Deformation of Gabor systems

We introduce a new notion for the deformation of Gabor systems. Such deformations are in general nonlinear and, in particular, include the standard jitter error and linear deformations of phase space. With this new notion we prove a strong deformation result for Gabor frames and Gabor Riesz sequences that covers the known perturbation and deformation results. Our proof of the deformation theorem requires a new characterization of Gabor frames and Gabor Riesz sequences. It is in the style of Beurling's characterization of sets of sampling for bandlimited functions and extends significantly the known characterization of Gabor frames “without inequalities” from lattices to non-uniform sets.     

Imputation of mean of ratios for missing data and its application to PPSWR sampling

In practical survey sampling, nonresponse phenomenon is unavoidable. How to impute missing data is an important problem. There are several imputation methods in the literature. In this paper, the imputation method of the mean of ratios for missing data under uniform response is applied to the estimation of a finite population mean when the PPSWR sampling is used. The imputed estimator is valid under the corresponding response mechanism regardless of the model as well as under the ratio model regardless of the response mechanism. The approximately unbiased jackknife variance estimator is also presented. All of these results are extended to the case of non-uniform response. Simulation studies show the good performance of the proposed estimators.     

Error estimation for non-uniform sampling in shift invariant space

To reconstruct a function from its sampling value is not always exact, error may arise due to a lot of reasons, therefore error estimation is useful in reconstruction. For non-uniform sampling in shift invariant space, three kinds of errors of the reconstruction formula are discussed in this article. For every kind of error, we give an estimation. We find the accuracy of the reconstruction formula mainly depends on the decay property of the generator and the sampling function.     

Sampling based on timing: Time encoding machines on shift-invariant subspaces

Sampling information using timing is an approach that has received renewed attention in sampling theory. The question is how to map amplitude information into the timing domain. One such encoder, called time encoding machine, was introduced by Lazar and Tóth (2004 [23]) for the special case of band-limited functions. In this paper, we extend their result to a general framework including shift-invariant subspaces. We prove that time encoding machines may be considered as non-uniform sampling devices, where time locations are unknown a priori. Using this fact, we show that perfect representation and reconstruction of a signal with a time encoding machine is possible whenever this device satisfies some density property. We prove that this method is robust under timing quantization, and therefore can lead to the design of simple and energy efficient sampling devices.     

Weighted Uniform Sampling -- a Monte Carlo Technique for Reducing Variance

A new Monte Carlo technique for reducing variance is presentedand analysed. It obtains gains in accuracy comparable to thoserealized by importance sampling (Hammersley & Handscomb,1964), without requiring the use of non-uniform probabilitydistributions. Some numerical examples illustrate the theory.     

Stability analysis of a class of uncertain switched systems on time scale using Lyapunov functions

This paper deals with the stability analysis of a class of uncertain switched systems on non-uniform time domains. The considered class consists of dynamical systems which commute between an uncertain continuous-time subsystem and an uncertain discrete-time subsystem during a certain period of time. The theory of dynamic equations on time scale is used to study the stability of these systems on non-uniform time domains formed by a union of disjoint intervals with variable length and variable gap. Using the concept of common Lyapunov function, sufficient conditions are derived to guarantee the asymptotic stability of this class of systems on time scale with bounded graininess function. The proposed scheme is used to study the leader–follower consensus problem under intermittent information transmissions.     

Existence and stability of stationary solutions to spatially extended autocatalytic and hypercyclic systems under global regulation and with nonlinear growth rates

An analytical analysis of spatially extended autocatalytic and hypercyclic systems is presented. It is shown that spatially explicit systems in the form of reaction–diffusion equations with global regulation possess the same major qualitative features as the corresponding local models. In particular, using the introduced notion of the stability in the mean integral sense we prove the competitive exclusion principle for the autocatalytic system and the permanence for the hypercycle system. Existence and stability of stationary solutions are studied. For some parameter values it is proved that stable spatially non-uniform solutions appear.     

Bounds on entanglement dimensions and quantum graph parameters via noncommutative polynomial optimization

This paper studies the matrix completion problem under arbitrary sampling schemes. We propose a new estimator incorporating both max-norm and nuclear-norm regularization, based on which we can conduct efficient low-rank matrix recovery using a random subset of entries observed with additive noise under general non-uniform and unknown sampling distributions. This method significantly relaxes the uniform sampling assumption imposed for the widely used nuclear-norm penalized approach, and makes low-rank matrix recovery feasible in more practical settings. Theoretically, we prove that the proposed estimator achieves fast rates of convergence under different settings. Computationally, we propose an alternating direction method of multipliers algorithm to efficiently compute the estimator, which bridges a gap between theory and practice of machine learning methods with max-norm regularization. Further, we provide thorough numerical studies to evaluate the proposed method using both simulated and real datasets.     

Lyapunov theorems for systems described by retarded functional differential equations

Lyapunov-like characterizations for non-uniform in time and uniform robust global asymptotic stability of uncertain systems described by retarded functional differential equations are provided.     

The kinetic theory of gases in dispersed media

A generalization of the kinetic theory of gases to non-uniform and non-equilibrium systems is given, and this is used to construct a kinetic theory of dispersed media.     

