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The request for very accurate 3D reconstruction in several applications is leading to the development of very large motion capture systems. A good synchronization of all the cameras in the system is of fundamental importance to guarantee the effectiveness of the 3D reconstruction.
In this work, first, an approximation of the reconstruction error variance taking into account of synchronization errors is derived. Then, a Kalman filter approach is considered to estimate the cameras synchronization errors. The estimated delays can be used to compensate the synchronization error effect on the reconstruction of target positions. The results obtained in some simulations suggest that the proposed strategy allows to obtain a significant reduction of the 3D reconstruction error.

Ion Implantation is one of the most sensitiveprocesses in Semiconductor Manufacturing. It consists inimpacting accelerated ions with a material substrate and isperformed by an Implanter tool. The major maintenanceissue of such tool concerns the breaking of the tungstenfilament contained within the ion source of the tool. Thiskind of fault can happen on a weekly basis, and theassociated maintenance operations can last up to 3 hours.It is important to optimize the maintenance activities bysynchronizing the Filament change operations with otherminor maintenance interventions. In this paper, a PredictiveMaintenance (PdM) system is proposed to tackle such issue;the filament lifetime is estimated on a statistical basisexploiting the knowledge of physical variables acting onthe process. Given the high-dimensionality of the data,the statistical modeling has been based on RegularizationMethods: Lasso, Ridge Regression and Elastic Nets. Thepredictive performances of the aforementioned regularizationmethods and of the proposed PdM module have beentested on actual productive semiconductor data.

Silicon Epitaxial Deposition is a process strongly influenced by wafer temperature behaviour, that has to be constantly monitored to avoid the production of defective wafers. However, temperature measurements are not reliable and the sensors have to be appropriately calibrated with some dedicated procedure. A Predictive Maintenance (PdM) System is proposed here with the aim of predicting process behaviour and scheduling control actions on the sensors in advance. Two different prediction techniques have been employed and compared: the Kalman predictor and the Particle Filter with Gaussian Kernel Density Estimator. The accuracy of the PdM module has been tested on real industrial production datasets.

Mutations in the survival of motor neuron gene (SMN1) are responsible for spinal muscular atrophy (SMA), a fatal neuromuscular disorder. Mice carrying a homozygous deletion of Smn exon 7 directed to skeletal muscle (HSA-Cre, Smn^F7/F7 mice) present clinical features of human muscular dystrophies for which new therapeutic approaches are highly warranted. Herein we demonstrate that tail vein transplantation of mouse amniotic fluid stem (AFS) cells enhances the muscle strength and improves the survival rate of the affected animals. Secondly, after cardiotoxin injury of the Tibialis Anterior, only AFS-transplanted mice efficiently regenerate. Most importantly, secondary transplants of satellite cells (SC) derived from treated mice show that AFS cells integrate into the muscle stem cell compartment, and have long term muscle regeneration capacity indistinguishable from that of wild type-derived SC. This is the first study demonstrating the functional and stable integration of AFS cells into the skeletal muscle, highlighting their value as cell source for the treatment of muscular dystrophies.

Soft Sensors (SSs) are on-line estimators of “hardly to be measured” quantities of a process. The difficultyin measuring can be related to economic or temporal costs that cannot be afforded in a high-intensivemanufacturing production. In semiconductor manufacturing this technology goes with the name of Virtual Metrology(VM) systems. While a lot of efforts in research have been produced in the past years to identify the bestregression algorithms for these statistical modules, small amount of work has been done to develop algorithms fordata clustering of the entire production. This paper contains a new Information Theory-based approach to dataclustering for Virtual Metrology and Soft Sensors; the proposed algorithm allows to automatically split the datasetinto groups to be equally modeled. The proposed approach has been tested on real industrial dataset.

Virtual Metrology (VM) module have become popular in the past years and they are now widely adopted in the semiconductor plants. However, nowadays, still few works have been presented to deal with the interaction between VM and Run-to-Run (R2R), the most common control approach in the fabs. We present in this paper a new strategy to integrate VM with R2R based on Information Theory measure. The proposed control method penalizes statistical measure based on their statistical distance from the physical measure. This new approach also cope with the virtual loop control, where the R2R runs for several process iterations without in-situ measures, but based only on VM predictions. The results are compared with the actual state-of-the-art. 

