Singular Coefficients Research Articles

Directional pixel value ordering based secret sharing using sub-sampled image exploiting Lagrange polynomial

In this paper, we proposed secret sharing within the sub-sampled image using Directional Pixel Value Ordering (DPVO) for Reversible Data Hiding scheme. The original cover is partitioned into sub-sampled images which extend its size through image interpolation technique. The secret information has been converted through Lagrange interpolation polynomial techniques. These new secret data is embedded within pixels of each interpolated sub-sampled images using DPVO. In the decoder side, the secret data is taken out from the pixels of every stego image using reverse DPVO. After that, Lagrange’s interpolation is applied to generate the original secret message. The proposed scheme enhanced the security for the sharable characteristics of secrets amongst several images. It raises the data embedding capacity and improves the visual quality that is determined by peak signal to noise ratio (PSNR). The average PSNR value is over 60 dB. It shows the superiority of the new algorithm over the other existing data hiding methods regarding payload of data, quality, and security of the image. Also, the stego images have evaluated through standard deviation, regular singular analysis, correlation coefficient, normalized cross-correlation and structural similarity index in between original cover and marked (stego) image to display the robustness of our result among the several steganographic strikes.

Highly Accurate Method for Solving Singular Boundary-Value Problems Via Padé Approximation and Two-Step Quartic B-Spline Collocation

We propose a highly accurate method to solve a class of singular two-point boundary-value problems having singular coefficients. These problems often arise when a partial differential equation is reduced to an ordinary differential equation by physical symmetry. The proposed method is based on Pade approximation and two-step collocation. The resolution domain is divided, Pade approximant is used on the subdomain in the vicinity of the singular point and provides a new boundary condition, and then, two-step quartic B-spline is used on the remaining subdomain where the problem is transformed to a regular boundary-value problem. The subdomains are chosen optimally using a suitable optimization procedure. Some examples are presented along with a comparison of the numerical results with other methods.

A new high accuracy cubic spline method based on half-step discretization for the system of 1D non-linear wave equations

PurposeThis paper aims to develop a new 3-level implicit numerical method of order 2 in time and 4 in space based on half-step cubic polynomial approximations for the solution of 1D quasi-linear hyperbolic partial differential equations. The method is derived directly from the consistency condition of spline function which is fourth-order accurate. The method is directly applied to hyperbolic equations, irrespective of coordinate system, and fourth-order nonlinear hyperbolic equation, which is main advantage of the work.Design/methodology/approachIn this method, three grid points for the unknown function w(x,t) and two half-step points for the known variable x in spatial direction are used. The methodology followed in this paper is construction of a cubic spline polynomial and using its continuity properties to obtain fourth-order consistency condition. The proposed method, when applied to a linear equation is shown to be unconditionally stable. The technique is extended to solve system of quasi-linear hyperbolic equations. To assess the validity and accuracy, the method is applied to solve several benchmark problems, and numerical results are provided to demonstrate the usefulness of the method.FindingsThe paper provides a fourth-order numerical scheme obtained directly from fourth-order consistency condition. In earlier methods, consistency conditions were only second-order accurate. This brings an edge over other past methods. In addition, the method is directly applicable to physical problems involving singular coefficients. Therefore, no modification in the method is required at singular points. This saves CPU time, as well as computational costs.Research limitations/implicationsThere are no limitations. Obtaining a fourth-order method directly from consistency condition is a new work. In addition, being an implicit method, this method is unconditionally stable for a linear test equation.Practical implicationsPhysical problems with singular and nonsingular coefficients are directly solved by this method.Originality/valueThe paper develops a new fourth-order implicit method which is original and has substantial value because many benchmark problems of physical significance are solved in this method.

