Subspace Theory Research Articles

Marginally coupled designs for two-level qualitative factors

Computer experiments with qualitative and quantitative factors occur frequently in various applications in science, engineering and business. For choosing input settings of such computer experiments, marginally coupled designs have been proposed in Deng, Hung and Lin (2015) for economic reasons. Although the concept of marginally coupled designs is well understood, the results on design construction are scarce. In addition, the constructed designs may have clustered points for the quantitative factors or cannot accommodate many quantitative factors, especially for two-level qualitative factors. To address this issue, this paper focuses on constructing marginally coupled designs when all qualitative factors have two levels. The proposed construction uses subspace theory in algebra, and the resulting designs can accommodate more quantitative factors while maintaining attractive design properties than the existing approaches.

Invariant subspaces for H2 spaces of σ-finite algebras

We show that a Beurling type theory of invariant subspaces of noncommutative $H^2$ spaces holds true in the setting of subdiagonal subalgebras of $\sigma$-finite von Neumann algebras. This extends earlier work of Blecher and Labuschagne for finite algebras, and complements more recent contributions in this regard by Bekjan, and Chen, Hadwin and Shen in the finite setting, and Sager in the semifinite setting. We then also introduce the notion of an analytically conditioned algebra, and go on to show that in the class of analytically conditioned algebras this Beurling type theory is part of a list of properties which all turn out to be equivalent to the maximal subdiagonality of the given algebra.

Nondestructive Estimation of the Eigenvectors of Mixed State

Given n qubits prepared according to the same mixed state ρ, we propose a measuring scheme which approximately yields the eigenvectors of ρ. By performing a sequence of measurements, we can scan the Bloch sphere like radar to search for the eigenvectors. It is shown that this scheme is nondestructive when n is large. This result is based on the quantum typical subspace theory and it actually reveals an interesting mathematical structure of the n-fold Hilbert space.

Multiple-Parameter Estimation Method Based on Spatio-Temporal 2-D Processing for Bistatic MIMO Radar

A novel spatio-temporal 2-dimensional (2-D) processing method that can jointly estimate the transmitting-receiving azimuth and Doppler frequency for bistatic multiple-input multiple-output (MIMO) radar in the presence of spatial colored noise and an unknown number of targets is proposed. In the temporal domain, the cross-correlation of the matched filters’ outputs for different time-delay sampling is used to eliminate the spatial colored noise. In the spatial domain, the proposed method uses a diagonal loading method and subspace theory to estimate the direction of departure (DOD) and direction of arrival (DOA), and the Doppler frequency can then be accurately estimated through the estimation of the DOD and DOA. By skipping target number estimation and the eigenvalue decomposition (EVD) of the data covariance matrix estimation and only requiring a one-dimensional search, the proposed method achieves low computational complexity. Furthermore, the proposed method is suitable for bistatic MIMO radar with an arbitrary transmitted and received geometrical configuration. The correction and efficiency of the proposed method are verified by computer simulation results.

Model order reduction for transient simulation of active distribution networks

With the increasing penetration of distributed generation, distribution networks are evolving from passive to active. New transient simulation methods are required to study the detailed dynamic characteristics of large-scale active distribution networks. A model order reduction method based on Krylov subspace theory is introduced in this study to reduce the overall model scale of active distribution networks for transient simulations. A modified state-space model of linear distribution networks is developed to replace the conventional models for improvement in model reduction efficiency, while guaranteeing the passivity and stability of the reduced models. The proposed model order reduction method is validated with the IEEE 123-node test network. The results prove that the proposed method is effective for different applications to improve the simulation efficiency of large-scale active distribution networks.

Engineering of a quantum state by time-dependent decoherence-free subspaces

We apply the time-dependent decoherence-free subspace theory to a Markovian open quantum system in order to present a novel proposal for quantum-state engineering program. By quantifying the purity of the quantum state, we verify that the quantum-state engineering process designed via our method is completely unitary within any total engineering time. Even though the controls on the open quantum system are not perfect, the asymptotic purity is still robust. Owing to its ability to completely resist decoherence and the lack of restraint in terms of the total engineering time, our proposal is suitable for multitask quantum-state engineering program. Therefore, this proposal is not only useful for achieving the quantum-state engineering program experimentally, it also helps us build both a quantum simulation and quantum information equipment in reality.