In fusion devices, the accurate reconstruction of the boundary location and shape from magnetic diagnostics is of paramount importance for the efficient control of the plasma evolution and the safe running of the experiment. In addition to a good and consistent performance in the reconstruction, the task must be performed in real time as the input for the shape controller and more in general for the scenario optimization. To this aim, a statistical procedure for the evaluation of the reconstruction capability of different magnetic sensor sets is presented, which can drive the choice for an optimal set to be used for the reconstruction of plasma location and boundary shape during real time operation. In addition, an algorithm to approximately solve the free boundary problem and estimate the plasma shape starting from the magnetics is devised. Beyond representing a first step towards the definition of a boundary reconstruction code for plasma shape control, this tool is also used to cross validate and confirm the statistical analysis on the diagnostics.

We consider the distributed unconstrained minimization of separable convex costfunctions, where the global cost is given by the sum of several local and private costs, eachassociated to a specific agent of a given communication network. We specifically address anasynchronous distributed optimization technique called Newton-Raphson consensus. Besidehaving low computational complexity, low communication requirements and being interpretableas a distributed Newton-Raphson algorithm, the technique has also the beneficial properties ofrequiring very little coordination and naturally support time-varying topologies. In this workwe analytically prove that under some assumptions it shows local convergence properties, andcorroborate this result by means of numerical simulations.

Semiconductor manufacturing is one of the most technologically advanced industrial sectors. Process quality and control are critical for decreasing costs and increasing yield. The contribution of automatic control and statistical modeling in this area can drastically impact production performance. For this reason in the past decade major collaborative research projects have been undertaken between fab industries and academia in the areas of Virtual Metrology, Predictive Maintenance, Fault Detection, Run-to-Run control and modeling. In this paper we review some this research, discuss its impact on production and highlight current challenges. 

The paper considers the problem of reconstructing a probability density function from a finite set of samples independently drawn from it. We cast the problem in a Bayesian setting where the unknown density is modeled via a nonlinear transformation of a Bayesian prior placed on a Reproducing Kernel Hilbert Space. The learning of the unknown density function is then formulated as a minimum variance estimation problem. Since this requires the solution of analytically intractable integrals, we solve this problem by proposing a novel algorithm based on the Markov chain Monte Carlo framework. Simulations are used to corroborate the goodness of the new approach.

In this paper exponential stabilizability
of continuous-time positive switched systems is in-
vestigated. For two-dimensional systems, exponential
stabilizability by means of a switching control law can
be achieved if and only if there exists a Hurwitz convex
combination of the (Metzler) system matrices. In the
higher dimensional case, it is shown by means of an
example that the existence of a Hurwitz convex combi-
nation is only sufficient for exponential stabilizability,
and that such a combination can be found if and only
if there exists a smooth, positively homogeneous and
co-positive control Lyapunov function for the system.
In the general case, exponential stabilizability ensures
the existence of a concave, positively homogeneous and
co-positive control Lyapunov function, but this is not
always smooth. The results obtained in the first part
of the paper are exploited to characterize exponential
stabilizability of positive switched systems with delays,
and to provide a description of all the “switched equi-
librium points” of an affine positive switched system.

To maintain and organize distributed systems it is necessary to have a certain degree of knowledge of their status like the number of cooperating agents. The estimation of this number, usually referred as the network size, can pose challenging questions when agents' identification information cannot be disclosed, since the exchanged information cannot be associated to who originated it. In this paper we propose a totally distributed network size estimation strategy based on statistical inference concepts that can be applied under anonymity constraints. The scheme is based on the following paradigm: agents locally generate some Bernoulli trials, then distributedly compute averages of these generated data, finally locally compute the Maximum Likelihood estimate of the network size exploiting its probabilistic dependencies with the previously computed averages. In this work we study the statistical properties of this estimation strategy, and show how the probability of returning a wrong evaluation decreases exponentially in the number of locally generated trials. Finally, we discuss how practical implementation issues may affect the estimator, and show that there exists a neat phase transition between insensitivity to numerical errors and uselessness of the results.

We consider the problem of commanding the electronic power interfaces of the microgenerators in a low voltage microgrid for the task of optimal reactive power compensation. In this work, we analyze the convergence of the strategy proposed by Tenti et al. in 2012. The proof of convergence gives some additional insight on the behavior of the algorithm and allows the characterization of its rate of convergence as a function of the microgrid parameters.

