Schur Decomposition Research Articles

Robust Color Image Watermarking Using Dual Embedding via Schur Decomposition

This research introduces a Schur-based watermarking technique that employs a 4´4 matrix block and a two-phase embedding process to enhance robustness performance. The watermark bits are scrambled with a key to obtain scrambled watermark. The two watermark bits are embedded into the c-th vector of Schur decomposition. In addition to achieving a remarkable trade-off between imperceptibility and robustness, this study investigates the relationship between c-th vector of Schur decomposition to achieve balancing between invisibility and robust against several attacks. Evaluation results demonstrate outstanding performance in preserving the integrity and resilience of the watermark across various colour images. The method achieves high PSNR values, indicating excellent watermark extraction quality with minimal distortion to the images. Furthermore, comparisons with earlier approaches highlight its superior resistance to attacks on test images. The experimental results show the proposed scheme has an ability to maintain image quality and withstand attacks, the proposed method shows significant potential for copyright protection with average SSIM value of 0.9939 and an average NC of 0.9118 under various attacks.

Data-driven Analysis on the Spatio-temporal Characteristics and Influencing Factors of Carbon Emissions in Guangdong Province

Scientific analysis of carbon emission characteristics is a key link for precise policy implementation and orderly achievement of Carbon Peak and Carbon Neutrality. Taking 21 prefecture level cities in Guangdong Province as research objects, data including their economic activity, energy consumption, and population distribution from 2013 to 2022 was collected. First, the carbon emission accounting method was used to analyze the time changes and spatial distribution characteristics of carbon emissions in the past decade. Second, the principal component analysis and K-means method was used to reduce the dimensionality and cluster division for Guangdong urban characteristics. Third, the Generalized Schur Decomposition model was applied to analyze the influencing factors. Furthermore, carbon reduction strategies for four typical cities in Guangdong Province were proposed. The results showed that： ① From 2013 to 2022, the total carbon emissions in Guangdong Province continued to increase and showed a significant regional imbalance in its spatial distribution. The carbon emissions contribution in the Pearl River Delta region was as high as 42.50%, whereas that in northern Guangdong was the lowest. ② Each city in Guangdong Province performed differently in green development, economic development, and industrial development. All 21 cities could be broadly divided into four categories, namely, Demonstration City, Resource-dependent City, Traditional Industrial City, and Emission-reducing City, accounting for approximately 14.29%, 28.57%, 19.05%, and 38.09%, respectively. ③ Economic activities were the common driving factors for carbon emissions in all cities, whereas energy consumption and lifestyle were the core driving factors for carbon emissions in resource-dependent cities and traditional industrial cities. After the proposal of Carbon Peak and Carbon Neutrality, energy composition became a key driving force for demonstration cities and emission-reducing cities.

The role of normal and non-normal contributions to enstrophy production in the near-wall region of a turbulent channel flow

The turbulent boundary layer is a region where both preferential dissipation of energy and the production of significant vorticity arises as a consequence of the strong velocity gradients. Previous work has shown that, following a Reynolds decomposition, the purely fluctuating component of the enstrophy production is the dominant term. Near the wall this varies in a complex manner with height. In this study, we additionally decompose the strain rate and vorticity terms into normal and non-normal components using a Schur decomposition and are able to explain all these features in terms of contributions at different heights from constituents involving different combinations of normal and non-normal quantities. What is surprising about our results is that, while the mean shear and the action of larger-scale structures should mean that non-normal effects are of over-riding importance at the wall, the most important individual term involves the fluctuating normal strain rate in the transverse direction. In part, this is because of a strong correlation between this term and the non-normal vorticity with a transverse axis, but it is also the case that individual components of the purely non-normal enstrophy production are negative in the mean. Hence, a local strain rate that is orthogonal to the direction of the dominant mean and fluctuating shear plays a crucial role in amplifying vorticity that is yet to have developed a local component. These conclusions support the emphasis in the control literature on the transverse velocity components at the wall.

On computing the zeros of Laguerre–Sobolev polynomials

Abstract In this work, we focus on the computation of the zeros of a monic Laguerre–Sobolev orthogonal polynomial of degree n. Taking into account the associated four–term recurrence relation, this problem can be formulated as a generalized eigenvalue problem, involving a lower bidiagonal matrix and a 2–banded lower Hessenberg matrix of order n. Unfortunately, the considered generalized eigenvalue problem is very ill–conditioned, and classical balancing procedures do not improve it. Therefore, customary techniques for solving the generalized eigenvalue problem, like the QZ method, yield unreliable results. Here, we propose a novel balancing procedure that drastically reduces the ill–conditioning of the eigenvalues of the involved matrix pencil. Moreover, we propose a fast and reliable algorithm, with $$ \mathcal {O}(n^2) $$ O ( n 2 ) computational complexity and $$ \mathcal {O}(n) $$ O ( n ) memory, exploiting the structure of the considered matrix pencil.

