Singular Components Research Articles

Spectral Hausdorff dimensions for a class of Schrödinger operators in bounded intervals

Exact Hausdorff dimensions are computed for singular continuous components of the spectral measures of a class of Schr\"odinger operators in bounded intervals.

A robust and secure image watermarking scheme using SURF and improved Artificial Bee Colony algorithm in DWT domain

To improve the security, robustness and imperceptibility of watermark scheme, a novel watermark scheme is devised by fusing multiple techniques, including DWT-SVD, Speed-Up Robust Feature (SURF), Random Sample Consensus (RANSAC) algorithm and improved Artificial Bee Colony algorithm named as IABC. To solve the false positive problem in various SVD-based watermark schemes, DWT is employed on the host image and then block wise singular components (maximum) are used for watermark’s principal component insertion. On the one hand, IABC algorithm is applied to optimize the scaling factors to balance invisibility and robustness and on the other hand to improve the convergence speed. Moreover, SURF and RANSAC algorithm are used to correct probable geometrical distortion to maintain watermark synchronization. Experimental outcomes demonstrated that compared with the related existing schemes, this scheme exhibits an overall superior performance in the aspects of security, imperceptibility and robustness.

New approaches to constructing quadrature formulas for functions with large gradients

The problem of numerical integration of functions with large gradients in the boundary layer is studied. The use of the Newton-Cotes formulas on a uniform grid to integrate such functions can lead to significant errors. The article investigates approaches to the construction of quadrature formulas, the error of which does not increase due to large gradients of the function in the boundary layer. The following approaches are considered: the construction of quadrature formulas that are exact on the singular component responsible for the growth of the function in the boundary layer, the application of the Newton-Cotes formulas on the Shishkin and Bakhvalov meshes, minimization the error of the quadrature formula due to the construction of nodes. The investigated approaches are new, partially reflected in the publications. The results of computational experiments are presented.

A generalization of Archimedean and Marshall-Olkin copulas family

In this paper, we consider a new family of multivariate copulas described by a sequence of functions, named as AMO copula. The set of AMO copulas corresponds to a class of multivariate shock models with the Archimedean type of dependence. Sufficient conditions on the involved sequence of functions to obtain a multivariate copula are given and the probabilistic structure of the AMO copula is provided. We show that the family of AMO copulas is a generalization of Archimedean and Marshall-Olkin copulas family, and it includes some well-known copulas as specific cases. An alternative method for generating random vectors from AMO copulas via distortion functions is provided. In addition, a singular component along the main diagonal of the AMO copula is also verified. Finally, the tail behaviors of AMO copulas are discussed and some numerical illustrations are provided.

The lemniscate tree of a random polynomial

À chaque polynôme complexe générique p(z) est associé un arbre binaire étiqueté (appelé dans cet aticle, “arbre de lemniscate”) qui code le type topologique du graphe de |p(z)|. La structure de ramification de l’arbre de lemniscate est déterminée par la configuration (c’est-à-dire la disposition dans le plan) des composants singuliers de ces ensembles de niveaux |p(z)|=t passant par un point critique.

To the theory of solvability of degenerate integro-differential equations in Banach spaces

The paper considers a new approach to constructing generalized solutions of degenerate integro-differential equations of convolution type in Banach spaces. The principal idea of the method proposed implies the refusal of the condition of existence of the full Jordan set for the Fredholm operator of the higher derivative with respect to the operator bundle formed by the rest of operator coefficients of the differential part and by the operator kernel of the integral component of the equation. The conditions are superimposed upon the values of the operator function specially constructed on the basis elements of the Fredholm operator kernel. Under such an approach, the differential part of the equation may include not only the higher derivative but also any combination of lower derivatives, what allows one to consider the convolution integral-differential equations from universal positions, without any special account of the structure of the operator bundle. The method proposed represents a form of generalization of the technique based on the application of Jordan sets of Fredholm operators, and, in the case of existence of the latter, the method coincides with this technique. A generalized solution is constructed in the form of a convolution of the fundamental operator function, which corresponds to the equation under investigation, and the function, which includes both the right-hand side of the equation and the initial data. The conditions, under which such a generalized solution does not contain any singular component, and the regular component converts the initial equation into an identity and satisfies the initial data will provide for the resolvability of the initial problem in the class of functions characterized by the respective smoothness. In this case, the generalized solution constructed will be classical. The theorem on the form of fundamental operator function has been proved. The abstract results have been illustrated via examples of initial-boundary value problems of applied character (from the theory electromagnetic fields, the theory of oscillations in visco-elastic media, the theory of vibrations of thermal-elastic plates).