Abstract—In this paper the concepts of controllability and
zero-controllability of a variable w, appearing either in a
standard or in a latent variable description (as manifest variable),
are introduced and characterized. By assuming this perspective,
the dead-beat control (DBC) problem is posed as the problem of
designing a controller, involving both w and the latent variable
c, such that, for the resulting controlled behavior, the variable w
goes to zero in a finite number of steps in every trajectory. Zerocontrollability
of w turns out to be a necessary and sufficient
condition for the existence of “admissible” DBC’s as well as for
the existence of regular DBC’s. The class of minimal DBC’s,
namely DBC’s with the least possible number of rows, is singledout
and a parametrization of such controllers is provided. Finally,
a necessary and sufficient condition for the existence of DBC’s
that can be implemented via a feedback law, for which w is
the input and the latent variable c the corresponding output, is
provided.

We consider how a set of collaborating agents can distributedly infer some of theproperties of the communication network that they form. We specifically focus on estimatingquantities that can characterize the performance of other distributed algorithms, namely theeccentricities of the nodes, and the radius and diameter of the network. We propose a strategythat can be implemented in any network, even under anonymity constraints, and has thedesirable properties of being fully distributed, parallel and scalable. We analytically characterizethe statistics of the estimation error, and highlight how the performance of the algorithmdepends on a parameter tuning the communication complexity.

In this paper we address the problem of exploitingthe distributed energy resources (DER) available in a smartmicro-grid to minimize the power distribution losses via optimalreactive power compensation. Due to their typically smallsize, the amount of reactive power provided by each micro-generator is subject to tight saturation constraints. As aconsequence, it might be impossible to achieve convergence tothe global optimum based on algorithms that rely on short-range, gossip-type communication. We therefore propose arandomized multi-hop protocol that guarantees convergence ofthe distributed optimization algorithm also when only short-range communications are possible, at the expense of someadditional communication overhead.

In this paper we focus on collaborative wireless sensor networks, where agents are randomly distributed over a region of interest and collaborate to achieve a common estimation goal. In particular, we introduce two consensus-based distributed linear estimators. The first one is designed for a Bayesian scenario, where an unknown common finite-dimensional parameter vector has to be reconstructed,  while the second one  regards the nonparametric reconstruction of an unknown function sampled at different locations by the sensors. Both of the algorithms are characterized in terms of the trade-off between estimation performance, communication, computation and memory complexity. In the finite-dimensional setting, we derive mild sufficient conditions which ensure that distributed estimator performs better than the local optimal ones in terms of estimation error variance. In the nonparametric setting, we introduce an on-line algorithm that allows the agents both to compute the function estimate with small computational, communication and data storage efforts, and to quantify its distance from the centralized estimate given by a Regularization Network, one of the most powerful regularized kernel methods. These results are obtained by deriving bounds on the estimation error that provide insights on how the uncertainty inherent in a sensor network, such as imperfect knowledge on the number of agents and the measurement models used by the sensors, can degrade the performance of the estimation process. Numerical experiments are included to support the theoretical findings.

We consider the problem of estimating the size of dynamic anonymous networks, motivated by network maintenance. The proposed algorithm is based on max-consensus information exchange protocols, and extends a previous algorithm for static anonymous networks. A regularization term is accounting for a-priori assumptions on the smoothness of the estimate, and we specifically consider quadratic regularization terms since they lead to closed-form solutions and intuitive design laws. We derive an explicit estimation scheme for a particular peer-to-peer service network, starting from its statistical model. To validate the accuracy of the algorithm, we perform numerical experiments and show how the algorithm can be implemented using finite precision arithmetics as well as small communication burdens.

A new boundary reconstruction procedure is presented and validated against ITER nominal equilibria. An approxima- tion of the plasma with an equivalent filamentary current model is employed, which is computed iteratively and allows to describe a wide variety of plasma current distributions (from the peaked ones, to the pedestal current ones). One of the specific features of the procedure is how the filaments are switched on and how the total current is distributed over the entire set, being the filaments independently considered: this allows more degrees of freedom to the model to adapt to particular current distributions, yielding better performances with a negligible additional computational burden. The code also implements a special points search making it well suited for both diverted (be they top or bottom x-point) and limiter configurations. In addition also the reconstruction in presence of noise has been explored.