New imputation methods for missing data using quantiles

The problem of missing values commonly arises in data sets, and imputation is usually employed to compensate for non-response. We propose a novel imputation method based on quantiles, which can be implemented with or without the presence of auxiliary information. The proposed method is extended to unequal sampling designs and non-uniform response mechanisms. Iterative algorithms to compute the proposed imputation methods are presented. Monte Carlo simulations are conducted to assess the performance of the proposed imputation methods with respect to alternative imputation methods. Simulation results indicate that the proposed methods perform competitively in terms of relative bias and relative root mean square error.     

Exponential decay for semilinear distributed systems with damping

The stability of second order abstract distributed systems with damping and nonlinear perturbations is considered. Sufficient conditions, including unique continuation property assumptions, are formulated to obtain (local, non-uniform and uniform) exponential stability. Applications to the wave and Euler-Bernoulli equations are given.     

Density of wavelet frames

Density conditions for wavelet systems with arbitrary sampling points to be frames are studied. We show that for a wavelet system generated by admissible functions with irregular affine lattices to be a frame, the sampling points must have a positive lower affine Beurling density. The same is true for wavelet systems with arbitrary sampling points and nice generating functions.     

Localization of Frames,Banach Frames,and the Invertibility of the Frame Operator

We introduce a new concept to describe the localization of frames. In our main result we show that the frame operator preserves this localization and that the dual frame possesses the same localization property. As an application we show that certain frames for Hilbert spaces extend automatically to Banach frames. Using this abstract theory, we derive new results on the construction of nonuniform Gabor frames and solve a problem about non-uniform sampling in shift-invariant spaces.     

Necessary and sufficient conditions for robust global asymptotic stabilization of discrete-time systems

We give Lyapunov-like conditions for non-uniform in time output stabilization of discrete-time systems. Particularly, it is proved that for a discrete-time control system there exists a (continuous) output stabilizing feedback if and only if there exists a (strong) output control Lyapunov function (OCLF). Moreover, strategies for the construction of continuous robust feedback stabilizers are presented.     

Sampling of vector-valued transforms associated with Green's matrix of Dirac systems

Recently a sampling theorem associated with regular one-dimensional Dirac systems has been developed. Integral transforms whose kernels are solutions of the systems have been reconstructed from their values at the eigenvalues. In the present article the situation is considered where the kernels are replaced by Green's matrix of the problem. Thus a vector-valued sampling theorem is established for vector-valued transforms. A family of examples where the systems have variable coefficients and a vector-valued Kramer-type sampling lemma are given at the end of the paper.     

Fuzinsumo - Fuzzy investigation with surrogate models

In this contribution, we investigate the accuracy of surrogate models for fuzzy systems. Within a virtual tolerance analysis, unknown uncertainties are considered by fuzzy numbers. These fuzzy numbers consist of higher-dimensional intervals, where each interval is assigned a membership value. To reproduce fuzzy numbers correctly, a large number of system evaluations is needed. Instead of evaluating numerically expensive full systems, we employ surrogate models with radial basis functions. In order to find sufficiently accurate surrogate models with as few as possible sampling points, we investigate the influence of the sampling points on the respective fuzzy number intervals. As an example, we investigate the tilting of an X-ray aperture, which shows different accuracy for different sampling strategies. The reference results for the nonlinear multi-body system are calculated by a Matlab optimizer and are compared to the results of the surrogate models. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Semi-uniform Sub-additive Ergodic Theorems for Skew-product Quasi-flows

The aim of this paper is to extend the semi-uniform ergodic theorem and semi-uniform sub-additive ergodic theorem to skew-product quasi-flows. Furthermore, more strict inequalities about these two theorems are established. By making use of these results, it is feasible to get uniform estimation of the Lyapunov exponent of some special systems even under non-uniform hypotheses     

Non-uniform robust global asymptotic stability for discrete-time systems and applications to numerical analysis

The notion of non-uniform Robust Global Asymptotic Stability(RGAS) presented in this paper generalizes the notion of non-uniformin time RGAS for finite- or infinite-dimensional discrete-timesystems. Lyapunov characterizations for this stability notionare provided. The results are applied to finite-dimensionaldiscrete-time systems obtained by time discretization of continuous-timesystems by the explicit Euler method.     

Discrete model reference adaptive control of continuous-time linear multi-input multi-output systems via multirate sampled-data controllers

The use of multirate sampled-data controllers for linear multivariable time-invariant systems with unknown parameters is investigated. Such controllers contain periodically time-varying elements and a multirate sampling mechanism with different sampling periods at each system input. Their application to unknown continuous-time linear multi-input, multi-output systems results in a sampled closedloop system for which an arbitrary discrete-time transfer function matrix can be assigned, as is shown in the present paper. The contribution of the present paper is twofold: the use of multirate sampled-data controllers in the area of model reference adaptive control; and the application, for the first time, of periodically varying controllers for model reference adaptive control of multi-input, multi-output systems.The work described in this paper has been partialy funded by the General Secretariat for Research and Technology of the Greek Ministry of Industry, Research, and Technology and by the Heracles General Cement Company of Greece.