Vibration under variable magnitude moving distributed masses of non-uniform Bernoulli-Euler beam resting on Pasternak elastic foundation

The dynamic response to variable magnitude moving distributed masses of simply supported non-uniform Bernoulli-Euler beam resting on Pasternak elastic foundation is investigated in this paper. The problem is governed by fourth-order partial differential equation with variable and singular coefficients. The main objective of this work is to obtain closed form solution to this class of dynamical problem. In order to obtain the solution, a technique based on the method of Galerkin with the series representation of Heaviside function is first used to reduce the equation to second order ordinary differential equations with variable coefficients. Thereafter the transformed equations are simplified using (i) The Laplace transformation technique in conjunction with convolution theory to obtain the solution for moving force problem and (ii) finite element analysis in conjunction with Newmark method to solve the analytically unsolvable moving mass problem because of the harmonic nature of the moving load. The finite element method is first used to solve the moving force problem and the solution is compared with the analytical solution of the moving force problem in order to validate the accuracy of the finite element method in solving the analytically unsolvable moving mass problem. The numerical solution using the finite element method is shown to compare favorably with the analytical solution of the moving force problem. The displacement response for moving distributed force and moving distributed mass models for the dynamical problem are calculated for various time t and presented in plotted curves.

Directional Association Measurement in Contingency Tables: Genomic Case.

Analysis of large data sets is currently a major challenge. Strong efforts are being undertaken to tackle this problem by developing new methods or modifying existing ones. The Z association method is a new method for describing directional association in contingency tables. It allows to arbitrarily group categories for each of the two variables, for which the contingency table is analyzed. The Z coefficient was calculated on a sample data set with gene mutations in different cancer types. Results showed some association with both gene mutations and annotation groups. Detailed results obtained for particular cancer types versus particular genes and annotation groups were in line with well-known facts in cancer genomics. The “MEUSassociation” R library allows to analyze the directional association between two categorical variables, and the mutual relationship is summarized in a contingency table, by means of the Z association coefficient. The method implemented in the library allows to compute the standard Z coefficient and to apply it in a case, where all possible singular coefficients Z(A:B) are computed at the same time, giving information of association between particular rows and columns. Investigating the ranked list of the highest singular coefficients allows to reduce the complexity of a large-scale data set. Both the Z coefficient and its R implementation are important tools in categorical data analysis.

Liouville-type theorem for the Lamé system with singular coefficients

Liouville-type theorem for the Lamé system with singular coefficients

Complexity-action of subregions with corners

In the past, the study of the divergence structure of the holographic entanglement entropy on singular boundary regions uncovered cut-off independent coefficients. These coefficients were shown to be universal and to encode important field theory data. Inspired by these lessons we study the UV divergences of subregion complexity-action (CA) in a region with corner (kink). We develop a systematic approach to study all the divergence structures, and we emphasize that the counter term that restores reparameterization invariance on the null boundaries plays a crucial role in simplifying the results and rendering them more transparent. We find that a general form of subregion CA contains a part dependent on the null generator normalizations and a part that is independent of them. The former includes a volume contribution as well as an area contribution. We comment on the origin of the area term as entanglement entropy, and point out that its presence constitutes a robust difference between the two prescriptions to calculate subregion complexity (-action vs. -volume). We also find universal log δ divergence associated with the kink feature of the subregion. Similar flat angle limit as the subregion-CV result is obtained.

Correction to: Sturm Attractors for Quasilinear Parabolic Equations with Singular Coefficients

The original version of this article unfortunately contained a mistake in Eq. 1.4. The correct equation is given below.

The Unique Solvability of Boundary Value Problems for a 3D Elliptic Equation With Three Singular Coefficients

We state and study several boundary value problems for an elliptic equation with three singular coefficients in a rectangular parallelepiped. Using the method of energy integrals, we prove the uniqueness of solutions of the stated problems. For proving the existence of solutions, we use the spectral Fourier method based on separation of variables. We construct solutions to the stated problems in the form of the sum of the double Fourier—Bessel series. We justify the uniform convergence of the constructed series with the help of asymptotic bounds for Bessel functions of real and imaginary arguments. Using these bounds, we estimate each term of the series, which allows us to prove the convergence of the series and its derivatives up to the second order inclusive and to justify the existence theorem in the class of regular solutions.

