The typically qualitative treatment of evidence underlying different causal hypotheses that attempt to explain singular historic events is criticized. Instead of using ad hoc qualitative arguments, it is here advocated to use Bayes theorem to assess how well evidence either support or weaken a specific causal hypothesis. It is argued that such a practice will enable scientists who study singular historic events to discuss, compare and aggregate different beliefs in a meaningful way.

A rigorous perturbation analysis of the singular value decomposition of a real matrix of full column rank is presented. It is shown that the SVD perturbation problem is well posed only in case of distinct singular values. The analysis involves the solution of coupled systems of linear equations and produces asymptotic (local) componentwise perturbation bounds of the entries of the orthogonal matrices participating in the decomposition of the given matrix and of its singular values. Local bounds are derived for the sensitivity of the singular subspaces measured by the angles between the unperturbed and perturbed subspaces. An iterative scheme is described to find global bounds on the respective perturbations. The analysis implements the same methodology used previously to determine componentwise perturbation bounds of the Schur form and the QR decomposition of a matrix.

This study develops a new definition of fractional derivative that mixes the definitions of fractional derivatives with singular and non-singular kernels. Such developed definition encompasses many types of fractional derivatives, such as the Riemann-Liouville and Caputo fractional derivatives for singular kernel type as well as the Caputo-Fabrizio, the Atangana-Baleanu and the generalized Hattaf fractional derivatives for non-singular kernel type. The associate fractional integral of the new mixed fractional derivative is rigorously introduced. Furthermore, newly numerical scheme is developed to approximate the solutions of a class of fractional differential equations (FDEs) involving the mixed fractional derivative. Finally, an application to computational biology is presented.

The usual equation for both motions of a single planet around the sun and electrons in the deterministic Rutherford-Bohr atomic model is conservative with a singular potential at the origin. When a dissipation is added, new phenomena appear which were investigated thoroughly by R. Ortega and his co-authors between 2014 and 2017, in particular all solutions are bounded and tend to $0$ for $t$ large, some of them with asymptotically spiraling exponentially fast convergence to the center. We provide explicit estimates for the bounds in the general case that we refine under specific restrictions on the initial state, and we give a formal calculation which could be used to determine practically some special asymptotically spiraling orbits. Besides, a related model with exponentially damped central charge or mass gives some explicit exponentially decaying solutions which might help future investigations. An atomic contraction hypothesis related to the asymptotic dying off of solutions proven for the dissipative model might give a solution to some intriguing phenomena observed in paleontology, familiar electrical devices and high scale cosmology

The usual equation for both motions of a single planet around the sun and electrons in the deterministic Rutherford-Bohr atomic model is conservative with a singular potential at the origin. When a dissipation is added, new phenomena appear. It is shown that whenever the momentum is not zero, the moving particle does not reach the center in finite time and its displacement does not blow-up either, even in the classical context where arbitrarily large velocities are allowed. Moreover we prove that all bounded solutions tend to $0$ for $t$ large, and some formal calculations suggest the existence of special orbits with an asymptotically spiraling exponentially fast convergence to the center.

This short communication makes use of the principle of singular perturbation to approximate the ordinary differential equation (ODE) of prompt neutron (in the point kinetics model) as an algebraic equation. This approximation is shown to yield a large gain in computational efficiency without compromising any significant accuracy in the numerical simulation of primary coolant system dynamics in a PWR nuclear power plant. The approximate (i.e., singularly perturbed) model has been validated with a numerical solution of the original set of neutron point-kinetic and thermal–hydraulic equations. Both models use variable-step Runge–Kutta numerical integration.

In this lecture notes, we will introduce Asymptotics, then we will give a short glimpse on Perturbation theory (regular versus singular), which plays a crucial role especially in theoretical physics. Our goal is to find Asymptotic series that approximates the values of integrals depending on some parameter or the solutions of differential equations. These methods that lead to obtain more effective algorithms of numerical evaluation are called: Asymptotic Methods.

