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In recent years, there has been a substantial surge in the application of image watermarking, which has evolved into an essential tool for identifying multimedia material, ensuring security, and protecting copyright. Singular value decomposition (SVD) and discrete cosine transform (DCT) are widely utilized in digital image watermarking despite the considerable computational burden they involve. By combining block-based direct current (DC) values with matrix norm, this research article presents a novel, robust zero-watermarking approach. It generates a zero-watermark without attempting to modify the contents of the image. The image is partitioned into non-overlapping blocks, and DC values are computed without applying DCT. This sub-image is further partitioned into non-overlapping blocks, and the maximum singular value of each block is calculated by matrix norm instead of SVD to obtain the binary feature matrix. A piecewise linear chaotic map encryption technique is utilized to improve the security of the watermark image. After that, the feature image is created via XOR procedure between the encrypted watermark image and the binary feature matrix. The proposed scheme is tested using a variety of distortion attacks including noise, filter, geometric, and compression attacks. It is also compared with the other relevant image watermarking methods and outperformed them in most cases. [ABSTRACT FROM AUTHOR]

In the present work, we define harmonic complex balancing numbers by considering well-known balancing numbers and inspiring harmonic numbers. Mainly, we investigate some of their basic characteristic properties such as the Binet formula and Cassini identity, etc. In addition, one type of symmetric matrix family whose entries are harmonic complex balancing numbers is constructed. Additionally, some linear algebraic properties are obtained. Furthermore, some inequalities are stated by exploiting the well-known inequalities between various matrix norms. Finally, we illustrate the results with some numerical examples. [ABSTRACT FROM AUTHOR]

For G ∈ R m × n and g ∈ R m , the minimization min ∥ G ψ − g ∥ 2 , with ψ ∈ R n , is known as the Tykhonov regularization. We transport the Tykhonov regularization to an infinite-dimensional setting, that is min ∥ T (h) − k ∥ , where T : H → K is a continuous linear operator between Hilbert spaces H , K and h ∈ H , k ∈ K . In order to avoid an unbounded set of solutions for the Tykhonov regularization, we transform the infinite-dimensional Tykhonov regularization into a multiobjective optimization problem: min ∥ T (h) − k ∥ and min ∥ h ∥ . We call it bounded Tykhonov regularization. A Pareto-optimal solution of the bounded Tykhonov regularization is found. Finally, the bounded Tykhonov regularization is modified to introduce the precise Tykhonov regularization: min ∥ T (h) − k ∥ with ∥ h ∥ = α . The precise Tykhonov regularization is also optimally solved. All of these mathematical solutions are optimal for the design of Magnetic Resonance Imaging (MRI) coils. [ABSTRACT FROM AUTHOR]

In this study, a new modified group iterative scheme for solving the two-dimensional (2D) fractional hyperbolic telegraph differential equation with Dirichlet boundary conditions is obtained from the 2h-spaced standard and rotated Crank–Nicolson FD approximations. The findings of new four-point modified explicit group relaxation method demonstrates the rapid rate of convergence of proposed method as compared to the existing schemes. Numerical tests are performed to test the capability of the group iterative scheme in comparison with the point iterative scheme counterparts. The stability of the derived modified group method is proven by the matrix norm algorithm. The obtained results are tabulated and concluded that exact solutions are exactly symmetric with approximate solutions. [ABSTRACT FROM AUTHOR]

We propose a new iterative method for solving a generalized Sylvester matrix equation A 1 X A 2 + A 3 X A 4 = E with given square matrices A 1 , A 2 , A 3 , A 4 and an unknown rectangular matrix X. The method aims to construct a sequence of approximated solutions converging to the exact solution, no matter the initial value is. We decompose the coefficient matrices to be the sum of its diagonal part and others. The recursive formula for the iteration is derived from the gradients of quadratic norm-error functions, together with the hierarchical identification principle. We find equivalent conditions on a convergent factor, relied on eigenvalues of the associated iteration matrix, so that the method is applicable as desired. The convergence rate and error estimation of the method are governed by the spectral norm of the related iteration matrix. Furthermore, we illustrate numerical examples of the proposed method to show its capability and efficacy, compared to recent gradient-based iterative methods. [ABSTRACT FROM AUTHOR]

In this manuscript we provide an exact solution to the maxmin problem max ∥ A x ∥ subject to ∥ B x ∥ ≤ 1 , where A and B are real matrices. This problem comes from a remodeling of max ∥ A x ∥ subject to min ∥ B x ∥ , because the latter problem has no solution. Our mathematical method comes from the Abstract Operator Theory, whose strong machinery allows us to reduce the first problem to max ∥ C x ∥ subject to ∥ x ∥ ≤ 1 , which can be solved exactly by relying on supporting vectors. Finally, as appendices, we provide two applications of our solution: first, we construct a truly optimal minimum stored-energy Transcranian Magnetic Stimulation (TMS) coil, and second, we find an optimal geolocation involving statistical variables. [ABSTRACT FROM AUTHOR]