Seismic data denoising under the morphological component analysis framework by dictionary learning

Traditional denoising methods based on fixed transforms are not suited for exploiting their complicated characteristics and attenuating noise due to their lack of adaptability. Recently, a novel method called morphological component analysis (MCA) was proposed to separate different geometrical components by amalgamating several irrelevance transforms. For studying the local singular and smooth linear components characteristics of seismic data, we propose a novel method that excels particularly in attenuating random and coherent noise while preserving effective signals. The proposed method, which combines MCA, dictionary learning (DL), and deep noise reduction consists of three steps: first, we separate the local singular and smooth linear components from the seismic signal using MCA. Second, we apply a DL method on these two components to suppress noise and obtain the denoised signal and noise. In the final step, we apply the DL method to the noise to obtain a little of the seismic signal. Afterwards, we integrate the two seismic signals to obtain the final denoised seismic signal. Numerical results indicate that the proposed method can effectively suppress the undesired noise, maximally preserve the information of geologic bodies and structures, and improve the signal-to-noise ratio (S/N) of the data. We also demonstrate the superior performance of this approach by comparing with other novel dictionaries such as discrete cosine transforms (DCTs), undecimated discrete wavelet transforms (UDWTs), or curvelet transforms. This algorithm provides new ideas for data processing to advance quality and S/N of seismic data.

Symplectic quaternion singular mode decomposition with application in gear fault diagnosis

Multivariate signals contain more abundant and accurate fault features than univariate signal, so it is beneficial to fault diagnosis with processing the multivariate signals simultaneously. Symplectic singular mode decomposition (SSMD) is an adaptive phase space reconstruction method based on symplectic geometry aiming at processing univariate signal. Quaternion singular spectrum analysis (QSSA) is a multivariate signal processing method in traditional Euclidean geometry, so basic features of original multivariate signals may be destroyed. Therefore, symplectic quaternion singular mode decomposition (SQSMD) is proposed to decompose multivariate signals to a series of independent meaningful components, meanwhile the method keeps essential features of raw multivariate time series unchanged. SQSMD applies symplectic similarity transformation to the constructed quaternion Hamilton matrix by selecting embedding dimension automatically without user-defined parameter, then the transformed trajectory matrix is decomposed by quaternion singular mode decomposition to obtain quaternion eigenvectors and singular values, and finally symplectic quaternion singular spectrum components (SQSSCs) are obtained by taking fault information from multivariate signals as a whole to enhance fault characteristics. Simulated and experimental multivariate signals results indicate the effectiveness and superiority of the proposed method.

Application a cubic spline to calculate derivatives in the presence of a boundary layer

The problem of calculating the derivatives of a function with large gradients given at the grid nodes is considered. The decomposition of the function into regular and singular components is assumed. Such decomposition takes place in the presence of an exponential boundary layer. The case of the Bakhvalov mesh is considered. The error in calculating the derivatives based on the differentiation of the interpolating cubic spline is estimated. Error estimates are obtained taking into account uniformity in a small parameter.

Investigation of the Fracture of a Semibounded Body Compressed Along a Near-Surface Interface Crack

We consider the problem of compression of a piecewise homogeneous half plane by forces directed along a near-surface crack located on the interface of two materials. This problem belongs to the class of nonclassical problems of fracture mechanics because, for the analyzed loading scheme, the stressstrain state realized in the body is homogeneous and the corresponding expressions for stresses and displacements in the vicinity of the crack do not contain singular components. Due to the fact that the stress intensity factors are equal to zero, for the analyzed problem, the classical Griffith–Irwin fracture criteria are inapplicable. In the indicated situation, the onset of crack propagation is associated with the local loss of stability of the equilibrium state of a part of the material located in the region adjacent to the crack. By using the approaches of the three-dimensional linearized theory of stability of deformable bodies, we propose a mathematical statement of the problem and an approach to its investigation. By using the representations of stresses and displacements in terms of complex potentials, we analyze the case where the roots of the corresponding characteristic equation are identical for each material.