The features of a class of cubic curves with a shape factor are analyzed by means of the theory of envelope and topological mapping. The effects of the shape factor on the cubic curves are made clear. Necessary and sufficient conditions are derived for the curve to have one or two inflection points, a loop or a cusp, or to be locally or globally convex. Those conditions are completely characterized by the relative position of the edge vectors of the control polygon and the shape factor. The results are summarized in a shape diagram, which is useful when the cubic parametric curves are used for geometric modeling. Furthermore we discussed the influences of the shape factor on the shape diagram and the ability for adjusting the shape of the curve.

In this paper, we proposed the fusion of two projection based face recognition algorithms: local binary Patterns in DCT domain and singular value decomposition (SVD) characterized by its simplicity and efficiently. Standard databases ORL are used to test the experimental results which prove that proposed system achieves more accurate face recognition as compared to individual method.

This research work deals with the formulation of numerical scheme with algorithm to compute singularly perturbed partial differential equations(SP-PDEs) involving a significant negative shift. The problem involves small parameter which causes a rapidly changing boundary layers in the vicinity of a body, and the negative shift term causes interior layer. Such appearance of an abruptly varying layers causes difficulties to find the exact solution and it is not adequate to employ classical numerical methods. The technique presented in this research work simplifies these challenges and yields accurate numerical solution.The stability and convergence analyses of the methods are examined and proven. To test the developed technique, numerical experiments are carried out and confirmed with theoretical analysis.

A method for studying properties of the earth's surface tremor, measured by means of GPS, is proposed. Four characteristics of tremor are considered: the entropy of the distribution of wavelet coefficients, the Donoho-Johnston wavelet index, and two estimates of the spectral slope. The anomalous areas of tremor are determined by estimating the probability densities of extreme values of the studied properties. The criteria for abnormal tremor behavior are based on the proximity to, or the difference between, tremor properties and white noise. The greatest deviation from the properties of white noise is characterized by entropy minima and spectral slope and DJ index maxima. This behavior of the tremor is called "active". The "passive" tremor behavior is characterized by the maximum proximity to the properties of white noise. The principal components approach provides weighted averaged density maps of these 2 variants of extreme distributions of parameters in a sliding time window of 3 years are considered. Singular points are the points of maximum average densities. The method is applied to the analysis of daily time series from a GPS network in the West USA in 2009-2022. It turned out that singular points of tremor form well-defined clusters.

A method for studying properties of the earth's surface tremor, measured by means of GPS, is proposed. Four characteristics of tremor are considered: the entropy of the distribution of wavelet coefficients, the Donoho-Johnston wavelet index, and two estimates of the spectral slope. The anomalous areas of tremor are determined by estimating the probability densities of extreme values of the studied properties. The criteria for abnormal tremor behavior are based on the proximity to, or the difference between, tremor properties and white noise. The greatest deviation from the properties of white noise is characterized by entropy minima and spectral slope and DJ index maxima. This behavior of the tremor is called "active". The "passive" tremor behavior is characterized by the maximum proximity to the properties of white noise. The principal components approach provides weighted averaged density maps of these 2 variants of extreme distributions of parameters in a sliding time window of 3 years are considered. Singular points are the points of maximum average densities. The method is applied to the analysis of daily time series from a GPS network in the West USA in 2009-2022. It turned out that singular points of tremor form well-defined clusters.