Kinematic vorticity decomposition for generic two-dimensional flow

The present paper begins with evaluating the matrix representation of the velocity gradient tensor A by applying the method of differential geometry to the intrinsic streamline triad. Then, a direction-dependent kinematic decomposition of the vorticity is proposed for a generic two-dimensional (2D) flow, where the corresponding expressions are derived for the two elementary vorticity modes (R(t),s(t)), physically accounting for the orbital rotation (as a specific form of the rigid rotation mode) and the shear under the field description. We show that (R(t),s(t)) are expressible in terms of the angular velocities of a pair of orthogonal material line (surface) elements, naturally resulting in a physically intuitive representation of A. For the first time, the vorticity constituents (RN,sN) in the established invariant vorticity decomposition, previously deduced from the real Schur form of A at the vortex-structure level, are reformulated based on the direction-dependent variables (R(t),s(t)). It is proved that the disparity between the magnitudes of sN and s(t) is solely attributed to the potential flow, being determined by the difference between the relative stretching rates along the tangential and principal normal directions of a streamline. Next, a thorough analysis is performed for all the possible cases of a generic 2D flow by which the physical understanding of the distinct vorticity modes is, thus, deepened. Finally, the significance of the theoretical results are further illustrated by analyzing the velocity fields of the viscous flow inside a lid-driven cavity and the intriguing coherent structures in the Jupiter's Great Red Spot (GRS) extracted from the Galileo cloud images. The analysis demonstrates the generality of the proposed kinematic vorticity decomposition and its superiority in capturing the details of flow structures. The present exposition could be valuable in separating the elementary vorticity modes at different levels of description, as well as understanding their kinematic evolution mechanisms in a wide range of vortical-flow phenomena.

Low-rank decomposition on the antisymmetric product of geminals for strongly correlated electrons

We investigated some variational methods to compute a wavefunction based on antisymmetric product of geminals (APG). The Waring decomposition on the APG wavefunction leads a finite sum of antisymmetrized geminal power (AGP) wavefunctions, each for which the variational principle can be applied. We call this as AGP-CI method which provides a variational solution of the APG wavefunction efficiently. However, number of AGP wavefunctions in the exact AGP-CI formalism become exponentially large in case of many-electron systems. Therefore, we also investigate the low-rank APG wavefunction, in which the coefficient matrices of geminals are factorized by the Schur decomposition. Interestingly, only a few non-zero eigenvalues (up to half number of electrons) were found from the Schur decomposition on the APG wavefunction. We developed some methods to approximate the APG wavefunction by lowering the ranks of coefficient matrices of geminals, and demonstrate their performance. Our new geminal method based on the low-rank decomposition can drastically reduce the number of variational parameters, although no efficient formalism has been elaborated so far, due to the difficulty of optimization caused by the inability to compute analytical derivatives.

INSTABILITY OF THERMOSOLUTAL CONVECTION OF KELVIN-VOIGT FLUID IN A POROUS MEDIUM USING DARCY-BRINKMAN MODEL

In this article, we investigate the problem of thermosolutal convection of a class of viscoelastic fluids in a porous medium of Darcy-Brinkman type. This phenomenon takes place when a layer is heated from beneath while also being exposed to salt either from the upper or lower side. Both linear instability and conditional nonlinear stability analyses are applied in this study. The eigenvalue system have been solved using the Chebyshev collocation technique and the QZ algorithm. The computation of instability boundaries is undertaken for the occurrence of thermosolutal convection in a fluid containing dissolved salt, where the fluid is of a complex viscoelastic nature resembling the Navier-Stokes-Voigt type. Notably, the Kelvin-Voigt parameter emerges as a critical factor in maintaining stability, particularly for oscillatory convection. In instances where the layer is heated from below and salted from above, the thresholds of stability align with those of instability, substantiating the appropriateness of the linear theory in predicting the thresholds for convection initiation. Conversely, when the layer is subjected to salting from the bottom while being heated, the thresholds of stability remain constant even with variations in the salt Rayleigh number. This leads to a significant disparity between the thresholds of linear instability and those of nonlinear stability.

