes: the process of interest and the noise process. In particular, the main performance criterion is the signal-to-noise ratio (SNR). After linear filtering, the optimal SNR corresponds to the maximal value of a Rayleigh quotient which can be interpreted as the largest generalized eigenvalue of the covariance matrices associated with the signal and noise processes, which are block multilevel Toeplitz structured for m-dimensional vector-valued second-order stationary p-dimensional random processes xi1, i2,…, ip ϵ.ℝm. In this paper, an extension of Szego's theorem to the generalized eigenvalues of Hermitian block multilevel Toeplitz matrices is given, providing information about the asymptotic distribution of those generalized eigenvalues and in particular of the optimal SNR after linear filtering. A simple proof of this theorem, under the hypothesis of absolutely summable elements is given. The proof is based on the notion of multilevel asymptotic equivalence between block multilevel matrix sequences derived from the celebrated Gray approach. Finally, a short example in wideband space-time beamforming is given to illustrate this theorem. [ABSTRACT FROM AUTHOR]

This paper deals with covariance matrix estimates in impulsive noise environments. Physical models based on compound noise modeling [spherically invariant random vectors (SIRV), compound Gaussian processes] allow to correctly describe reality (e.g., range power variations or clutter transitions areas in radar problems). However, these models depend on several unknown parameters (covariance matrix, statistical distribution of the texture, disturbance parameters) that have to be estimated. Based on these noise models, this paper presents a complete analysis of the main covariance matrix estimates used in the literature. Four estimates are studied: the well-known sample covariance matrix MSCM and a normalized version MN, the fixed-point (FP) estimate MFP, and a theoretical benchmark MTFP. Among these estimates, the only one of practical interest in impulsive noise is the FP. The three others, which could be used in a Gaussian context, are, in this paper, only of academic interest, i.e., for comparison with the FP. A statistical study of these estimates is performed through bias analysis, consistency, and asymptotic distribution. This study allows to compare the performance of the estimates and to establish simple relationships between them. Finally, theoretical results are emphasized by several simulations corresponding to real situations. [ABSTRACT FROM AUTHOR]

In a typical array processing scenario, noise acting on the array can not be assumed spatially white. It is in many cases necessary to use quiet periods, when only noise is received, to estimate the noise covariance. If estimation of the signal parameters and noise covariance is performed jointly, performance can be improved. This is especially true when stationarity considerations limit the amount of available valid noise-only data. An asymptotically valid approximative maximum likelihood method (AML) for the estimation problem is derived in this work. The resulting criterion is, when concentrated with respect to the signal parameters, relatively simple. In numerical experiments, AML shows promising small-sample performance compared to earlier methods. The criterion function is also well suited for numerical optimization. The new criterion function allows for the development of a novel, MODE-like, non-iterative estimation procedure if the array belongs to the important class of uniform linear arrays. The resulting procedure retains the asymptotic properties of maximum likelihood, and numerical simulations indicate superior threshold performance when compared to an optimally weighted subspace fitting (WSF) formulation of MODE. For the detection problem, no method has been presented that takes the unknown noise covariance into account. Here, a well known detection scheme for WSF is extended to work in this scenario as well. The derivations of this scheme further stress the importance of using the correct weighting in WSF when the noise covariance is unknown. It is also shown that the minimum value of the criterion function associated with AML can be used for the detection purpose. Numerical experiments indicate very promising performance for the AML-detection scheme. [ABSTRACT FROM AUTHOR]

Errors in the secondary path model of the filtered-x LMS (FXLMS) algorithm will lead to its divergence when the eigenvalues of the cross-correlation matrix between the estimated filtered reference and the true filtered reference signals are not all located in the right half plane. This cross-correlation matrix has a (block) Toeplitz structure whose dimension is determined by the number of adaptive filter coefficients. Using results on the asymptotic pseudospectrum of Toeplitz matrices, a frequency-domain condition on the model is derived to ensure stability. The condition is sufficient and necessary for a large number of filter coefficients. A transient analysis shows that the sufficient condition given by Wang and Ren [‘Convergence Analysis of the Multi-Variable Filtered-x LMS Algorithm with Application to Active Noise Control,’ IEEE Trans. Signal Process., vol. SP-47, no.4, pp. 1166–1169, Apr. 1999] is only necessary to prevent an initial increase of the error in the adaptive filter coefficients (critical behavior). [ABSTRACT FROM AUTHOR]

This paper derives and analyzes the asymptotic performances of the maximum-likelihood (ML) estimator and the generalized likelihood ratio test (GLRT) derived under the assumption of independent identically distribution (i.i.d.) samples, where in the actual model the signal samples are m-dependent. The ML and GLRT under such a modeling mismatch are based on the marginal likelihood function, and they are referred to as marginal maximum likelihood (MML) and ‘generalized (sum) marginal log-likelihood ratio test’ (GMLRT), respectively. Under some regularity conditions, the asymptotical distributions of the MML and GMLRT are derived. The asymptotical distributions in some signal processing examples are analyzed. Simulation results support the theory via several examples. [ABSTRACT FROM AUTHOR]

This paper examines the asymptotic performance of MUSIC-like algorithms for estimating directions of arrival (DOA) of narrowband complex noncircular sources. Using closed-form expressions of the covariance of the asymptotic distribution of different projection matrices, it provides a unifying framework for investigating the asymptotic performance of arbitrary subspace- based algorithms valid for Gaussian or non-Gaussian and complex circular or noncircular sources. We also derive different robustness properties from the asymptotic covariance of the estimated DOA given by such algorithms. These results are successively applied to four algorithms: to two attractive MUSIC-like algorithms previously introduced in the literature, to an extension of these algorithms, and to an optimally weighted MUSIC algorithm proposed in this paper. Numerical examples illustrate the performance of the studied algorithms compared to the asymptotically minimum variance (AMY) algorithms introduced as benchmarks. [ABSTRACT FROM AUTHOR]

A formula for the second-order expansion of the input-output mutual information of multidimensional channels as the signal-to-noise ratio (SNR) goes to zero is obtained. While the additive noise is assumed to be Gaussian, we deal with very general classes of Input and channel distributions. As special cases, these channel models include fading channels, channels with random parameters, and channels with almost Gaussian noise. When the channel is unknown at the receiver, the second term in the asymptotic expansion depends not only on the covariance matrix of the Input signal but also on the fourth mixed moments of its components. The study of the second-order asymptotics of mutual Information finds application in the analysis of the bandwidth-power tradeoff achieved by various signaling strategies in the wideband regime. [ABSTRACT FROM AUTHOR]

Presents information on a study which re-examined the asymptotic performance analysis of second-order methods for parameter estimation in a general context. Data model; Constraints on the differential of the algorithm; Asymptotic distribution of the sample covariance matrix; Asymptotic distribution of the estimated parameter.

Computes the first three coefficients of the asymptotic expansion of Tian-Yau-Zelditch. Introduction of the concept of peak sections initiated by Zelditch; Utilization of the peak section method in computing the coefficients; Provision of the iteration process; Analysis of the asymptotic expansion of the Szego kernel.

The article presents research which analyzed the asymptotic expansions for the correlation matrix statistics distributions. The researcher applied different statistics concepts in this study including latent root, principal component analysis and perturbation method. This study focused on the α-th correlation matrix largest latent root which can be helpful in the reduction of dimensionality in principal component analysis.