Risk-averse areas such as the medical, aerospace and energy sectors have been somewhat slow towards accepting and applying Additive Manufacturing (AM) in many of their value chains. This is partly because there are still signicant uncertainties concerning the quality of AM builds. This paper introduces a machine learning algorithm for the automatic detection of faults in AM products. The approach is semi-supervised in that, during training, it is able to use data from both builds where the resulting components were certied and builds where the quality of the resulting components is unknown. This makes the approach cost ecient, particularly in scenarios where part certication is costly and time consuming. The study specically analyses Selective Laser Melting (SLM) builds. Key features are extracted from large sets of photodiode data, obtained during the building of 49 tensile test bars. Ultimate tensile strength (UTS) tests were then used to categorise each bar as `faulty' or `acceptable'. A fully supervised approach identied faulty specimens with a 77% success rate while the semi-supervised approach was able to consistently achieve similar results, despite being trained on a fraction of the available certication data. The results show that semi-supervised learning is a promising approach for the automatic certication of AM builds that can be implemented at a fraction of the cost currently required.

In this paper, we study a differential operator of parabolic type with a variable and unbounded coefficient, defined on an infinite strip. Sufficient conditions for the existence and compactness of the resolvent are established, and an estimate for the maximum regularity of solutions of the equation Lu=f∈L_2(Ω) is obtained. Two-sided estimates for the distribution function of approximation numbers are obtained. As is known, estimates of approximation numbers show the rate of best approximation of the resolvent of an operator by finite-dimensional operators. The paper proves the assertion about the existence of positive eigenvalues among the eigenvalues of the given operator and finds two-sided estimates for them.

: In this article, the authors propose an algorithmic approach to building a model of the dynamics of economic and, in particular, innovation processes. The approach under consideration is based on a complex algorithm that includes (1) decomposition of the time series into components using singular spectrum analysis; (2) recognition of the optimal component model based on fuzzy rules, and (3) creation of statistical models of individual components with their combination. It is shown that this approach corresponds to the high uncertainty characteristic of the tasks of the dynamics of innovation processes. The proposed algorithm makes it possible to create effective models that can be used both for analysis and for predicting the future states of the processes under study. The advantage of this algorithm is the possibility to expand the base of rules and components used for modeling. This is an important condition for improving the algorithm and its applicability for solving a wide range of problems.

We propose a novel data hiding method in an audio host with a compressive sampling technique. An over-complete dictionary represents a group of the watermark. Each row of the dictionary is a Hadamard sequence representing multiple bits of the watermark. Then, the singular values of segment-based host audio in a diagonal matrix multiply by the over-complete dictionary producing a lower size matrix. At the same time, we embed the watermark into the compressed audio. In the detector, we detect the watermark and reconstruct the audio. This proposed method offers not only hiding the information but also compressing the audio host. The application of the proposed method is a broadcast monitoring and biomedical signal recording. We can mark and secure the signal content by hiding the watermark inside the signal while we compress the signal for memory efficiency. We evaluate the performance in terms of payload, compression ratio, audio quality, and watermark quality. The proposed method can hide the data imperceptibly, in range 729-5292 bps with compression ratio 1.47-4.84 and perfect detected watermark.

In the paper, we develop some numerical schemes including finite difference scheme and finite volume scheme for the fractional Laplacian operator, and apply the resulted numerical schemes to solve some fractional diffusion equations. First, the fractional Laplacian operator can be characterized as the weak singular integral by a integral operator with zero boundary condition. Second, because of the solution of fractional diffusion equation is usually singular near the boundary, we apply finite difference scheme to discrete fractional Laplacian operator and fractional diffusion equation by fractional interpolation function. Moreover, it is find that the differential matrix of the above scheme is symmetric matrix, and strictly row-wise diagonally dominant in special fractional interpolation function. Third, we show finite volume scheme to discrete fractional diffusion equation by fractional interpolation function, and analyze the properties of differential matrix. Finally, the numerical experiments are given, and verify the correctness of theoretical results and the efficiency of the schemes.