Linear and nonlinear stability analysis of double-diffusion convection in an inclined Brinkman porous media with a concentration-based internal heat source

The onset of double-diffusion convection in an inclined porous medium with a concentration-based internal heat source is investigated by performing linear and nonlinear stability analysis. The Brinkman model is employed to model the momentum equation. Effects of different parameters, such as the thermal Rayleigh number (RaT) and solutal Rayleigh number (Ras), the angle of inclination (ϕ), the Lewis number (Le), the Darcy number (Da), and the concentration-based internal heat source (Q), are shown. A normal mode technique has been employed on the disturbance equations to get the generalized eigenvalues problem, which is solved by the Chebyshev-tau method coupled with the QZ algorithm in MATLAB. It was observed that increasing the solutal Rayleigh number stabilizes the system due to the higher concentration at the lower boundary than the upper boundary. It has also been observed that decreasing the Darcy number has a destabilizing effect, which means that decreasing permeability advances the onset of double-diffusion convection. Furthermore, it was observed that an increase in the concentration-based internal heat source destabilizes the system. Our numerical results show that for Ras>0 and ϕ>0°, for all Q values, the subcritical instability only exists for transverse rolls.

Application of the Krylov-Schur method in three-dimensional nuclear reactor discrete ordinates criticality calculations

Application of the Krylov-Schur method in three-dimensional nuclear reactor discrete ordinates criticality calculations

Enhancing color image watermarking via fast quaternion Schur decomposition: a high-quality blind approach

Enhancing color image watermarking via fast quaternion Schur decomposition: a high-quality blind approach

PSNR and SSIM Performance Analysis of Schur Decomposition for Imperceptible Steganography

Purpose: This research examines how well Schur decomposition-based steganography can hide data in digital images without being noticed, while also keeping the image quality high and keeping the hidden information safe. Methods: The study starts by choosing regular test images (Lena, Plane, Peppers, Cameraman, Baboon) to use for hiding messages in. The Schur decomposition is used to hide information within images in a subtle way. To test how well the new method works, we added Gaussian noise and Salt & Pepper noise after embedding. The quality of the image is determined by looking at the Peak Signal to Noise Ratio (PSNR) and Structural Similarity Index (SSIM) metrics. Result: The research shows that Schur decomposition results in very good SSIM values (greater than 0.92) and high PSNR scores (as high as 90.27 dB) for various image sizes (64x64, 128x128, 256x256). This means that the quality of the images is not greatly reduced even after steganography is applied. Novelty: This research introduces a unique use of Schur decomposition for hiding data in images without affecting their quality. The study highlights how this method can securely hide information in digital media, which could be really useful for improving steganography techniques in the future. Future studies should concentrate on making improvements to Schur decomposition-based steganography, especially for bigger images. One possibility is to create adaptive methods that can change how images are hidden based on their content. This could make it harder for others to detect and analyze hidden information in the images.

Onset of Darcy–Brinkman convection with thermal anisotropy in an inclined porous layer

In this article, we study the linear instability and the nonlinear stability (through energy functional) analyses of thermal convection in an inclined Darcy–Brinkman porous layer considering uniformly heated horizontal rigid, impermeable walls from below and above. The effects of a uniform internal heat source and anisotropy in effective thermal diffusivity on heat transfer are also considered. Heating the porous layer from below yields the temperature gradient, influencing the buoyancy and making the convection happen. This temperature gradient also impacts the base state. The basic solution for velocity incorporates both hyperbolic and polynomial functions, significantly increasing the complexity of linear and nonlinear analyses. The Chebyshev-tau method, together with the QZ algorithm, is used to solve the linear and nonlinear perturbed system of equations numerically. The region of subcritical instability is obtained by comparing the linear and nonlinear thresholds for the longitudinal and transverse rolls, respectively. We found that perturbations for longitudinal and transverse rolls do not grow after inclination is more than 30.3° and 31.3°, respectively. It has been noted that in transverse roll scenarios, the flow becomes stabilized when the inclination angle, ϕ, is equal to or exceeds 60°, where ϕ plays a leading role in surpassing the impact of internal heating. However, when the inclination angle is ϕ<60°, then internal heating dominates and destabilizes the flow. For the longitudinal rolls, the internal heating dominates the whole range of ϕ, destabilizing the system. Furthermore, it can be seen that the Darcy number (Da) and the anisotropic thermal diffusivity (ξ) delay the onset of convection.

Onset of double-diffusive convection in a Poiseuille flow with a uniform internal heat source

The linear stability analysis of the onset of double-diffusive convection in a Poiseuille flow system is investigated. In addition, a volumetric uniform internal heat source is taken into account. In this problem, the horizontal fluid channel is bounded by two plates which are isothermal and isosolutal. The governing parameters are thermal Rayleigh number RaT, solutal Rayleigh number Ras, internal heat source parameter RaI, Prandtl number Pr, and Reynolds number Re. The eigenvalue problem arising from the linear perturbed system of equations is solved numerically using the Chebyshev–Tau method coupled with the QZ algorithm. It is found that the positive solutal Rayleigh number Ras destabilizes the system. Furthermore, it is observed that an increase in the Prandtl number Pr stabilizes the system. Additionally, at Ras = −60, the critical values of the thermal Rayleigh number Rac decreases with R=Re cos ϕ up 2; and increases with R beyond R=2.

