Wavelet Multi-Resolution Interpolation Galerkin Method for Linear Singularly Perturbed Boundary Value Problems

In this study,a wavelet multi-resolution interpolation Galerkin method(WMIGM)is proposed to solve linear singularly perturbed boundary value problems.Unlike conventional wavelet schemes,the proposed algorithm can be readily extended to special node generation techniques,such as the Shishkin node.Such a wavelet method allows a high degree of local refinement of the nodal distribution to efficiently capture localized steep gradients.All the shape functions possess the Kronecker delta property,making the imposition of boundary conditions as easy as that in the finite element method.Four numerical examples are studied to demonstrate the validity and accuracy of the proposedwavelet method.The results showthat the use ofmodified Shishkin nodes can significantly reduce numerical oscillation near the boundary layer.Compared with many other methods,the proposed method possesses satisfactory accuracy and efficiency.The theoretical and numerical results demonstrate that the order of theε-uniform convergence of this wavelet method can reach 5.

A Color Image Encryption Scheme Based on Singular Values and Chaos

The security of digital images transmitted via the Internet or other public media is of the utmost importance.Image encryption is a method of keeping an image secure while it travels across a non-secure communication medium where it could be intercepted by unauthorized entities.This study provides an approach to color image encryption that could find practical use in various contexts.The proposed method,which combines four chaotic systems,employs singular value decomposition and a chaotic sequence,making it both secure and compression-friendly.The unified average change intensity,the number of pixels’change rate,information entropy analysis,correlation coefficient analysis,compression friendliness,and security against brute force,statistical analysis and differential attacks are all used to evaluate the algorithm’s performance.Following a thorough investigation of the experimental data,it is concluded that the proposed image encryption approach is secure against a wide range of attacks and provides superior compression friendliness when compared to chaos-based alternatives.

Robust Damage Detection and Localization Under Complex Environmental Conditions Using Singular Value Decomposition-based Feature Extraction and One-dimensional Convolutional Neural Network

Ultrasonic guided wave is an attractive monitoring technique for large-scale structures but is vulnerable to changes in environmental and operational conditions(EOC),which are inevitable in the normal inspection of civil and mechanical structures.This paper thus presents a robust guided wave-based method for damage detection and localization under complex environmental conditions by singular value decomposition-based feature extraction and one-dimensional convolutional neural network(1D-CNN).After singular value decomposition-based feature extraction processing,a temporal robust damage index(TRDI)is extracted,and the effect of EOCs is well removed.Hence,even for the signals with a very large temperature-varying range and low signal-to-noise ratios(SNRs),the final damage detection and localization accuracy retain perfect 100%.Verifications are conducted on two different experimental datasets.The first dataset consists of guided wave signals collected from a thin aluminum plate with artificial noises,and the second is a publicly available experimental dataset of guided wave signals acquired on a composite plate with a temperature ranging from 20℃to 60℃.It is demonstrated that the proposed method can detect and localize the damage accurately and rapidly,showing great potential for application in complex and unknown EOC.

A SINGULAR DIRICHLET PROBLEM FOR THE MONGE-AMPÈRE TYPE EQUATION

We consider the singular Dirichlet problem for the Monge-Ampère type equation■=0,whereΩis a strictly convex and bounded smooth domain in■is positive and strictly decreasing in(0,∞)with■is positive inΩ.We obtain the existence,nonexistence and global asymptotic behavior of the convex solution to such a problem for more general b and g.Our approach is based on the Karamata regular variation theory and the construction of suitable sub-and super-solutions.

Contourlet watermarking algorithm based on Arnold scrambling and singular value decomposition

A new digital watermarking algorithm based on the contourlet transform is proposed to improve the robustness and anti-attack performances of digital watermarking. The algorithm uses the Arnold scrambling technique and the singular value decomposition （SVD） scheme. The Arnold scrambling technique is used to preprocess the watermark, and the SVD scheme is used to find the best suitable hiding points. After the contourlet transform of the carrier image, intermediate frequency sub-bands are decomposed to obtain the singularity values. Then the watermark bits scrambled in the Arnold rules are dispersedly embedded into the selected SVD points. Finally, the inverse contourlet transform is applied to obtain the carrier image with the watermark. In the extraction part, the watermark can be extracted by the semi-blind watermark extracting algorithm. Simulation results show that the proposed algorithm has better hiding and robustness performances than the traditional contourlet watermarking algorithm and the contourlet watermarking algorithm with SVD. Meanwhile, it has good robustness performances when the embedded watermark is attacked by Gaussian noise, salt- and-pepper noise, multiplicative noise, image scaling and image cutting attacks, etc. while security is ensured.