Blockchain technology has been recognized as a promising solution to enhance the security and privacy of Internet of Things (IoT) and Edge Computing scenarios. Taking advantage of the Proof-of-Work (PoW) consensus protocol, which solves a computation intensive hashing puzzle, Blockchain assures the security of the system by establishing a digital ledger. However, the computation intensive PoW favors members possessing more computing power. In the IoT paradigm, fairness in the highly heterogeneous network edge environments must consider devices with various constraints on computation power. Inspired by the advanced features of Digital Twins (DT), an emerging concept that mirrors the lifespan and operational characteristics of physical objects, we propose a novel Miner-Twins (MinT) architecture to enable a fair PoW consensus mechanism for blockchains in IoT environments. MinT adopts an edge-fog-cloud hierarchy. All physical miners of the blockchain are deployed as microservices on distributed edge devices, while fog/cloud servers maintain digital twins that periodically update miners’ running status. By timely monitoring miner’s footage that is mirrored by twins, a lightweight Singular Spectrum Analysis (SSA) based detection achieves to identify individual misbehaved miners that violate fair mining. Moreover, we also design a novel Proof-of-Behavior (PoB) consensus algorithm to detect byzantine miners that collude to compromise a fair mining network. A preliminary study is conducted on a proof-of-concept prototype implementation, and experimental evaluation shows the feasibility and effectiveness of proposed MinT scheme under a distributed byzantine network environment.

We are concerned with the use of some classical spectral collocation methods as well as with the new software system Chebfun in order to compute high order (index) eigenpairs of singular as well as regular Schrodinger eigenproblems. We want to highlight both the qualities as well as the shortcomings of these methods and evaluate them vis-a-vis the usual ones. In order to resolve a boundary singularity we use Chebfun with the simple domain truncation technique. Although this method is equally easy to apply with spectral collocation, things are more nuanced in the case of these methods. A special technique to introduce boundary conditions as well as a coordinate transform which maps an unbounded domain to a nite one are the ingredients. A challenging set of "hard" benchmark problems, for which usual numerical methods (f. d., f. e. m., shooting etc.) fail, are analysed. In order to separate "good"and "bad"eigenvalues we estimate the drift of the set of eigenvalues of interest with respect to the order of approximation and/or scaling of domain parameter. It automatically provides us with a measure of the error within which the eigenvalues are computed and a hint on numerical stability. We pay a particular attention to problems with almost multiple eigenvalues as well as for problems with a mixed (continuous) spectrum. In the latter case we try to numerically highlight its existence. Special attention will be paid to the higher eigenpairs (the pair of eigenvalue and the corresponding eigenfunction approximated by an eigenvector spanning its nodal values).

While the importance of continuous monitoring of electrocardiographic (ECG) or photoplethysmographic (PPG) signals to detect cardiac anomalies is generally accepted in preventative medicine, there remain major barriers to its actual widespread adoption. Most notably, current approaches tend to lack real-time capability, exhibit high computational cost, and be based on restrictive modeling assumptions or require large amounts of training data. We propose a lightweight and model-free approach for the online detection of cardiac anomalies such as ectopic beats in ECG or PPG signals based on the change detection capabilities of Singular Spectrum Analysis (SSA) and nonparametric rank-based cumulative sum (CUSUM) control charts. The procedure is able to quickly detect anomalies without requiring the identification of fiducial points such as R-peaks and is computationally significantly less demanding than previously proposed SSA-based approaches. Therefore, the proposed procedure is equally well suited for standalone use and as an add-on to complement existing (e.g. heart rate (HR) estimation) procedures.

We examine a nonlinear initial value problem both singularly perturbed in a complex parameter and singular in complex time at the origin. The study undertaken in this paper is the continuation of a joined work with A. Lastra published in 2015. A change of balance between the leading and a critical subdominant term of the problem considered in our previous work is performed. It leads to a phenomenon of coalescing singularities to the origin in the Borel plane w.r.t time for a finite set of holomorphic solutions constructed as Fourier series in space on horizontal complex strips. In comparison to our former study, an enlargement of the Gevrey order of the asymptotic expansion for these solutions relatively to the complex parameter is induced.