Two-level implicit high order method based on half-step discretization for 1D unsteady biharmonic problems of first kind

In this work, compact difference scheme based on half-step discretization is proposed to solve fourth order time dependent partial differential equations subject to Dirichlet and Neumann boundary conditions. The difference method reported here is second order accurate in time and fourth order accurate in space. The scheme employs only three grid points at each time-level and the given boundary conditions are exactly satisfied with no further approximations at the boundaries. The linear stability of the presented method has been examined and it is shown that the proposed two-level finite difference method is unconditionally stable for a model linear problem. The developed method is directly applicable to fourth order parabolic partial differential equations with singular coefficients which is the main highlight of our work. The method is successfully tested on singular problem. The proposed method is applied to find the numerical solution of the classical nonlinear Kuramoto–Sivashinsky equation and the extended Fisher–Kolmogorov equation. Comparison of the obtained results with those of some earlier known methods demonstrate the superiority of the present approach.

Holmgren problem for Helmholtz equation with the three singular coefficients

Abstract Fundamental solutions for a multidimensional Helmholtz equation with three singular coefficients have been constructed recently which are expressed in terms of the confluent hypergeometric function in four variables. In this paper, we study the Holmgren problem for a 3D elliptic equation with three singular coefficients. A unique solution of the problem is obtained in the explicit form.

On unique solvability of boundary-value problems for three-dimentional elliptic equation with three singular coefficients

On unique solvability of boundary-value problems for three-dimentional elliptic equation with three singular coefficients

Green’s function for second order elliptic equations with singular lower order coefficients

We construct Green’s function for second order elliptic operators of the form in a domain and obtain pointwise bounds, as well as Lorentz space bounds. We assume that the matrix of principal coefficients is uniformly elliptic and bounded and the lower order coefficients b, c, and d belong to certain Lebesgue classes and satisfy the condition . In particular, we allow the lower order coefficients to be singular. We also obtain the global pointwise bounds for the gradient of Green’s function in the case when the mean oscillations of the coefficients and b satisfy the Dini conditions and the domain is .

Evaluating seasonal and regional calibration of temperature-based methods for estimating potential evaporation in Ontario

Development of appropriate water management strategies in the context of projected air temperature increases under a changing climate necessitates assessing how increases in atmospheric demand for water might constrain future water availability. This is most easily assessed using estimates of potential evaporation (PE). Most temperature-based methods for estimating PE have been calibrated in different landscapes in more southern locations using a singular calibration coefficient. This study assessed whether regionally calibrated temperature-based models improve estimates of PE in Ontario and whether further improvements could be made by adjusting coefficients monthly to address a marked seasonality in Ontario or spatially to address the province’s size and varied landscape. Monthly calibration coefficients were calculated for four temperature-based methods (Thornthwaite, Blaney–Criddle, Hamon and Hargreaves–Samani) using climate data and evaporation estimates from 18 pan evaporation stations in Ontario. Calibrated models were then evaluated against an additional 19 pan evaporation stations by comparing estimates of PE calculated using the adjusted and literature-derived coefficients with pan evaporation estimates. Finally, PE estimated using adjusted coefficients was compared to evaporation (E) estimated using a water balance approach for 64 watersheds across Ontario. As anticipated, the regional calibration process increased the accuracy of the PE estimates relative to those calculated using published model coefficients. Use of monthly model coefficients addressed the marked seasonality of evaporation in Ontario and resulted in improved PE estimates. Adjusted model coefficients varied spatially but the paucity of measurement stations precluded further regionalization. PE estimates improved using adjusted coefficients, varied less between models, and were highly correlated with water balance estimates of E. The Hargreaves–Samani model produced the best estimates of PE using unadjusted and adjusted coefficients and was most highly correlated with long-term water balance estimates of E.

