A companion matrix and 2D compressive sensing based efficient image encryption method

We study sequences of bounded operators \((T_n)_{n \ge 0}\) on a complex separable Hilbert space \(\mathcal{H}\) that satisfy a linear recurrence relation of the form $$ T_{n+r} = A_0 T_n + A_1 T_{n+1} + \cdots + A_{r-1} T_{n+r-1} \quad(\textrm{for all } n\ge 0), $$ where the coefficients \(A_0, A_1, \dots, A_{r-1}\) are pairwise commuting bounded operators on \(\mathcal{H}\). \ Such relations naturally arise in the context of the operator-valued moment problem, particularly in the study of flat extensions of block Hankel operators. \ Our first goal is to derive an explicit combinatorial formula for \(T_n\). As a concrete application, we provide an explicit expression for the powers of an operator-valued companion matrix. \ In the special case of scalar coefficients $A_k=a_kI_\mathcal{H}$, with $a_k\in\mathbb{R}$, we recover a Binet-type formula that allows the explicit computation of the powers and the exponential of algebraic operators in terms of Bell polynomials.

The modeling of compressible multiphase flows is a decades-old area of study with many applications across various fields. Many of these application areas use stiff pressure relaxation. This process involves the solution of a nonlinear system with N+1 equations and N+1 unknowns, where N is the number of phases. The resolution of this system with general equations of state (EOSs) is difficult. Furthermore, nonlinear systems can admit multiple solutions, and current solution methods do not address this possibility. Very recently, a thermodynamic relaxation method was introduced, which effectively maps a relatively simple predictor equation of state onto a more complex target equation of state. In this context, the target EOSs are the chosen EOSs for the thermodynamic model. This thermodynamic relaxation has the benefit of simplifying the stiff pressure relaxation system of equations. In this article, we show this system reduces to a polynomial of degree N, which can be recast as an eigenvalue problem through the use of the associated companion matrix. We show that although this eigenvalue method is generally less efficient than Newton–Raphson iteration, it does not suffer from convergence issues and finds all N roots of the polynomial. Hence, the method provides a fail-safe for root-finding iterative methods and a way to address the issue of multiple solutions to the nonlinear system of equations in stiff pressure relaxation.

ABSTRACTAn open and fully reproducible numerical procedure to solve the Rayleigh stability equation is presented in this work for the spatial analysis of hydrodynamic instabilities in an inviscid shear flow using the Chebyshev spectral‐collocation method. The resulting cubic polynomial eigenvalue problem is linearized via a companion matrix and solved efficiently, from which the most unstable spatial growth rate and its frequency are identified. Verification against the classical reference [1] at dimensionless temporal frequency demonstrates spectral (exponential) convergence of the dispersion relation ; in practice, a resolution of yields grid‐independent predictions. Across the unstable band, the maximum relative difference is for the real component of the wavenumber and for the dimensionless imaginary component of the complex streamwise wavenumber , with the peak growth rate and its frequency matching the reference. The computed real and imaginary eigenfunctions closely reproduce the reported mode shapes. An additional comparison with a shooting method implementation (method used by Michalke) indicated that the spectral formulation attains the same accuracy with superior conditioning and reduced parameter tuning. The proposed workflow thus offers a computationally practical and fully reproducible route to assess shear‐flow stability — complementary to large eddy and direct numeric simulation LES/DNS — while enabling straightforward verification against canonical benchmarks.

Abstract The pro-isomorphic zeta function of a finitely generated nilpotent group is a Dirichlet generating series that enumerates all finite-index subgroups whose profinite completion is isomorphic to that of the ambient group. We study the pro-isomorphic zeta functions of ℚ-indecomposable D *-groups of even Hirsch length. These groups are building blocks of finitely generated class-two nilpotent groups with rank-two centre, up to commensurability. Due to a classification by Grunewald and Segal, they are parameterised by primary polynomials whose companion matrices define commutator relations for an explicit presentation. For Grunewald–Segal representatives of even Hirsch length of type f ( t ) = t m , we give a complete description of the algebraic automorphism groups of associated Lie lattices. Utilising the automorphism groups, we determine the local pro-isomorphic zeta functions of groups associated to t 2 and t 3 . In both cases, the local zeta functions are uniform in the prime p and satisfy functional equations. The functional equations for these groups, not predicted by the currently available theory, prompt us to formulate a conjecture which prescribes, in particular, information about the symmetry factor appearing in local functional equations for pro-isomorphic zeta functions of nilpotent groups. Our description of the local zeta functions also yields information about the analytic properties of the corresponding global pro-isomorphic zeta functions. Some of our results for the D *-groups associated to t 2 and t 3 generalise to two infinite families of class-two nilpotent groups that result naturally from the initial groups via ‘base extensions’.