This study develops a predictor-corrector algorithm for the numerical simulation of IVPs involving singular generalized fractional derivatives with Mittag-Leffler kernels. The proposed algorithm converts the considered IVP into a Volterra-type integral equation and then uses Trapezoidal rule to obtain approximate solutions. Numerical approximate solutions of some singular generalized fractional derivative with Mittag-Leffler kernels models have been presented to demonstrate the efficiency and accuracy of the proposed algorithm. The algorithm describes the influence of the fractional derivative parameters on the dynamics of the studied models. The suggested method is expected to be effectively employed in the field of simulating generalized fractional derivative models

We examine a family of linear partial differential equations both singularly perturbed in a complex parameter and singular in complex time at the origin. These equations entail forcing terms which combine polynomial and logarithmic type functions in time and that are bounded holomorphic on horizontal strips in one complex space variable. A set of sectorial holomorphic solutions are built up by means of complete and truncated Laplace transforms w.r.t time and parameter and Fourier inverse integral in space. Asymptotic expansions of these solutions relatively to time and parameter are investigated and two distinguished Gevrey type expansions in monomial and logarithmic scales are exhibited.

Densimetric Froude (Fr) is the minimum velocity required to prevent sediment deposition in pipes. Prediction of Fr is of utmost importance in numerous applications in civil engineering. In this paper through using a new hybrid method. Genetic Algorithm (GA) is used for optimum selection of membership functions of Adaptive Neuro-Fuzzy Inference System (ANFIS), and Singular Value Decomposition (SVD) method is used to compute the linear parameters of ANFIS’s results section (ANFIS-GA/SVD). Also, two different target functions are known as training error (TE) and prediction error (PE) by Pareto curve, the trade-off between these functions is selected as the optimal modeling point. First, different models will be presented using the parameters affecting Fr prediction, classifying them in different groups; then the Fr parameter will be predicted for all the different models through utilizing three different sets of data and the ANFIS-GA/SVD technique. The results of the models indicate that the best Fr prediction is obtained when independent parameters such as volumetric sediment concentration (CV), ratio of median diameter of particle size to pipe diameter (d/D), ratio of median diameter of particle size to hydraulic radius (d/R) and overall friction factor of sediment (λs) use as input variables in prediction of Fr. A sensitivity analysis is also conducted for the purpose of examining the effect of each of the dimensionless parameters on Fr prediction accuracy. Comparing the results of the suggested models with the existing regression-based equations shows that ANFIS-GA/SVD (R²=0.986, MAPE=4.397, RMSE=0.206, SI=0.053, ρ=0.026, BIAS=-0.025) is more accurate than the rest of the models.

For more than half a century, Manfred Deistler has been contributing to the construction of the rigorous theoretical foundations of the statistical analysis of time series and more general stochastic processes. Half a century of unremitting activity is not easily summarized in a few pages. In this short note, we chose to concentrate on a relatively little-known aspect of Manfred's contribution which nevertheless had quite an impact on the development of one of the most powerful tools of contemporary time series and econometrics: dynamic factor models.

New modied Hayward metric of magnetically charged non-singular black hole spacetime in the framework of nonlinear electrodynamics is constructed. When the fundamental length introduced, characterising quantum gravity effects, vanishes one comes to the general relativity coupled with the Bronnikov model of nonlinear electrodynamics. The metric can have one (an extreme) horizon, two horizons of black holes, or no horizons corresponding to the particle-like solution. Corrections to the Reissner-Nordström solution are found as the radius approaches to innity. As r -> 0 the metric has a de Sitter core showing the absence of singularities. The asymptotic of the Ricci and Kretschmann scalars are obtained and they are nite everywhere. The thermodynamics of black holes, by calculating the Hawking temperature and the heat capacity, is studied. It is demonstrated that phase transitions take place when the Hawking temperature possesses the maximum. Black holes are thermodynamically stable at some range of parameters.

