Singularity Exponent Research Articles

Radar Sea Clutter Reconstruction Based on Statistical Singularity Power Spectrum and Instantaneous Singularity Exponents Distribution

The modeling and reconstruction of radar sea clutter (RSC) based on fractal theory is studied in this article. With theoretical derivation and quantitative analysis, the radar clutter reconstruction model based on singularity power spectrum (SPS) and instantaneous singularity exponent (ISE) distribution is proposed. The proposed ISE–SPS model consists of two coupling channels: power measure channel and dimension measure channel. The former is controlled by SPS to characterize the power distribution of signals in the SE domain, while the latter is determined by ISE function to characterize the statistical distribution of ISE. The multiscale wavelet coefficients are estimated based on the coupling of the two channels to reconstruct the RSC. The proposed ISE–SPS method is tested on RSC data in different sea state, from the ice multiparameter imaging X-band (IPIX) radar. Experimental and simulation based on the IPIX radar data set indicates that the proposed method performs better than the $N$ -partitioned random multiplicative model (NRMM) method and SPS-multifractal spectrum (MFS) method, and exceeds the traditional methods by nearly 35% in terms of correlation coefficient and normalized mean square error of MFS and SPS.

On the pressure and stress singularities induced by steady flows of a pair of nonmiscible, incompressible, viscous fluids contacting a wall with slip

An earlier asymptotic analysis identified the flow-induced power and log singularities present when two nonmiscible, incompressible, viscous fluids contact a wall with no slip permitted. Here a parallel asymptotic analysis identifies such singularities in pressures and stresses when slip is admitted. While the singularity exponents for the two types of contact conditions are similar, the angles of interface impingement that have either power or log singularities are usually quite distinct.

Numerical and applied results of the analysis of singular solutions for a closed wedge consisting of two dissimilar materials

This paper considers solutions that can determine the nature of the stress singularity in the vicinity of a singular point for two-dimensional and three-dimensional closed composite wedges. A comparative analysis of stress singularity exponents is carried out for closed composite wedges with perfectly bonded interfaces and homogeneous wedges with stress-free edges. Using analysis results, it is possible to evaluate a change in the nature of the stress singularity near the crack tip when the crack cavity is filled with a certain material. Results were obtained for different ratios of the wedge component’s opening angles, different values of elastic moduli, and different Poisson ratios of the wedge’s constituent materials. The anomalous nature of changes in singular solutions for a composite closed wedge is established for Poisson’s ratios of the wedge part corresponding to a weakly compressible or incompressible material. In the context of the model and applied problems, it is shown that the use of eigensolutions for composite wedges makes it possible to find material characteristics that ensure maximum reduction in the stress concentration near the crack tip in the case of the crack cavity being filled with a certain material. The dependence of these solutions on the Poisson ratio demonstrates a significant reduction in stress concentration at sufficiently large crack opening angles and at low rigidity of the material.

Crack-Tip Strain Field in Supershear Crack of Elastomers.

We characterize the crack-tip strain field in the high-speed (supershear) crack in the elastomers propagating faster than the shear wave speed of sound (Cs). The dependence of steady-state crack velocity (V) on input tearing energy exhibits a crossover at V ≈ Cs between the subsonic (V < Cs) and supershear cracks (V > Cs). Several features of the crack-tip strain field such as strain-magnitude, extent boundary, and singularity exponent also change substantially accompanying the transition from subsonic to supershear cracks. The definite crossover of these characteristics at V ≈ Cs reflects the variations in the crack-growth mechanism: The inertia effect comes into play in the supershear crack. We also demonstrate that the azimuthal distribution of the local crack-tip strain has a close correlation with the macroscopic crack-tip shape, regardless of the regime of V.

Positive Solutions for Elliptic Problems with the Nonlinearity Containing Singularity and Hardy-Sobolev Exponents

In this paper, we study multiplicity of positive solutions for a class of semilinear elliptic equations with the nonlinearity containing singularity and Hardy-Sobolev exponents. Using variational methods, we establish the existence and multiplicity of positive solutions for the problem.

Determination of high‐order fields for multianisotropic material antiplane V‐notches and inclusions by the asymptotic expansion technique and an overdeterministic method

AbstractIn this paper, the singular behavior for anisotropic multimaterial V‐notched plates is investigated under antiplane shear loading condition. Firstly, the elastic governing equations are transformed into eigen ordinary differential equations through introducing the asymptotic expansions of displacements near the notch tip. The stress singularity exponents, including the higher‐order terms, and the corresponding eigen angular functions are then obtained by solving the established equations by using the interpolating matrix method. Thus, using the combination of the results from finite element analyses and the derived asymptotic expansion, an overdeterministic method is employed to calculate the amplitudes of the coefficients in the asymptotic expansions. Finally, the stress and displacement fields in the vicinity of the notch tip, consisting of both singular terms and higher‐order terms, are determined. The effects of material properties and geometry characteristic on the singular behaviour of the notch tip are discussed in detail.