The Singular Value Decomposition Analysis between Summer Precipitation in the Dongting Lake Region and the Global Sea Surface Temperature

By dint of the summer precipitation data from 21 stations in the Dongting Lake region during 1960-2008 and the sea surface temperature(SST) data from NOAA,the spatial and temporal distributions of summer precipitation and their correlations with SST are analyzed.The coupling relationship between the anomalous distribution in summer precipitation and the variation of SST has between studied with the Singular Value Decomposition(SVD) analysis.The increase or decrease of summer precipitation in the Dongting Lake region is closely associated with the SST anomalies in three key regions.The variation of SST in the three key regions has been proved to be a significant previous signal to anomaly of summer rainfall in Dongting region.

IMPROVED SINGULAR VALUE DECOMPOSITION TECHNIQUE FOR DETECTING AND EXTRACTING PERIODIC IMPULSE COMPONENT IN A VIBRATION SIGNAL

Vibration acceleration signals are often measured from case surface of arunning machine to monitor its condition. If the measured vibration signals display to have periodicimpulse components with a certain frequency, there may exist a corresponding local fault in themachine, and if further extracting the periodic impulse components from the vibration signals, theseverity of the local fault can be estimated and tracked. However, the signal-to-noise ratios (SNRs)of the vibration acceleration signals are often so small that the periodic impulse components aresubmersed in much background noises and other components, and it is difficult or inconvenient for usto detect and extract the periodic impulse components with the current common analyzing methods forvibration signals. Therefore, another technique, called singular value decomposition (SVD), istried to be introduced to solve the problem. First, the principle of detecting and extracting thesignal periodic components using singular value decomposition is summarized and discussed. Second,the infeasibility of the direct use of the existing SVD based detecting and extracting approach ispointed out. Third, the approach to construct the matrix for SVD from the signal series is improvedlargely, which is the key program to improve the SVD technique; Other associated improvement is alsoproposed. Finally, a simulating application example and a real-life application example ondetecting and extracting the periodic impulse components are given, which showed that the introducedand improved SVD technique is feasible.

MULTIPLE POSITIVE SOLUTIONS OF SINGULAR THIRD-ORDER PERIODIC BOUNDARY VALUE PROBLEM

This paper deals with the singular nonlinear third-order periodic boundary value problem u'' + p(3)u = f (t, u), 0 less than or equal to t less than or equal to 2pi, with u((i)) (0) = u((i)) (2pi), i = 0, 1, 2, where p is an element of (Graphics) and f is singular at t = 0, t = 1 and u = 0. Under suitable weaker conditions than those of [1], it is proved by constructing a special cone in C[0, 2pi] and employing the fixed point index theory that the problem has at least one or at least two positive solutions.

MULTIPLE POSITIVE SOLUTIONS TO SINGULAR BOUNDARY VALUE PROBLEMS FOR SUPERLINEAR SECOND ORDER ODES

This paper deals with the existence of positive solutions to the singular boundary value problemwhere q(t) may be singular at t = 0 and t = 1, f(t,y) may be superlinear at y =∞ and singular, at y = 0.

Estimation of fracture density and orientation from azimuthal elastic impedance difference through singular value decomposition

Accurate estimation of fracture density and orientation is of great significance for seismic characterization of fractured reservoirs.Here,we propose a novel methodology to estimate fracture density and orientation from azimuthal elastic impedance(AEI)difference using singular value decomposition(SVD).Based on Hudson's model,we first derive the AEI equation containing fracture density in HTI media,and then obtain basis functions and singular values from the normalized AEI difference utilizing SVD.Analysis shows that the basis function changing with azimuth is related to fracture orientation,fracture density is the linearly weighted sum of singular values,and the first singular value contributes the most to fracture density.Thus,we develop an SVD-based fracture density and orientation inversion approach constrained by smooth prior elastic parameters.Synthetic example shows that fracture density and orientation can be stably estimated,and the correlation coefficient between the true value and the estimated fracture density is above 0.85 even when an S/N ratio of 2.Field data example shows that the estimated fracture orientation is consistent with the interpretation of image log data,and the estimated fracture density reliably indicates fractured gas-bearing reservoir,which could help to guide the exploration and development of fractured reservoirs.