We investigate a family of nonlinear partial differential equations which are singularly perturbed in a complex parameter and singular in a complex time variable at the origin. These equations combine differential operators of Fuchsian type in time and space derivatives on horizontal strips in the complex plane with a nonlocal operator acting on the complex parameter known as the formal monodromy around 0. Their coefficients and forcing terms comprise polynomial and logarithmic type functions in time and are bounded holomorphic in space. A set of logarithmic type solutions are shaped by means of Laplace transforms relatively to time and parameter and Fourier integrals in space. Furthermore, a formal logarithmic type solution is modeled which represents the common asymptotic expansion of Gevrey type of the genuine solutions with respect to the complex parameter on bounded sectors at the origin.

In this work, an analytical and numerical analysis of the transition to chaos in five nonlinear systems of ordinary and partial differential equations, which are models of the autocatalytic chemical processes and the numbers of interacting populations, is carried out. It is analytically and numerically shown that in all considered systems of equations, further complication of the dynamics of solutions and the transition to chemical and biological turbulence is carried out in full accordance with the universal Feigenbaum-Sharkovsky-Magnitskii bifurcation theory through subharmonic and homoclinic cascades of bifurcations of stable limit cycles. In this case, irregular (chaotic) attractors in all cases are exclusively singular attractors in the sense of the FShM theory.

A family of linear singularly perturbed Cauchy problems is studied. The equations defining the problem combine both partial differential operators together with the action of linear fractional transforms. The exotic geometry of the problem in the Borel plane, involving both sectorial regions and strip-like sets, gives rise to asymptotic results relating the analytic solution and the formal one through Gevrey asymptotic expansions. The main results lean on the appearance of domains in the complex plane which remain intimately related to Lambert W function, which turns out to be crucial in the construction of the analytic solutions. On the way, an accurate description of the deformation of the integration paths defining the analytic solutions and the knowledge of Lambert W function are needed in order to provide the asymptotic behavior of the solution near the origin, regarding the perturbation parameter. Such deformation varies depending on the analytic solution considered, which lies in two families with different geometric features.

The work is devoted to the study of a family of linear initial value problems of partial differential equations in the complex domain, dealing with two complex time variables. The use of a truncated Laplace-like transformation in the construction of the analytic solution allows to overcome a small divisor phenomenon arising from the geometry of the problem and represents an alternative approach to the one proposed in a recent work by the first two authors. The result leans on the application of a fixed point argument and the classical Ramis-Sibuya theorem.

A nonlinear singularly perturbed Cauchy problem with confluent fuchsian singularities is examined. This problem involves coefficients with polynomial dependence in time. A similar initial value problem with logarithmic reliance in time has been investigated by the author in a recent work, for which sets of holomorphic inner and outer solutions were built up and expressed as a Laplace transform with logarithmic kernel. Here, a family of holomorphic inner solutions are constructed by means of exponential transseries expansions containing infinitely many Laplace transforms with special kernel. Furthermore, asymptotic expansions of Gevrey type for these solutions relatively to the perturbation parameter are established.

In this work the existence and nonexistence of stationary radial solutions to the elliptic partial differential equation arising in the molecular beam epitaxy are studied. Since we are interested in radial solutions we arrive at the following fourth-order differential equation

This paper is a slightly modified, abridged version of a previous work ``Parametric Gevrey asymptotics in two complex time variables through truncated Laplace transforms'' motivated by our contribution in the conference ``Formal and Analytic Solutions of Diff. (differential, partial differential, difference, q-difference, q-difference-differential) Equations on the Internet'' (FASnet20). It aims to clarify and give further detail at some crucial points concerning the asymptotic behavior of the solutions of the problems studied in that work.

This paper is a slightly modified, abridged version of a previous work "Parametric Gevrey asymptotics in two complex time variables through truncated Laplace transforms'' motivated by our contribution in the conference "Formal and Analytic Solutions of Diff. (differential, partial differential, difference, q-difference, q-difference-differential) Equations on the Internet'' (FASnet20). It aims to clarify and give further detail at some crucial points concerning the asymptotic behavior of the solutions of the problems studied in that work.