Insect pests pose significant challenges to crop production. The heavy reliance on insecticides for pest control has led to numerous issues, including pesticide residues in the food chain, phytotoxicity, resistance development, pest resurgence, bioaccumulation, and outbreaks of secondary pests. These issues also negatively impact the environment and non target beneficial organisms. Trap cropping-a specialized form of companion planting-offers a sustainable alternative by attracting insect pests away from main crops during vulnerable growth stages through the use of more preferred alternative host plants. Beyond pest diversion, trap crops can also attract and conserve natural enemies, providing an ecological benefit that conventional pest control methods often lack. As a component of Integrated Pest Management (IPM), trap cropping can reduce the need for chemical insecticides, thereby lessening environmental disruption and supporting beneficial organisms. Despite its potential, trap cropping remains underutilized and under-researched. This review compiles current knowledge on trap crops used in cereals, legumes, oilseeds, pulses, and fiber crops, highlighting their functions and potential to enhance sustainable pest management strategies globally.

Twisted period integrals are ubiquitous in theoretical physics and mathematics, where they inhabit a finite-dimensional vector space governed by an inner product known as the intersection number. In this work, we uncover the associated tensor structures of intersection numbers and integrate them with the fibration method to develop a novel evaluation scheme. Companion matrices allow us to cast the computation of the intersection numbers in terms of a matrix operator calculus within the ambient tensor space. For illustrative purposes, our algorithm has been successfully applied to the numerical decomposition of a sample of two-loop integrals, coming from planar five-point massless functions, representing a significant advancement for the direct projection of Feynman integrals to master integrals via intersection numbers.

This study develops a robust recursive least-squares (RLS) Wiener fixed-interval (FI) smoother by exploiting covariance information for linear continuous-time systems that face uncertainties in both their system and observation matrices. Uncertainties in the state-space model cause degradations in the signal and observed values. The robust FI smoothing and filtering methods introduced do not assume that the system and the observation matrix have norm-bounded uncertainties. An observable companion form represents the state space model of the degraded signal. Robust RLS FI smoothing is to minimize the mean-square value of the smoothing errors of the system state over a fixed interval. Section 3 introduces an integral equation satisfied by the impulse response function that is optimal for robust FI smoothing estimation of the system state. An integral equation for the impulse response function, which provides a filtering estimate of the state of the degraded system, is also shown. Theorem 1 presents the robust RLS FI smoothing and filtering algorithm for the signal and the system state using covariance information. Theorem 2 presents the robust RLS Wiener (RLSW) FI smoothing and filtering algorithm for the signal and the system state. Robust RLS FI smoother outperforms robust RLS filter in estimation accuracy, as shown by the FI smoothing error covariance function in Section 5. Numerical simulation examples demonstrate that the robust RLSW FI smoother achieves superior signal estimation accuracy compared to the robust RLSW filter.

In this paper, the classical notion of the companion matrix of a polynomial is generalized to higher dimensions.

IntroductionEngaging in physical activity is essential for a healthy pregnancy. A reliable tool is necessary to enhance the assessment and counseling of safe physical activity. This study aimed to translate the original English Get Active Questionnaire for Pregnancy (GAQ-P) and its companion Health Care Provider Consultation Form for Prenatal Physical Activity (cHCP-CF-PPA) into simplified Chinese language and evaluate the psychometric properties in Chinese pregnant women.MethodsThe Brislin’s model of translation was employed to translate the GAQ-P/cHCP-CF-PPA tool. We conducted a cross-sectional study at a tertiary women’s hospital in Shanghai, China, enrolling a convenience sample of 325 pregnant women across all trimesters to evaluate the psychometric properties of the GAQ-P/cHCP-CF-PPA. Reliability was assessed through test-retest reliability and inter-rater reliability, while validity was examined using content validity, known-groups validity, and criterion validity. Sensitivity, specificity, positive predictive value, and negative predictive value were calculated using PARmed-X for Pregnancy as the gold standard.ResultsRegarding content validity, the GAQ-P had an average S-CVI/UA of 0.81 (I-CVIs: 0.83-1.0), while the cHCP-CF-PPA exhibited an average S-CVI/UA of 0.87 (I-CVIs: 0.83-1.0). The GAQ-P/cHCP-CF-PPA scores effectively distinguished women recommended for physical activity from those with contraindications. The Spearman’s correlations between the GAQ-P/cHCP-CF-PPA and the PARmed-X for Pregnancy were 0.851 for absolute contraindications and 0.847 for relative contraindications. The test-retest reliability score was 0.759 for physical activity contraindications, and 0.953 for inter-rater reliability. The sensitivity of the GAQ-P/cHCP-CF-PPA was determined to be 90.00%, with a specificity of 98.31%. The positive predictive value was 78.26%, while the negative predictive value reached 99.32%.ConclusionThe Chinese version of the GAQ-P/cHCP-CF-PPA is a reliable and valid tool for assessing physical activity readiness in pregnant women.

Statistical analysis shows that the most common errors in the transmission of information consist of single errors and transposition errors. Error detection and correction methods are often desired, particularly when the accuracy of information is of crucial importance. Inspired by a check digit system constructed from the companion matrix of a primitive polynomial over the integers Zp and that focused on error detection, this work develops error-correction formulas for single errors and transposition errors for that check digit scheme. We also propose an application to DNA sequences.