Restriction formula and subadditivity property related to multiplier ideal sheaves

Abstract We give a restriction formula on jumping numbers which is a reformulation of Demailly–Ein–Lazarsfeld’s important restriction formula for multiplier ideal sheaves and a generalization of Demailly–Kollár’s important restriction formula on complex singularity exponents, and then we establish necessary conditions for the extremal case in the reformulated formula; we pose the subadditivity property on the complex singularity exponents of plurisubharmonic functions which is a generalization of Demailly–Kollár’s fundamental subadditivity property, and then we establish necessary conditions for the extremal case in the generalization. We also obtain two sharp relations on jumping numbers, introduce a new invariant of plurisubharmonic singularities and get its decreasing property for consecutive differences.

Numerical analysis of stresses near the vertices of a single and several radial spatial cracks

In problems of the theory of elasticity, there are singular solutions associated with the presence of infinite values of stresses at separate points (or lines), which are called singular. The great interest due to practical importance is those singular points that define the vertices of plane or spatial cracks. The algorithm for the numerical analysis of singular solutions in two-dimensional and three-dimensional problems of elasticity theory is considered. The numerical algorithm is based on extracting from the finite element solution the power law dependence of stresses near the singular points of the considered region, where singular solutions are possible. Based on the found power-law dependences, a conclusion about the absence or presence of a stress singularity and its nature is made. This algorithm is verified for two-dimensional and three-dimensional problems of the theory of elasticity by comparing the stress singularity exponents found by the proposed numerical method and obtained from known analytical and numerical solutions. The considered algorithm is applicable for variants in which the stress behavior in the vicinity of singular points is described by one power law dependence. One of the purposes of the article is to study the stress behavior near the common vertex of several radial spatial cracks in an isotropic material. In this case, the evaluation of stress behavior in the vicinity of the singular points is based on the construction of a correct stress field pattern by the finite element method when approaching the singular points. The results of numerical studies have shown that the highest stress level near the common vertex is observed for configurations with the equal angles between radial cracks.

Quasilinear nonlocal elliptic problems with variable singular exponent

In this article, we provide existence results to the following nonlocal equation \begin{document}$ \begin{align*} \begin{cases} (-\Delta)_p^{s} u = g(x,u),\;u>0\; \text{in}\; u = 0 \; \text{in}\; \mathbb{R}^N\setminus \Omega, \end{cases} \end{align*}\quad\quad(P_ \lambda)$ \end{document} where \begin{document}$ (-\Delta)_{p}^{s} $\end{document} is the fractional \begin{document}$ p $\end{document} -Laplacian operator. Here \begin{document}$ \Omega \subset \mathbb R^N $\end{document} is a smooth bounded domain, \begin{document}$ s\in(0,1) $\end{document} , \begin{document}$ p>1 $\end{document} and \begin{document}$ N>sp $\end{document} . We establish existence of at least one weak solution for \begin{document}$ (P_ \lambda) $\end{document} when \begin{document}$ g(x,u) = f(x)u^{-q(x)} $\end{document} and existence of at least two weak solutions when \begin{document}$ g(x,u) = \lambda u^{-q(x)}+ u^{r} $\end{document} for a suitable range of \begin{document}$ \lambda>0 $\end{document} . Here \begin{document}$ r\in(p-1,p_{s}^{*}-1) $\end{document} where \begin{document}$ p_s^{*} $\end{document} is the critical Sobolev exponent and \begin{document}$ 0 .

A semi-analytical stress correction method for bi-material Mode-III V-notches

This article proposes a method to correct the numerical near singularity stress field in vicinity of reentrant corners by a semi-analytical solution. The approach pertains to perfectly bonded bi-material interface V-notches with arbitrary opening angles under Mode-III loading. The method addresses dissimilar joints with sharp corners in large-scale numerical engineering models where high-resolution discretization becomes computationally expensive. The method is therefore particularly useful for fatigue lifetime analysis of shear-bending dominated large utility multi material structures such as compound castings, aircraft wings and wind turbine rotor blades to name a few. The solution for the governing Laplace equation is obtained by separation of variables in conjunction with the eigenvalue expansion method yielding the bi-material notch singularity exponent. The notch stress intensity factor is then used to fit the analytical singular near field to the numerical far field utilizing the fact that the latter is more accurately predicted. The analytical anti-plane shear stress correction method is valid for any geometry subject to arbitrary remote loading conditions. The accuracy of the approach is investigated by numerical stress analysis of a slender beam subject to axial torsion for different notch opening angles.