A NOTE ON SINGULAR VALUE DECOMPOSITION FOR RADON TRANSFORM IN R^n

The singular value decomposition is derived when the Radon transform is restricted to functions which are square integrable on the unit ball in R-n with respect to the weight W-lambda(x). It fulfilles mainly by means of the projection-slice theorem. The range of the Radon transform is spanned by products of Gegenbauer polynomials and spherical harmonics. The inverse transform of the those basis functions are given. This immediately leads to an inversion formula by series expansion and range characterizations.

Super-resolution reconstruction of synthetic-aperture radar image using adaptive-threshold singular value decomposition technique

A super-resolution reconstruction approach of （SVD） technique was presented, and its performance was radar image using an adaptive-threshold singular value decomposition analyzed, compared and assessed detailedly. First, radar imaging model and super-resolution reconstruction mechanism were outlined. Then, the adaptive-threshold SVD super-resolution algorithm, and its two key aspects, namely the determination method of point spread function （PSF） matrix T and the selection scheme of singular value threshold, were presented. Finally, the super-resolution algorithm was demonstrated successfully using the measured synthetic-aperture radar （SAR） images, and a Monte Carlo assessment was carried out to evaluate the performance of the algorithm by using the input/output signal-to-noise ratio （SNR）. Five versions of SVD algorithms, namely 1 ） using all singular values, 2） using the top 80% singular values, 3） using the top 50% singular values, 4） using the top 20% singular values and 5） using singular values s such that S2≥/max（s2）/rinsNR were tested. The experimental results indicate that when the singular value threshold is set as Smax/（rinSNR）1/2, the super-resolution algorithm provides a good compromise between too much noise and too much bias and has good reconstruction results.

TWO POSITIVE SOLUTIONS TO THREE-POINT SINGULAR BOUNDARY VALUE PROBLEMS

In this article, we consider the existence of two positive solutions to nonlinear second order three-point singular boundary value problem： -u′′（t） = λf（t, u（t）） for all t ∈ （0, 1） subjecting to u（0） = 0 and αu（η） = u（1）, where η ∈ （0, 1）, α ∈ [0, 1）, and λ is a positive parameter. The nonlinear term f（t, u） is nonnegative, and may be singular at t = 0, t = 1, and u = 0. By the fixed point index theory and approximation method, we establish that there exists λ＊ ∈ （0, ＋∞], such that the above problem has at least two positive solutions for any λ ∈ （0, λ＊） under certain conditions on the nonlinear term f.

Existence of Positive Solution for Superlinear Semipositone Singular Second-order m-point Boundary Value Problem

By applying fixed point theorem, the existence of positive solution is considered for superlinear semipositone singular m-point boundary value problem -（Lφ）（x）=（p（x）φ′（x））′＋q（x）φ（x） and ξi ∈ （0,1）with 0〈ξ1〈ξ2……〈ξm-2〈1,αi ∈ R^＋,f ∈C[（0,1）×R^＋,R^＋],f（x,φ） may be singular at x=0 and x=1,g（x）：（0,1）→R is Lebesgue measurable, g may tend to negative infinity and have finitely many singularities.

Analysis of heart rate variability based on singular value decomposition entropy

Assessing the dynamics of heart rate fluctuations can provide valuable information about heart status. In this study, regularity of heart rate variability （HRV） of heart failure patients and healthy persons using the concept of singular value decomposition entropy （SvdEn） is analyzed. SvdEn is calculated from the time series using normalized singular values. The advantage of this method is its simplicity and fast computation. It enables analysis of very short and non-stationary data sets. The results show that SvdEn of patients with congestive heart failure （CHF） shows a low value （SvdEn： 0.056±0.006, p 〈 0.01） which can be completely separated from healthy subjects. In addition, differences of SvdEn values between day and night are found for the healthy groups. SvdEn decreases with age. The lower the SvdEn values, the higher the risk of heart disease. Moreover, SvdEn is associated with the energy of heart rhythm. The results show that using SvdEn for discriminating HRV in different physiological states for clinical applications is feasible and simple.