Estimating DOA and polarization with spatially spread loop and dipole pair array

The nonuniform L-shaped spatially spread loop and dipole (SSLD) array whose inter-element spacing is greater than half a wavelength is studied. A joint parameter estimation algorithm of direction of arrival (DOA), frequency and polarization is presented for plane-wave signals. The direct sampling and the corresponding delayed sampling data are used to construct the data correlation matrix. On the basis of the subspace theory and the least square method, the frequency and the steering vector of the whole array are obtained. According to the relationship of the array manifold vector between electric dipoles and magnetic loops, the polarization parameters are given. The unambiguous phase estimates are acquired by applying virtual baseline array transformation to the spatial steering vectors, and they are used as coarse references to disambiguate the cyclic phase ambiguities in phase differences between two adjacent array elements on the array, then the high accuracy DOA estimates are obtained. Closed-form solutions for each parameter are obtained. This method has advantages of lower calculation complexity and no parameter matching. The experiment results verify the effectiveness and feasibility of the presented algorithm.

Switching Stabilization for a Class of Slowly Switched Systems

In this technical note, the problem of switching stabilization for slowly switched linear systems is investigated. In particular, the considered systems can be composed of all unstable subsystems. Based on the invariant subspace theory, the switching signal with mode-dependent average dwell time (MDADT) property is designed to exponentially stabilize the underlying system. Furthermore, sufficient condition of stabilization for switched systems with all stable subsystems under MDADT switching is also given. The correctness and effectiveness of the proposed approaches are illustrated by a numerical example.

Invariant subspaces with no generator and a problem of H. Helson

In the almost periodic context, any $H_0^2-$space cannot be generated by one of its elements. Together with cocycle argument, this derives that there exist all kinds of invariant subspaces without single generator, from which we can answer some questions on invariant subspace theory.

Absolutely continuous spectrum of fourth order difference equations

We have studied the absolutely continuous spectrum of a selfadjoint subspace extension generated by a three‐term fourth order difference equation with bounded coefficients using subspace theory. In particular, we have shown that the absolutely continuous spectrum exists outside a certain bounded interval. In addition, we have computed the spectral multiplicity as well as the location of absolutely continuous spectrum of selfadjoint subspace extension under certain asymptotic conditions.

On the predictable representation property of martingales associated with Lévy processes

We investigate the predictable representation property (PRP) in the frame of Lévy processes. To give a general definition of the PRP, we make use of the theory of stable subspaces. Let L be a Lévy process with Lévy measure . The main result is that any total system in leads to a family of martingales with the PRP.

An efficient algorithm for direction finding against unknown mutual coupling.

In this paper, an algorithm of direction finding is proposed in the presence of unknown mutual coupling. The preliminary direction of arrival (DOA) is estimated using the whole array for high resolution. Further refinement can then be conducted by estimating the angularly dependent coefficients (ADCs) with the subspace theory. The mutual coupling coefficients are finally determined by solving the least squares problem with all of the ADCs utilized without discarding any. Simulation results show that the proposed method can achieve better performance at a low signal-to-noise ratio (SNR) with a small-sized array and is more robust, compared with the similar processes employing the initial DOA estimation and further improvement iteratively.

Quantum entropy-typical subspace and universal data compression

The quantum entropy-typical subspace theory is specified. It is shown that any \(\rho ^{\otimes n}\) with von Neumann \(\hbox {entropy}\le h\) can be preserved approximately by the entropy-typical subspace with \(\hbox {entropy}=h\). This result implies an universal compression scheme for the case that the von Neumann entropy of the source does not exceed \(h\).