Positive solutions for a Schrödinger-Poisson system with singularity and critical exponent

We study multiplicity of positive solutions for a class of Schrödinger-Poisson system with singularity and critical exponent, and obtain two positive solutions via the variational and perturbation methods.

Checking complex networks indicators in search of singular episodes of the photochemical smog.

A set of indicators derived from the analysis of complex networks have been introduced to identify singularities on a time series. To that end, the Visibility Graphs (VG) from three different signals related to photochemical smog (O3, NO2 concentration and temperature) have been computed. From the resulting complex network, the centrality parameters have been obtained and compared among them. Besides, they have been contrasted to two others that arise from a multifractal point of view, that have been widely used for singularity detection in many fields: the Hölder and singularity exponents (specially the first one of them). The outcomes show that the complex network indicators give equivalent results to those already tested, even exhibiting some advantages such as the unambiguity and the more selective results. This suggest a favorable position as supplementary sources of information when detecting singularities in several environmental variables, such as pollutant concentration or temperature.

On some clarifications of estimates of Lyapunov’s exponents in the method of freezing

Method of V.M. Alekseev was used to obtain the clarifications of lower estimates for the lower singular exponent and of upper estimates for the upper singular exponent of a linear system of differential equations. In contrast to the classical results of the freezing method, which assume that the derivatives of the elements of the matrix of the coefficients of the system are small, the new estimates admit that these derivatives are unbounded.

Convergence behaviour of the enriched scaled boundary finite element method

SummaryIn this work, a very efficient numerical solution of three‐dimensional boundary value problems of linear elasticity including stress singularities is discussed, focussing on its convergence behaviour. For the employed scaled boundary finite element method, a discretization is only needed at the boundary, while the solution is considered analytically in a scaling coordinate. This presents a major advantage for two‐dimensional problems, when the scaling center is placed at a stress singularity. Unfortunately, three‐dimensional problems usually do not only include point singularities but also line singularities, which results in singular gradients in the boundary coordinates and thereby diminishes the method's original advantages. To alleviate this drawback, this work discusses an enrichment of the separation of variables representation with analytical asymptotic near fields of the line singularities. In contrast to previous works, besides the near‐field functions with λ=0.5, also those with λ=1.5 were determined and used for enrichment. This leads to a high accuracy and it is shown that this approach is required to recover the convergence properties of smooth boundary value problems without singularities when using quadratic Lagrange shape functions. In order to recover the convergence rates for higher order shape functions, near‐field functions with higher singularity exponent have to be included for enrichment.

Crossover multiparameter equation of state: General procedure and demonstration with carbon dioxide

The multiparameter equation of state represents experimental data in wide ranges of temperature and density with superb accuracy. However, at the critical point, all classical models fail in describing the asymptotic behavior of thermodynamic properties, which is governed by the renormalization group (RG) theory. Using the crossover method, the classical models far from the critical point can be transformed to the RG theory at the critical point. Here we validate the procedure of combining the crossover method with the multiparameter equation of state through comparison with experimental data and the original multiparameter equation of state, and testing against the qualitative criterion of characteristic curves. We select carbon dioxide as a demonstration due to its data availability and qualification as a benchmark fluid for thermodynamic property modeling. We describe in detail the contribution of different types of terms in the crossover method and original multiparameter equation of state to thermodynamic properties. Furthermore, we propose twin Gaussian terms to compensate for the loss of accuracy near the coexistence curve upon removing the non-analytical terms and leave the analytical part of the formulation (terms and coefficients) unaltered. With a slight and acceptable loss of accuracy in the crossover region, the crossover method enforces the asymptotic singular behavior and critical exponents at the critical point. Far from the critical point, the present model transforms to the original model and retains the superb accuracy of the latter.

Multiple positive solutions for Schrödinger‐Poisson system involving singularity and critical exponent

In this paper, the existence and multiplicity of positive solutions is established for Schrödinger‐Poisson system of the form urn:x-wiley:mma:media:mma5519:mma5519-math-0001 where 0 ∈ Ω is a smooth bounded domain in , , and λ &gt; 0 is a real parameter. Combining with the variational method and Nehari manifold method, two positive solutions of the system are obtained.