Backward stability of the Schur decomposition under small perturbations

Backward stability of the Schur decomposition under small perturbations

Impact of Phase Angle Jump on a Doubly Fed Induction Generator under Low-Voltage Ride-Through Based on Transfer Function Decomposition

During the fault period, a phase angle jump may occur at the stator or the point of common coupling, which will deteriorate the low-voltage ride-through (LVRT) characteristics of a doubly fed induction generator (DFIG). The existing LVRT studies focus on the impact of a voltage drop on DFIGs but often ignore that of a phase angle jump. The time-domain simulation is accurate in describing the response of a DFIG during the LVRT process, but it is time-consuming for a DFIG with the full-order model. In this paper, by using the voltage magnitude and phase angle of the stator or the point of common coupling as the inputs, and the state variables as the outputs, the transfer function of a DFIG is derived to analyze its response and find the LVRT measures against the voltage drop and, especially, the phase angle jump. Firstly, the differential-algebraic equations of the DFIG are linearized to propose their transfer function model. Secondly, considering its high-order characteristic, a model reduction method for the transfer function of the DFIG using the Schur decomposition is proposed, and the analytical expression of the output variables of the DFIG with the phase angle jump is derived by the inverse Laplace transformation to judge the necessity of the LVRT measures. Finally, the simulation results of the DFIG are provided to verify the accuracy of the transfer function model and its reduced-order form and validate the feasibility of the LVRT against the phase angle jump with the proposed models.

Gustafson, Jeribi, Weidmann and Wolf essential spectra and generator properties for one-sided coupled operator matrices

This paper is concerned with the closedness (closability), essential spectra and generator properties for a one-sided coupled operator matrix A 0 acting in a product of Banach spaces X and Y. By combining the Browder Resolvent with the Schur factorization, some basic properties related to the operator entries of A 0 are collected, including perturbations, closedness and spectra. Utilizing these results, we present a necessary and sufficient condition for A 0 to be closed (closable), and the essential spectra as well as generator properties of A 0 are further described.

Inelastic fluid models with an objective stretch rate parameter

Inelastic fluid models with an objective stretch rate parameter

Probabilistic perturbation bounds of matrix decompositions

AbstractIn this article, we determine probabilistic approximations of the entries of random perturbation matrices implementing the Markoff inequality. These approximations are used to derive with prescribed probability asymptotic componentwise perturbation bounds of some orthogonal and unitary matrix decompositions. We show that the probabilistic asymptotic bounds are significantly less conservative than the corresponding deterministic perturbation bounds. As case studies, we consider the determining of probabilistic perturbation bounds of the QR decomposition, the singular value decomposition and the Schur decomposition of a matrix using an unified method for asymptotic componentwise perturbation analysis of these decompositions. It is demonstrated that the probability bounds of the orthogonal transformations, singular values and eigenvalues are much tighter than the corresponding deterministic asymptotic bounds. The probabilistic bounds derived are appropriate for perturbation analysis of large‐order matrix problems.

A modified block Hessenberg method for low-rank tensor Sylvester equation

A modified block Hessenberg method for low-rank tensor Sylvester equation

Computing the Mittag-Leffler function of a matrix argument

It is well-known that the two-parameter Mittag-Leffler (ML) function plays a key role in Fractional Calculus. In this paper, we address the problem of computing this function, when its argument is a square matrix. Effective methods for solving this problem involve the computation of higher order derivatives or require the use of mixed precision arithmetic. In this paper, we provide an alternative method that is derivative-free and works entirely using IEEE standard double precision arithmetic. If certain conditions are satisfied, our method uses a Taylor series representation for the ML function; if not, it switches to a Schur-Parlett technique that will be combined with the Cauchy integral formula. A detailed discussion on the choice of a convenient contour is included. Theoretical and numerical issues regarding the performance of the proposed algorithm are discussed. A set of numerical experiments shows that our novel approach is competitive with the state-of-the-art method for IEEE double precision arithmetic, in terms of accuracy and CPU time. For matrices whose Schur decomposition has large blocks with clustered eigenvalues, our method far outperforms the other. Since our method does not require the efficient computation of higher order derivatives, it has the additional advantage of being easily extended to other matrix functions (e.g., special functions).