Vertex-algebraic structure of principal subspaces of standard $A^{(2)}_2$-modules, I

Extending earlier work of the authors, this is the first in a series of papers devoted to the vertex-algebraic structure of principal subspaces of standard modules for twisted affine Kac–Moody algebras. In this part, we develop the necessary theory of principal subspaces for the affine Lie algebra [Formula: see text], which we expect can be extended to higher rank algebras. As a "test case", we consider the principal subspace of the basic [Formula: see text]-module and explore its structure in depth.

Blind carrier frequency offset estimation for interleaved orthogonal frequency division multiple access uplink with multi‐antenna receiver: algorithms and performance analysis

In this study, for interleaved orthogonal frequency division multiple access uplink with multi-antenna receiver, we propose two generalised carrier frequency offset estimators which, respectively, exploit the subspace theory and maximum-likelihood (ML) criterion. We find that, as long as the numbers of multipaths from two of the users are smaller than the number of antennas at the receiver, both the proposed estimators support fully loaded transmissions. We also derive the theoretical performance lower bound for the proposed ML estimator. The numerical results are then provided, which corroborate the proposed studies.

Robust stability analysis of uncertain switched linear systems with unstable subsystems

The problem of robust stability for switched linear systems with all the subsystems being unstable is investigated. Unlike the most existing results in which each switching mode in the system is asymptotically stable, the subsystems may be unstable in this paper. A necessary condition of stability for switched linear systems is first obtained with certain hypothesis. Then, under two assumptions, sufficient conditions of exponential stability for both deterministic and uncertain switched linear systems are presented by using the invariant subspace theory and average dwell time method. Moreover, we further develop multiple Lyapunov functions and propose a method for constructing multiple Lyapunov functions for the considered switched linear systems with certain switching law. Several examples are included to show the effectiveness of the theoretical findings.

Identification of Tandem-Extrusion System's Dynamic Characteristics Based on MATLAB and State Subspace Theory

In view of the complexity of twin screw and melt pump tandem extrusion systems, the identification of the dynamic characteristics of these systems was performed by using MATLAB and state subspace method. Three input one output model was obtained and the controllability and observability of the system was identified by MATLAB. The fitting degree of simulated and actual melt pressure curve increased with increasing the output variables, which indicated that the tandem extrusion system was an interrelated coupled nonlinear system.

Enabling polarisation filtering in wireless communications: models, algorithms and characteristics

To suppress co-channel interference in polarisation-enabled wireless communication systems, this work aims to provide an interference suppression scheme by exploiting polarisation domain, besides the state-of-the-art temporal, frequency, spatial and code domains. System models, algorithms, characteristics and applications of polarisation filtering (PF) for co-channel interference suppressions for polarisation-enabled (e.g. orthogonal dually polarised antennas) wireless communications are investigated. Specifically, four system models for PF using subspace analysis are established and discussed. The four proposed system models are categorised based on different statistic characteristics of the target signal and that of the interfering signal: both the target signal and interference are temporal deterministic, the target signal is deterministic whereas interference is temporal random, the target signal is random whereas interference is deterministic and both the target signal and interference are random, respectively. Based on the statistic characteristics and subspace theory, the detailed PF implementation for each model is analysed and the closed-form filtering operator is given. It is also shown that the PF implementation for each model can be attained by using one of the zero-forcing matched subspace processing, decorrelating matched subspace processing or Wiener subspace processing. Furthermore, relationship among these four models indicates that, under certain conditions, the implementation of the other three models can be fulfilled by using the implementation of the first model. Numerical and simulation results show the effectiveness of the proposed scheme.

Deficiency indices and spectrum of fourth order difference equations with unbounded coefficients

AbstractUsing subspace theory together with appropriate smoothness and decay conditions, we calculated the deficiency indices and absolutely continuous spectrum of fourth order difference equations with unbounded coefficients. In particular, we found the absolutely continuous spectrum to be \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathbb {R}}$\end{document} with a spectral multiplicity one.

Applications of fixed point theorems in the theory of invariant subspaces

We survey several applications of fixed point theorems in the theory of invariant subspaces. The general idea is that a fixed point theorem applied to a suitable map yields the existence of invariant subspaces for an operator on a Banach space.MSC:47A15, 47H10.