Time-singularity Power Spectral Distribution and Its Application in the Analysis of Radar Sea Clutter

For multifractal time series, singularity power spectrum (SPS) provides the power distribution of fractal subsets with respect to singularity exponent, however fails to trace the time-varying singularity characteristics. Aiming at this shortcoming, this paper has two main objectives. First, it develops SPS method in time-singularity 2D plane, and proposes a novel time-singularity power spectrum distribution (TSPSD) for both deterministic as well as non-stationary random series, thereby emphasizing an intuitive approach to trace the local SPS distribution, derived by the inner dynamics mechanism. Second, it deduces the implementation of TSPSD, based on the self-correlation and SPS algorithm, and particular attention is paid to application of TSPSD in the analysis of radar sea clutter which combines the attractive features of non-stationary and multifractal self-similar processes. Statistical properties as well as tracking capability of TSPSD for radar sea clutter are addressed.

Multifractal analysis of uterine electromyography signals to differentiate term and preterm conditions.

In this study, an attempt has been made to identify the origin of multifractality in uterine electromyography signals and to differentiate term (gestational age > 37 weeks) and preterm (gestational age ≤ 37 weeks) conditions by multifractal detrended moving average technique. The signals obtained from a publicly available database, recorded from the abdominal surface during the second trimester, are used in this study. The signals are preprocessed and converted to shuffle and surrogate series to examine the source of multifractality. Multifractal detrended moving average algorithm is applied on all the signals. The presence of multifractality is verified using scaling exponents, and multifractal spectral features are extracted from the spectrum. The variation of multifractal features in term and preterm conditions is analyzed statistically using Student's t-test. The results of scaling exponents show that the uterine electromyography or electrohysterography signals reveal multifractal characteristics in term and preterm conditions. Further investigation indicates the existence of long-range correlation as the primary source of multifractality. Among all extracted features, strength of multifractality, exponent index, and maximum and peak singularity exponents are statistically significant ( p < 0.05) in differentiating term and preterm conditions. The coefficient of variation is found to be lower for strength of multifractality and peak singularity exponent, which reveal that these features exhibit less inter-subject variance. Hence, it appears that multifractal analysis can aid in the diagnosis of preterm or term delivery of pregnant women.

Data-driven monitoring of the gearbox using multifractal analysis and machine learning methods

Data-driven diagnostic methods allow to obtain a statistical model of time series and to identify deviations of recorded data from the pattern of the monitored system. Statistical analysis of time series of mechanical vibrations creates a new quality in the monitoring of rotating machines. Most real vibration signals exhibit nonlinear properties well described by scaling exponents. Multifractal analysis, which relies mainly on assessing local singularity exponents, has become a popular tool for statistical analysis of empirical data. There are many methods to study time series in terms of their fractality. Comparing computational complexity, a wavelet leaders algorithm was chosen. Using Wavelet Leaders Multifractal Formalism, multifractal parameters were estimated, taking them as diagnostic features in the pattern recognition procedure, using machine learning methods. The classification was performed using neural network, k-nearest neighbours’ algorithm and support vector machine. The article presents the results of vibration acceleration tests in a demonstration transmission system that allows simulations of assembly errors and teeth wear.

Recognizing geochemical anomalies via stochastic simulation-based local singularity analysis

The complexity of geochemical variation patterns and the limited number of samples that can be collected jointly make uncertainty an element that could not be ignored in modeling the geochemical distributions and further identifying geochemical anomalies. Local singularity analysis (LSA) of stochastically simulated geochemical distributions is carried out in this paper to identify geochemical anomalies in southwestern Fujian province of China, with the uncertainty at unsampled locations and its propagation being considered. A supervised way by analyzing the sensitivity of area under curve (AUC) to threshold values to determine an appropriate threshold is also presented. The case study further illustrated and validated the procedure, and several insights can be obtained thereby: (1) the uncertainty of values at unsampled locations could significantly affect the grid-based LSA result, especially for those locations with weak singularities. An explanation from the LSA algorithm is that the estimation of singularity exponent involves multiple local neighborhoods of the location to study, uncertain values at any location within the neighborhood would affect the result. Locations with weak singularities are quite sensitive to such uncertainties in that enrichment or depletion patterns identified are likely to change from one to the other. (2) Sensitivity analysis of AUC to threshold values provides a feasible way to determine an appropriate threshold to derive the anomaly probability map, which can enhance the geochemical signatures with respect to the interpolation-based singularity map. The anomaly probability accounts for the local distribution structure of singularity exponents, hence providing a more discriminative measure to enhance geochemical signatures with respect to the single singularity exponent from interpolation-based LSA, which mainly reveals the trend of local singularities. (3) Several targets have been preliminarily delineated based on the anomaly probabilities and favorable geological conditions in the study area, which should be validated by other types of information available in near future.