Hyperspectral Image Enhancement by Two Dimensional Quaternion Valued Singular Spectrum Analysis for Object Recognition

This paper proposes a two dimensional quaternion valued singular spectrum analysis based method for enhancing the hyperspectral image. Here, the enhancement is for performing the object recognition, but neither for improving the visual quality nor suppressing the artifacts. In particular, the two dimensional quaternion valued singular spectrum analysis components are selected in such a way that the ratio of the interclass separation to the intraclass separation of the pixel vectors is maximized. Next, the support vector machine is employed for performing the object recognition. Compared to the conventional two dimensional real valued singular spectrum analysis based method where only the pixels in a color plane is exploited, the two dimensional quaternion valued singular spectrum analysis based method fuses four color planes together for performing the enhancement. Hence, both the spatial information among the pixels in the same color plane and the spectral information among various color planes are exploited. The computer numerical simulation results show that the overall classification accuracy based on our proposed method is higher than the two dimensional real valued singular spectrum analysis based method, the three dimensional singular spectrum analysis based method, the multivariate two dimensional singular spectrum analysis based method, the median filtering based method, the principal component analysis based method, the Tucker decomposition based method and the hybrid spectral convolutional neural network (hybrid SN) based method.

Determination of non-axisymmetric stresses in the bodies of revolution based on regulized intergral equations

A solution to the 3D elasticity theory problem in the form of a Fourier series expansion by an angular coordinate, the coefficients of which are determined from the system of one-dimensional integral equations, is constructed. The relation for determining the kernels of these equations for harmonics of arbitrary order is written in analytical form. The regularization of the obtained equations is carried out. For this purpose, the kernels of the equations are represented as the sum of regular functions and singular components, which are determined through the kernels of the plane problem of the elasticity theory in the Cartesian coordinate system. In the stress tensor, the components that contain logarithmic functions as multiplier, are additionally highlighted. The test calculations are performed, which confirm the reliability of the analytically obtained relations and integral equations built on their basis. Testing of the developed numerical algorithm for solving integral equations is carried out and examples of calculation of non-axisymmetric stress concentration near cavities in bodies of various shapes are resulted. The obtained results allow to expand the application of boundary elements method to study of practically important non-axisymmetric problems for the bodies of rotation.

An optimized robust watermarking technique using CKGSA in frequency domain

The existing SVD based watermarking techniques insert watermark into the singular component which causes the false positive problem (FPP). To overcome this problem, a robust watermarking technique using a chaotic kbest gravitational search algorithm (CKGSA) in DCT and SVD domain is proposed. To eliminate the FPP, the watermark is embedded into the principal component (PC) of the cover image rather than the singular value. The objective to use CKGSA is to optimize the value of multiple embedding factors so that robustness and imperceptibility can be balanced. Arnold transform (AT) is applied to the watermark image before embedding, to enhance security and robustness. The performance of the proposed technique has been experimented on a standard dataset and compared with the six recent state-of-the-art techniques including DS-DE, DS-PSO, DCT-SVD, DDS-PSO, DS-GSA, and DS-TLBO in terms of imperceptibility (PSNR, SSIM) and robustness (PSNR, NC). The experimental results demonstrate that the proposed technique outperforms the other existing techniques.

Electrophysiological Mapping During Brain Tumor Surgery: Recording Cortical Potentials Evoked Locally, Subcortically and Remotely by Electrical Stimulation to Assess the Brain Connectivity On-line