The note provides a new formula for the companion matrix of the superposition of two polynomials over a commutative ring. The results obtained are used to provide a constructive proof of Plans’ theorem for two-bridge knots, which states that the first homology group of an odd-sheeted cyclic covering of a three-dimensional sphere branched over a given knot is the direct sum of two copies of some Abelian group. A similar result is also true for the homology of even-sheeted coverings factored by the reduced homology group of a two-sheeted covering. The structure of the above mentioned Abelian groups is described through Chebyshev polynomials of the second and fourth kind.

The current research designs an original robust recursive least-squares (RLS) finite impulse response (FIR) filter for linear continuous-time systems with uncertainties in both the system and observation matrices. These uncertainties in the state-space model generate the degraded signal and observed value. The robust RLS FIR filter does not account for the norm-bounded uncertainties in the system and observation matrices. This study uses an observable companion form to represent the degraded signal state-space model. The system and observation matrices are estimated based on the author's previous computational methods. The robust RLS FIR filtering problem aims to minimize the mean-square errors in FIR filtering for the system state. The robust FIR filtering estimate is formulated as an integral transformation of the degraded observations using an impulse response function. Section 3 obtains the integral equation satisfied by the optimal impulse response function. Theorem 1 presents the robust RLS FIR filtering algorithms for the signal and the system state. This integral equation derives the robust RLS-FIR filtering algorithms. Numerical simulation examples show the validity of the proposed robust RLS FIR filter.

In this paper, we present a matrix representation for the generalized Quadranacci number sequence, defined by the 4th-order generalized recurrence relation, initial terms , and constant coefficients , The parameters , and are arbitrarily chosen real numbers. Fundamental results based on this definition are established in general symbolic form. A generalized companion matrix, associated with the recurrence relation, is introduced to analyze the properties of Quadranacci numbers. Subsequently, is derived in a generalized form, enabling the application of matrix techniques to study the properties of Quadranacci number sequences. By appropriately specifying the initial values, and the constant coefficients,, several existing results are shown to be special cases of the derived results. Moreover, all the results obtained are implicitly applicable to the generalized Tribonacci and Fibonacci sequences, which are governed by lower-order recurrence relations.

A companion matrix-based efficient image encryption method

This note presents insights on the Jordan structure of a matrix which are derived from an extension of the and conditions in Johansen (1996). It is first observed that these conditions not only characterize, as it is well known, the size (1 or 2) of the largest Jordan block in the Jordan form of the companion matrix but more generally the geometric multiplicities, the algebraic multiplicities and the whole Jordan structure for eigenvalues of index 1 or 2. In the context of the Granger representation theorem, this means that the Johansen rank conditions do more than determine the order of integration of the process. It is then shown that an extension of these conditions leads to the characterization of the Jordan structure of any matrix.

This paper aims to find special cases of some inequalities for numerical radii and spectral radii of a bounded linear operator on a Hil-bert space, we focus on numerical radii inequalities for restricted linear operators on complex Hil-bert spaces for the case of one and two operators, and study the numerical range of an operator K on a complex Hil-bert space H, after that we present some inequalities for numerical radii and spectral radii and studied it to find new results. At the end of this paper we find several inequalities for numerical radii by using the spectral norm, this study is necessary to find other bound for zeros of polynomials and this study is necessary to find new bounds for the zeros of polynomials by a playing the new results to the companion matrix.

In this paper, we define a new quadra polynomial sequence by using Özkoç numbers as the coefficients. Then, we derive some properties for this polynomial sequence by the help of Fibonacci and Pell polynomials. Additionally, we attempt to define the companion matrix of this polynomial sequence.

In this work, we investigate Frobenius companion matrices with entries from the ring of integers modulo a prime number. We introduce a mapping 𝝍 to construct a directed graph, where vertices represent these matrices and edges are defined by 𝝍. Our analysis focuses on the structure and properties of this directed graph, including vertex degrees,cycles, and connected components, in relation to the eigenvalues of the matrices،

In this paper, we study linear codes of length [Formula: see text] that are invariant under an endomorphism [Formula: see text] ([Formula: see text] copies of [Formula: see text]), where [Formula: see text] is a cyclic endomorphism on [Formula: see text]. As each endomorphism can be represented by a matrix, we restrict our study on linear codes that are under a matrix [Formula: see text], where [Formula: see text] is an [Formula: see text] cyclic matrix, called quasi-[Formula: see text]-cyclic codes of index [Formula: see text], and quasi-[Formula: see text]-cyclic codes when [Formula: see text] is the companion matrix of a polynomial [Formula: see text]. We prove a one-to-one correspondence between quasi-[Formula: see text]-cyclic codes ofindex [Formula: see text] and [Formula: see text]-submodules of [Formula: see text], where [Formula: see text] and [Formula: see text] is the minimal polynomial of [Formula: see text]. We prove the BCH-like and Hartmann–Tzeng-like bounds for [Formula: see text]-generator quasi-[Formula: see text]-cyclic codes. In addition, we study the additive structure of quasi-[Formula: see text]-cyclic codes by mapping them to [Formula: see text] via an [Formula: see text]-module morphism. Finally, we provide examples of new quantum codes derived from quasi-[Formula: see text]-cyclic codes as an application of our results.