Distribution dependent SDEs with singular coefficients

Under integrability conditions on distribution dependent coefficients, existence and uniqueness are proved for distribution dependent SDEs with non-degenerate noise. When the coefficients are Dini continuous in the space variable, gradient estimates and Harnack type inequalities are derived. These generalize the corresponding results derived for classical SDEs, and are new in the distribution dependent setting.

Sturm Attractors for Quasilinear Parabolic Equations with Singular Coefficients

The goal of this paper is to construct explicitly the global attractors of parabolic equations with singular diffusion coefficients on the boundary, as it was done without the singular term for the semilinear case by Brunovský and Fiedler (1986), generalized by Fiedler and Rocha (1996) and later for quasilinear equations by Lappicy (2017). In particular, we construct heteroclinic connections between hyperbolic equilibria, stating necessary and sufficient conditions for heteroclinics to occur. Such conditions can be computed through a permutation of the equilibria. Lastly, an example is computed yielding the well known Chafee–Infante attractor.

The Convergence of Least-Squares Progressive Iterative Approximation for Singular Least-Squares Fitting System

Data fitting is an extensively employed modeling tool in geometric design. With the advent of the big data era, the data sets to be fitted are made larger and larger, leading to more and more leastsquares fitting systems with singular coefficient matrices. LSPIA (least-squares progressive iterative approximation) is an efficient iterative method for the least-squares fitting. However, the convergence of LSPIA for the singular least-squares fitting systems remains as an open problem. In this paper, the authors showed that LSPIA for the singular least-squares fitting systems is convergent. Moreover, in a special case, LSPIA converges to the Moore-Penrose (M-P) pseudo-inverse solution to the leastsquares fitting result of the data set. This property makes LSPIA, an iterative method with clear geometric meanings, robust in geometric modeling applications. In addition, the authors discussed some implementation detail of LSPIA, and presented an example to validate the convergence of LSPIA for the singular least-squares fitting systems.

A New Fast Algorithm Based on Half-Step Discretization for 3D Quasilinear Hyperbolic Partial Differential Equations

In this paper, we study a new numerical method of order 4 in space and time based on half-step discretization for the solution of three-space dimensional quasi-linear hyperbolic equation of the form [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] subject to prescribed appropriate initial and Dirichlet boundary conditions. The scheme is compact and requires nine evaluations of the function [Formula: see text] For the derivation of the method, we use two inner half-step points each in [Formula: see text]-, [Formula: see text]-, [Formula: see text]- and [Formula: see text]-directions. The proposed method is directly applicable to three-dimensional (3D) hyperbolic equations with singular coefficients, which is main attraction of our work. We do not require extra grid points for computation. The proposed method when applied to 3D damped wave equation is shown to be unconditionally stable. Operator splitting method is used to solve 3D linear hyperbolic equations. Few benchmark problems are solved and numerical results are provided to support the theory which is discussed in this paper.

Initial-Boundary Value Problem for Hyperbolic Equation with Singular Coefficient and Integral Condition of Second Kind

We research an initial-boundary value problem with integral condition of the second kind in a rectangular domain for a hyperbolic equation with singular coefficient. The solution is obtained in the form of the Fourier–Bessel series. There are proved theorems on uniqueness, existence and stability of the solution. In order to prove the existence of solution of the non-local problem we obtain sufficient conditions for the convergence of the series in terms of the initial values.

On the Existence of $$L^p$$ L p -Solution of Generalized Euler–Poisson–Darboux Equation in the Upper Half Space

We focus on the existence of solution of generalized Euler–Poisson–Darboux equation, which is elliptic in $$\mathbb R^{n+1}_+$$ and has a singular coefficient on its boundary. Based on Mikhlin’s multiplier theorem and Hardy inequalities, the well-posedness of its Dirichlet problem in the upper half space is established.