Direct electrical stimulation (DES) is used to perform functional brain mapping during awake surgery and in epileptic patients. DES may be coupled with the measurement of Evoked Potentials (EP) to study the conductive and integrative properties of activated neural ensembles and probe the spatiotemporal dynamics of short- and long-range networks. However, its electrophysiological effects remain by far unknown. We recorded ECoG signals on two patients undergoing awake brain surgery and measured EP on functional sites after cortical stimulations and were the firsts to record three different types of EP on the same patients. Using low-intensity (1-3mA) to evoke electrogenesis we observed that: (i) "true" remote EPs are attenuated in amplitude and delayed in time due to the divergence of white matter pathways; (ii) "false" remote EPs are attenuated but not delayed: as they originate from the same electrical source; (iii) Singular but reproducible positive components in the EP can be generated when the DES is applied in the temporal lobe or the premotor cortex; and (iv) rare EP can be triggered when the DES is applied subcortically: these can be either negative, or surprisingly, positive. We proposed different activation and electrophysiological propagation mechanisms following DES, based on the nature of activated neural elements and discussed important methodological pitfalls when measuring EP in the brain. Altogether, these results pave the way to map the connectivity in real-time between the DES and the recording sites; to characterize the local electrophysiological states and to link electrophysiology and function. In the future, and in practice, this technique could be used to perform electrophysiological mapping in order to link (non)-functional to electrophysiological responses with DES and could be used to guide the surgical act itself.

Frequency-Aware SVD Decomposition and its Application to Color Magnification and Motion Denoising

Videos are full of dynamic changes along both the spatial and temporal dimensions. Large, jerky short-term motions make it difficult to extract significant changes from videos such as subtle color changes and long-term motions occurring in time-lapse sequences. In this paper, we introduce two singular value decomposition (SVD)-based video decomposition schemes to clearly reveal such changes. The first scheme involves enhancing the visual characteristics of small subtle color changes in the presence of a wide variety of motion patterns by magnifying their pixel intensities. The second scheme removes short-term motions that visually distract attention from the underlying content of video sequences such as time-lapse videos, snowing scene, and maritime surveillance. Both schemes involve the decomposition of videos into spatiotemporal slices in which each slice is further decomposed into several singular components. The low-rank components that primarily represent background and color intensity information are then temporally processed to magnify the magnitude of the signal at the subtle color change target frequency. At the same time, an approach similar to that used in denoising time-lapse sequences is applied to temporally filter the singular components representing sparse information, thereby removing jittery short-term motions while preserving long-term motions, which are represented by both low-rank and unfiltered sparse components. We demonstrate promising color magnification and motion denoising results that can be obtained much faster than results estimated using state-of-the-art techniques.

Seismic data reconstruction method based on morphological component analysis

The real seismic data is usually under-sampled in space domain because of the physical or economic limitations. So the incomplete seismic data need to be reconstructed before the subsequent processing. A method based on morphological component analysis (MCA) is discussed in the paper, which uses the curvelet dictionary and local discrete cosine transform (LDCT) dictionary to reconstruct the smooth components and the singular components respectively. Block coordinate relaxation (BCR) algorithm is adopted to complete the sparse optimisation. The validity of the proposed method was tested by numerical experiments on synthetic and real data demonstrate. As the method based on curvelet combining with POCS is widely used in practice, we compare its reconstructed results with the MCA-based method. The numerical results validate that the proposed method has higher reconstruction performance.

A New Singular Spectrum Decomposition Method Based on Cao Algorithm and Amplitude Aware Permutation Entropy

Singular spectrum decomposition (SSD) is relatively affected by noise. The singular spectrum component (SSC) obtained in a strong noise environment not only contains a lot of noise, but also has over-decomposition phenomenon. One reason is that the selection of the SSD embedding dimension is subjective and cannot adapt well to signals with higher complexity. Another reason is that the iterative termination condition of the SSD is based on the normalized mean square error of the residual signal and the original signal. This will cause the SSD to produce a lot of pseudo components when SSD processes strong noisy signal. In order to enhance the decomposition performance of the SSD in strong noise environment, this paper starts from the selection of embedding dimension and the improvement of iteration termination conditions, and proposes an improved singular spectrum decomposition (ISSD) based on Cao algorithm and amplitude aware permutation entropy (AAPE). Firstly, the DC of the signal to be decomposed is taken out. Secondly, Cao algorithm is used to calculate the optimal embedding dimension of the signal to be decomposed, and then SSD is performed to obtain a SSC and a residual signal. Finally, the normalized AAPE value (NAAPE) of the SSC is calculated. If it is greater than the threshold, the residual signal at this time is considered to be a noise component, the decomposition is stopped, and the residual signal is output as the last SSC. Otherwise, the residual signal is used as the signal to be decomposed, and it continues to decompose. After the decomposition, the validity of the DC (Direct component) is determined by energy comparison. If the DC is a valid component, it will be output as the first SSC. Otherwise, the DC is discarded. The completeness of ISSD through theoretical analysis and equation derivation is proved. Its effectiveness is verified by analog signal, Lorenz signal and sunspot activity data. The comparative experiment of EMD, EEMD, VMD, SSD and ISSD is used to prove the advanced nature of ISSD. The simulation results show that ISSD has the ability to identify noise and can effectively avoid over-decomposition. At the same time, ISSD can accurately extract the DC contained in the original signal to avoid energy loss. In addition, ISSD can use Cao algorithm to select the appropriate embedding dimension to improve the quality of SSC. Finally, the rank sum test is used to prove the advanced nature of ISSD from a statistical point of view.