This article deals with the conformable double Laplace transforms and their some properties with examples and also the existence Condition for the conformable double Laplace transform is studied. Finally, in order to obtain the solution of nonlinear fractional problems, we present a modified conformable double Laplace that we call conformable double Laplace decomposition methods (CDLDM). Then, we apply it to solve, Regular and singular conformable fractional coupled burgers equation illustrate the effectiveness of our method some examples are given.

The article considers an internal boundary value problem of the distribution of an electrostatic field in a lens formed by two identical semi-infinite circular cylinders coaxially located inside an infinite external cylinder. The problem is reduced to solving a system of singular Wiener-Hopf integral equations, which is further solved by the Wiener-Hopf method using factorized Bessel functions. Solutions to the problem for each region inside the infinite outer cylinder are presented as exponentially converging series in terms of eigenfunctions and eigenvalues.

We consider a family of nonlinear singularly perturbed PDEs whose coefficients involve a logarithmic dependence in time with confluent Fuchsian singularities that unfold an irregular singularity at the origin and rely on a single perturbation parameter. We exhibit two distinguished finite sets of holomorphic solutions, so-called outer and inner solutions, by means of a Laplace transform with special kernel and Fourier integral. We analyze the asymptotic expansions of these solutions relatively to the perturbation parameter and show that they are (at most) of Gevrey order 1 for the first set of solutions and of some Gevrey order that hinges on the unfolding of the irregular singularity for the second.

A reduced order model is developed to monitor aeroengines condition (defining their degradation from a baseline state) in real-time, by using data collected in specific sensors. This reduced model is constructed by applying higher order singular value decomposition plus interpolation to appropriate data, organized in tensor form. Such data are obtained using a detailed engine model that takes the engine physics into account. Thus, the method synergically combines the advantages of data-driven (fast online operation) and model-based (the engine physics is accounted for) condition monitoring methods. Using this reduced order model as surrogate of the engine model, two gradient-like condition monitoring tools are constructed. The first tool is extremely fast and able to precisely compute `on the fly’ the turbine inlet temperature, which is a paramount parameter for the engine performance, operation, and maintenance, and can only be roughly estimated by the engine instrumentation in civil aviation. The second tool is not so fast (but still reasonably inexpensive) and precisely computes both, the engine degradation and the turbine inlet temperature at which sensors data have been acquired. These tools are robust in connection with random noise added to the sensors data and can be straight forwardly applied to other mechanical systems.

Synthetic Aperture RADAR (SAR) is a radar imaging technique in which the relative motion of the sensor is used to synthesize a very long antenna and obtain high spatial resolution. Standard SAR raw data processing techniques assume uniform motion of the satellite (or aerial vehicle) and a fixed antenna beam pointing sideway orthogonally to the motion path, assumed rectilinear. Despite SAR data processing is a well established imaging technology that has become fundamental in several fields and applications, in this paper a novel approach has been used to exploit coherent illumination, demonstrating the possibility of extracting a large part of the ancillary data information from the raw data itself, to be used in the focusing procedure. In this work an effort has been carried out to try to focus the raw SAR complex data matrix without the knowledge of anyof the parameters needed in standard focusing procedures as Range Doppler (RD) algorithm, ω - K algorithm and Chirp Scaling (CS) algorithm. All the literature references regarding the algorithms needed to obtain a precise image from raw data use such parameters that refer both to the SAR system acquisition geometry and its radiometric specific parameters. In [12], authors introduced a preliminary work dealing with this problem and able to obtain, in the presence of a strong point scatterer in the observed scene, good quality images, if compared to the standard processing techniques. In this work the proposed technique is described and performances parameters are extracted to compare the proposed approach to RD.