A new boundary reconstruction procedure is presented and validated against ITER nominal equilibria. An approximation of the plasma with an equivalent filamentary current model is employed, which is computed iteratively and allows to describe a wide variety of plasma current distributions (from the peaked ones, to the pedestal current ones). One of the specific features of the procedure is how the filaments are switched on and how the total current is distributed over the entire set, being the filaments independently considered: this allows more degrees of freedom to the model to adapt to particular current distributions, yielding better performances with a negligible additional computational burden. The code also implements a special points search making it well suited for both diverted (be they top or bottom x-point) and limiter configurations. In addition also the reconstruction in presence of noise has been explored.

Many modeling problems require to estimate ascalar output from one or more time series. Such problemsare usually tackled by extracting a fixed number of featuresfrom the time series (like their statistical moments), with aconsequent loss in information that leads to suboptimal predictivemodels. Moreover, feature extraction techniques usuallymake assumptions that are not met by real world settings (e.g.uniformly sampled time series of constant length), and failto deliver a thorough methodology to deal with noisy data.In this paper a methodology based on functional learningis proposed to overcome the aforementioned problems; theproposed Supervised Aggregative Feature Extraction (SAFE)approach allows to derive continuous, smooth estimates oftime series data (yielding aggregate local information), whilesimultaneously estimating a continuous shape function yieldingoptimal predictions. The SAFE paradigm enjoys severalproperties like closed form solution, incorporation of first andsecond order derivative information into the regressor matrix,interpretability of the generated functional predictor and thepossibility to exploit Reproducing Kernel Hilbert Spaces settingto yield nonlinear predictive models. Simulation studies areprovided to highlight the strengths of the new methodology withrespect to standard unsupervised feature selection approaches.

In semiconductor manufacturing plants, monitoringphysical properties of all wafers is fundamental in order tomaintain good yield and high quality standards. However, suchan approach is too costly and in practice only few wafers in a lotare actually monitored. Virtual Metrology (VM) systems allowto partly overcome the lack of physical metrology. In a VMscheme, tool data are used to predict, for every wafer, metrologymeasurements. In this paper, we present a VM system for aChemical Vapor Deposition (CVD) process. On the basis ofthe available metrology results and of the knowledge, for everywafer, of equipment variables, it is possible to predict CVDthickness. In this work we propose a VM module based onLARS to overcome the problem of high dimensionality andmodel interpretability. The proposed VM models have beentested on industrial production data sets.

In this paper we study the problem of real-time optimal distributed partitioning for perimeter patrolling in the context of multi-camera networks for surveillance. The objective is to partition a given segment into non-overlapping sub-segments, each assigned to a different camera to patrol. Each camera has both physical mobility range and limited speed, and it must patrol its assigned sub-segment by sweeping it back and forth at maximum speed. Here we first review the solution for the centralized optimal partitioning. Then we propose two different distributed control strategies to determine the extremes of the optimal patrolling areas of each camera. Both these strategies require only local communication with the neighboring cameras but adopt different communication schemes, respectively, symmetric gossip and asynchronous asymmetric broadcast. The first scheme is shown to be provably convergent to the optimal solution. Some theoretical insights are provided also for the second scheme whose effectiveness is validated through numerical simulations.

In this work the design and implementation of an application to track multiple agents in a indoor Wireless Sensor Actor Network (WSAN) is proposed. The adopted embedded hardware for the network nodes is theTmote Sky, an ultra low power IEEE 802.15.4 compliant wireless device, which has become a reference in the academia for the early development of algorithms and applications for Wireless Sensor Actor Networks (WSANs). These devices are based on the TinyOS operative system and are programmed in NesC a C-derived language specifically developed for embedded systems. NesC has become indispensable for low-level management ofindividual agents while Java was chosen to provide the user with a simple and intuitive graphical interface with whom showing and coordinating the tracking.

In this work an algorithm to track multiple agents in an indoor Wireless Sensor Actor Network (WSAN) is proposed. The algorithm falls into the category of the radio frequency localization methods, since it exploits the strength of the wireless communications among nodes to establish the position of a set of mobile nodes within a network of fixed nodes placed in known locations. In this sense, a radio channel model is introduced that allows to estimate the distances among nodes to attain localization and tracking (range-based approach). Moreover, to compensate for the scant robustness of power measurements, the loss effects induced by wireless communication,the intrinsic uncertainty of unstructured environments, the algorithm resorts to an Extended Kalman Filter to process the node measurements and reach a desired level of localization performance. Finally, the design phase is validated through the implementation and the experiments on a real testbed.