Ecosystem services and uses of dune systems of the coast of the Araucanía Region, Chile: A perception study

Coastal dunes can offer a series of Ecosystem Services (ES), especially to local communities. This study aimed to evaluate these benefits by exploring the perception of the Mapuche communities and representatives of the local government concerning the ES provided by the coastal dune fields of the Araucanía Region of Chile. Binary, multiple-choice, Likert scale and open questions were used to explore the general knowledge of 49 subjects about the ES provided by the dunes. Closed questions were analysed using Fisher's exact test, and tests of goodness of fit chi-square and G, while open questions were analysed by qualitative content analysis. The results showed that (1) cultural services were more important for Mapuche communities, while tourism was more valued for local government, (2) tourism and stock-raising were the most frequent use given to the dunes in the opinion of local government and Mapuche communities respondents, respectively, (3) both groups think that dunes offer habitat for biodiversity, and (4) the regulation of natural processes is one of the most important functions for both groups, although Mapuche communities consider that dune fields would not attenuate the effects of a tsunami. The conclusion is that dune fields offer a series of benefits, which are perceived in different ways by Mapuche communities and representatives of the local government. The lack of recognition of this knowledge is one of the main gaps in Chile's territorial planning instruments since it could make a decisive contribution to the management of social-ecological systems by zoning areas and identifying singular components, especially in Mapuche territories.

Fundamental equation of state developed on basis of Migdal’s phenomenological theory of critical point

In this work a fundamental equation of state (FEoS) with account taken of requirements of the scale theory of the critical point has been developed. The behavior of the FEOS in the asymptotic vicinity of the critical point is determined by the characteristics of the scaling function a(x) of the Helmholtz scale free energy F. The structure of the function a(x) is calculated on the basis of Migdal’s phenomenological theory and the hypothesis about the identical behavior of the isothermal compressibility coefficient at the critical and near-critical isohors near the critical point (Benedek’s hypothesis). The parameters of the singular component of the FEOS are calculated using the similarity theory. The proposed FEoS has been approved on the example of description of equilibrium properties of the refrigerating agent R1234yf in the range of state parameters: by a temperature from 220 K to 440 K and by pressure up to 40 MPa. The obtained results are discussed.

Proposal for Refugee Housing in Southeast Europe Utilizing Regionally Appropriate, Sustainable, and Affordable Construction Methods

This research focuses on the emergency-housing demand caused by the on-going refugee crisis in Southeastern Europe. The research and proposed solutions focus on Northern Serbia, as this region generally lacks permanent housing solutions for accommodating the increasing influx of immigrants. The outcome of this research is an architectural proposal for the refugee housing unit designed particularly for the situational factors of this focus region. 
 The study identifies earth architecture as the primary building technique, due to its ability to satisfy a range of defined end-product goals. These goals include: the sustainability of the material, quality and durability of the final product, skill level required for non-professional construction, final cost of material and execution, historical regional precedent, and opportunity for communal engagement of the immigrant population. The proposed architectural design uses earth-bag construction as the sub-method most suitable for this location and in keeping with the goals outlined above.
 The proposed housing unit is a singular component that could be duplicated to create larger communal housing communities. A broad overview of possible solutions is included, followed by the development of the earth-bag construction option. The development of this proposal includes material studies, sketches, and an architectural model as representation tools. The outcomes of this research serve as a guideline, rather than a precise construction model, in creating much needed refugee housing communities in North Serbia.