POSITIVE SOLUTIONS OF BOUNDARY VALUE PROBLEMS FOR SECOND-ORDER SINGULAR NONLINEAR DIFFERENTIAL EQUATIONS

New existence results are presented for the singular second-order nonlinear boundary value problems u ' + g(t)f(u) = 0, 0 < t < 1, au(0) - betau ' (0) = 0, gammau(1) + deltau ' (1) = 0 under the conditions 0 less than or equal to f(0)(+) < M-1, m(1) < f(infinity)(-)less than or equal to infinity or 0 less than or equal to f(infinity)(+)< M-1, m(1) < f (-)(0)less than or equal to infinity where f(0)(+) = lim(u -->0)f(u)/u, f(infinity)(-)= lim(u --> infinity)f(u)/u, f(0)(-)= lim(u -->0)f(u)/u, f(infinity)(+) = lim(u --> infinity)f(u)/u, g may be singular at t = 0 and/or t = 1. The proof uses a fixed point theorem in cone theory.

Audio Zero-Watermark Scheme Based on Discrete Cosine Transform-Discrete Wavelet TransformSingular Value Decomposition

Traditional watermark embedding schemes inevitably modify the data in a host audio signal and lead to the degradation of the host signal.In this paper,a novel audio zero-watermarking algorithm based on discrete wavelet transform(DWT),discrete cosine transform(DCT),and singular value decomposition(SVD) is presented.The watermark is registered by performing SVD on the coefficients generated through DWT and DCT to avoid data modification and host signal degradation.Simulation results show that the proposed zero-watermarking algorithm is strongly robust to common signal processing methods such as requantization,MP3 compression,resampling,addition of white Gaussian noise,and low-pass filtering.

An Intercomparison of Rules for Testing the Significance of Coupled Modes of Singular Value Decomposition Analysis

This paper clarifies the essence of the significance test of singular value decomposition analysis （SVD）, and investigates four rules for testing the significance of coupled modes of SVD, including parallel analysis, nonparametric bootstrap, random-phase test, and a new rule named modified parallel analysis. A numerical experiment is conducted to quantitatively compare the performance of the four rules in judging whether a coupled mode of SVD is significant as parameters such as the sample size, the number of grid points, and the signal-to-noise ratio vary. The results show that the four rules perform better with lower ratio of the number of grid points to sample size. Modified parallel analysis and nonparametric bootstrap perform best to abandon the spurious coupled modes, but the latter is better than the former to retain the significant coupled modes when the sample size is not much larger than the number of grid points. Parallel analysis and random-phase test are robust to abandon the spurious coupled modes only when either （1） the observations at the grid points are spatially uncorrelated, or （2） the coupled signal is very strong for parallel analysis and is not weak for random-phase test. The reasons affecting the accuracy of the test rules are discussed.

Source-Generated Noise Suppression Using the Singular Value Decomposition

Source-generated noise, such as air, refracted, guided waves, near-surface multiples, and radial ground roll, is one of the most challenging problems in the land seismic method. The interference of the noise with reflection events often results in a distorted representation of the subsurface and gives rise to interpretation uncertainties. To suppress the noise, geophysicists have devised various techniques in both acquisition and processing stages. Conventional processing methods, such as high-pass, f - k and hyperbolic velocity filters, however, have certain disadvantages when handling actual seismic data. In this study, we present a new hybrid method combining singular value decomposition (SVD) with a special linear transformation of the common-shot gather. The method is aimed at effectively removing the noise while minimizing harm to the signal. As compared with other methods, the SVD-based one gives a denser approximation to source-generated noise before its subtraction from the seismic data, due to the use of more appropriate basis functions. The special transformation applied in advance to the data is intended to align the source-generated noise events horizontally and thus to benefit the subsequent SVD. The effectiveness of the method in suppressing source-generated noise is demonstrated with a synthetic data set. Emphasis is put on the comparison of the performance of the method with that of conventional f - k filtering.

ON THE SINGULAR DIRECTIONS OF VALUE DISTRIBUTION OF HOLOMORPHIC CURVES IN P^n(C)

This article proves the existence of singular directions of value distribution theory for some transcendental holomorphic curves in the n-dimensional complex projective space P^n（C）.. An example is given to complement these results.