In this work we consider a multidimensional distributed optimization technique that is suitable for multiagents systems subject to limited communication connectivity. In particular, we consider a convex unconstrained additive problem, i.e. a case where the global convex unconstrained multidimensional cost function is given by the sum of local cost functions available only to the specific owning agents. We show how, by exploiting the separation of time-scales principle,the multidimensional consensus-based strategy approximates a Newton-Raphson descent algorithm. We propose two alternative optimization strategies corresponding to approximations of the main procedure. These approximations introduce tradeoffs between the required communication bandwidth and the convergence speed/accuracy of the results. We provide analytical proofs of convergence and numerical simulations supporting the intuitions developed through the paper.

In semiconductor manufacturing, state of the art for wafer quality control relies on product monitoring and feedback control loops; the involved metrology operations, performed by means of scanning electron microscopes, are particularly cost-intensive and time-consuming. For this reason, it is not possible to evaluate every wafer: in common practice, a small subset of a productive lot is measured at the metrology station and devoted to represent the whole lot. Virtual Metrology (VM) methodologies are able to obtain reliable predictions of metrology results at process time, without actually performing physical measurements; this goal is usually achieved by means of statistical models, linking process data and context information to target measurements. Since semiconductor manufacturing processes involve a high number of sequential operations, it is reasonable to assume that the quality features of a certain wafer (such as layer thickness, critical dimensions, electrical test results) depend on the whole processing and not only on the last step before measurement. In this paper, we investigate the possibilities to improve the Virtual Metrology quality relying on knowledge collected from previous process steps. We will present two different scheme of multistep VM, along with dataset preparation indications; special consideration will be reserved to regression techniques capable of handling high dimensional input spaces. The proposed multistep approaches will be tested against actual data from semiconductor manufacturing industry. 

This paper considers the problem of simulating the turbulence effect on ground telescope observations. The approach presented here is an evolution of a recently proposed approach [3]. The main contributions with respect to [3] are: First, the Haar transform at the basis of the multiscale model in [3] is shown to be equivalent to a local PCA representation. This equivalence allows to reduce the computational complexity of the simulation algorithm by neglecting the components in the signal with lower energy. Furthermore, the simulation of nonstationary turbulence is obtained by properly changing the values of the multiscale model: Such change is eased by the invariance of the PCA spatial basis with respect to the change of turbulence statistical characteristics. The proposed approach is validated by means of some simulations.

In this paper we consider a supervisory control scheme for non-homogenous multi-agent systems. Each
agent is modeled through an independent strictly proper SISO state space model, and the supervisory
controller, representing the information exchange among the agents, is implemented in turn via a linear
state-space model. Controllability and observability of the overall system are characterized, and some
preliminary results about stability and stabilizability are provided. The paper extends to non-homogenous
multi-agent systems some of the results obtained in Hara et al. (2007, 2009) [4,5,7] for the homogenous
case.

This paper reviews a number of recent results dealing with the zeros of rational  transfer functions, and including transfer functions of blocked discrete-time systems having more than one underlying sampling period.  A non-square system which is finite-dimensional, linear and time-invariant generically has  no zeros; in contrast, for a square system, generically all zeros will be distinct, finite and simple.  For  systems subjected to loss of some output values, so that with inputs  every T seconds,  some outputs available every T seconds and others only available every NT seconds, a blocked transfer function can be constructed. Similar conclusions hold, save that special attention nis needed for zeros at 0 and infinity.  The multi-frequency results are potentially very important in econometric modelling, where different time-series are available at different frequencies, e.g. monthly and quarterly, and offer the possibility of great simplification in the associated identification problem.

Geometric triangulation is at the basis of the estimation of the 3D position of a target from a set of camera measurements. The problem of optimal estimation (minimizing the L2 norm) of the target position from multi-view perspective projective measurements is typically a hard problem to solve. In literature there are different types of algorithms for this purpose, based for example on the exhaustive check of all the local minima of a proper eigenvalue problem [2], or branch- and-bound techniques [3]. However, such methods typically become unfeasible for real time applications when the number of cameras and targets become large, calling for the definition of approximate procedures to solve the reconstruction problem.
In the first part of this paper, linear (fast) algorithms, computing an approximate solution to such problems, are described and compared in simulation. Then, in the second part, a Gaussian approximation to the measurement error is used to express the reconstruction error’s standard deviation as a function of the position of the reconstructed point. An upper bound, valid over all the target domain, to this expression is obtained for a case of interest. Such upper bound allows to compute a number of cameras sufficient to obtain a user defined level of position estimation accuracy.

Epitaxy is a process strongly dependent onwafer temperature. Unfortunately, the performance ofthe pyrometers in charge of sensing wafer temperaturedeteriorate with the usage. This represents the majormaintenance issue for epitaxy process engineers who haveto frequently calibrate pyrometers emissivity coefficient. Atthe present state the change of the emissivity coefficient isheuristically based on fab tradition and process engineersexperience. We present a statistical tool to map therelationship between change in the temperature readingsand emissivity adjustments. The module has been testedon real industrial dataset.

This paper proposes a real-time distributed algorithm for a team of smart camerasto self-organize and perform video surveillance of an open boundary. In particular, our algorithmsimultaneously partitions the boundary among the cameras, and synchronizes the motion of thecameras to optimize the surveillance performance. We focus on the detection of smart intruders,who are aware of the cameras configuration at each time instant, and who schedule their motionto avoid detection for as long as possible. We consider both the worst-case and the averagedetection times of smart intruders. Our algorithm achieves minimum worst-case detection time,and, under some reasonable assumptions, constant-factor optimal average detection time.

In this paper we consider the class of discretetime
switched systems switching between p autonomous positive
subsystems. First, sufficient conditions for testing stability, based
on the existence of special classes of common Lyapunov functions,
are investigated, and these conditions are mutually related, thus
proving that if a linear copositive common Lyapunov function can
be found, then a quadratic positive definite common function can
be found, too, and this latter, in turn, ensures the existence of a
quadratic copositive common function. Secondly, stabilizability
is introduced and characterized. It is shown that if these
systems are stabilizable, they can be stabilized by means of a
periodic switching sequence, which asymptotically drives to zero
every positive initial state. Conditions for the existence of statedependent
stabilizing switching laws, based on the values of a
copositive (linear/quadratic) Lyapunov function, are investigated
and mutually related, too.
Finally, some properties of the patterns of the stabilizing
switching sequences are investigated, and the relationship between
a sufficient condition for stabilizability (the existence of
a Schur convex combination of the subsystem matrices) and an
equivalent condition for stabilizability (the existence of a Schur
matrix product of the subsystem matrices) is explored.

We consider the convergence rates of two peculiar2 convex optimization strategies in the context of multi agent3 systems, namely the Newton-Raphson consensus optimization4 and a distributed Gradient-Descent opportunely derived from5 the first. To allow analytical derivations, the convergence6 analyses are performed under the simplificative assumption of7 quadratic local cost functions. In this framework we derive8 sufficient conditions which guarantee the convergence of the9 algorithms. From these conditions we then obtain closed form10 expressions that can be used to tune the parameters for11 maximizing the rate of convergence. Despite these formulae12 have been derived under quadratic local cost functions13 assumptions, they can be used as rules-of-thumb for tuning14 the parameters of the algorithms in general situations.

Nowadays, the adaptive optics (AO) system is of fundamental importance to reduce the effect of atmospheric turbulence on the images formed on large ground telescopes. In this paper the AO system takes advantage of the knowledge of the current turbulence characteristics, that are estimated by data, to properly control the deformable mirrors. The turbulence model considered in this paper is based on two assumptions: considering the turbulence as formed by a discrete set of layers moving over the telescope lens, and each layer is modeled as a Markov-Random-Field. The proposed Markov-Random-Field approach is exploited for estimating the layers’ characteristics. Then, a linear predictor of the turbulent phase, based on the computed information on the turbulence layers, is constructed. Since scalability and low computational complexity of the control algorithms are important requirements for real AO systems, the computational complexity properties of the proposed model are investigated. Interestingly, the proposed model shows a good scalability and an almost linear computational complexity thanks to its block diagonal structure. Performances of the proposed method are investigated by means of some simulations.