Cayley-Hamilton Research Articles

On the positivity and asymptotic stability of the fractional 2D hybrid linear Roesser model

The aim of this paper is to introduce a new 2D fractional hybrid linear system described by the Roesser model with the conformable fractional derivative in the continuous time variables and the fractional difference in the discrete part. The solution of the considered model is derived. The positivity conditions are then extracted and the Cayley–Hamilton theorem is extended. A new necessary and sufficient conditions for the asymptotic stability of the proposed positive fractional 2D hybrid linear model are then established. Finally, some numerical examples are presented to show the applicability and the effectiveness of the presented methods.

Analogy and Generalization as a Driving Force of Learning Mathematics – the Case of a Matrix Analog of a Zero of a Polynomial

The concepts of polynomials and matrices essentially expand and enhance the elementary arithmetic of numbers. Once introduced, polynomials and matrices open up new interesting mathematical challenges which extend to new fields of mathematical explorations within university mathematics. We present an aspect of a rather elementary exploration of polynomials and matrices, which offers a new perspective and an interesting matrix analog to the concept of a zero of a polynomial. The discussion offers an opportunity for better comprehension of the fundamental concepts of polynomials and matrices. As an application, we are led to the meaningful questions of linear algebra and to an easy proof of the otherwise advanced and abstract Cayley–Hamilton theorem.

Exact closed forms for the transmittance of electromagnetic waves in one-dimensional anisotropic periodic media

In this work, we obtain closed expressions for the transfer matrix and the transmittance of electromagnetic waves propagating in finite one-dimensional anisotropic periodic stratified media with an arbitrary number of cells. By invoking the Cayley–Hamilton theorem on the transfer matrix for the electromagnetic field in a periodic stratified media formed by N cells, we obtain a fourth-order recursive relation for the matrix coefficients that defines the so-called Tetranacci polynomials (TPs). In the symmetric case, corresponding to a unit-cell transfer matrix with a characteristic polynomial where the coefficients of the linear and cubic terms are equal, closed expressions for the solutions to the recursive relation, known as symmetric TPs, have recently been derived, allowing us to write the transfer matrix and transmittance in a closed form. We show as sufficient conditions that the 4×4 differential propagation matrix of each layer in the binary unit cell, Δ , a) has eigenvalues of the form ±p1 , ±p2 , with p1≠p2 , and b) its off-diagonal 2×2 block matrices possess the same symmetric structure in both layers. Otherwise, the recursive relations are still solvable for any 4×4 matrix and provide an algorithm to compute the Nth power of the transfer matrix without carrying out explicitly the matrix multiplication of N matrices. We obtain analytical expressions for the dispersion relation and transmittance, in closed form, for two finite periodic systems: the first one consists of two birefringent uniaxial media with their optical axis perpendicular to the z-axis, and the second consists of two isotropic media subject to an external magnetic field oriented along the z-axis and exhibiting the Faraday effect. Our formalism applies also to lossy media, magnetic anisotropy or optical activity.

Relative controllability of linear state-delay fractional systems

AbstractIn this paper, our focus is on exploring the relative controllability of systems governed by linear fractional differential equations incorporating state delay. We introduce a novel counterpart to the Cayley-Hamilton theorem. Leveraging a delayed perturbation of the Mittag-Leffler function, along with a determining function and an analog of the Cayley-Hamilton theorem, we establish an algebraic Kalman-type rank criterion for assessing the relative controllability of fractional differential equations with state delay. Moreover, we articulate necessary and sufficient conditions for relative controllability criteria concerning linear fractional time-delay systems, expressed in terms of a new $$\alpha $$ α -Gramian matrix and define a control which transfer the system from any initial state to any final state within a given time. The theoretical findings are exemplified through the presentation of illustrative examples.

Exact closed forms for the transfer matrix of free oscillations in finite periodic Timoshenko–Ehrenfest beams

In this work, we obtain closed-form expressions for the transfer matrix of free oscillations in finite periodic Timoshenko–Ehrenfest beams with an arbitrary number of cells. By invoking the Cayley–Hamilton theorem on the transfer matrix for free oscillations of a beam composed of N cells, we obtain a fourth-order recursive relation for the matrix coefficients, which defines the so-called Tetranacci Polynomials. Such recursive relation provides an algorithm to compute the Nth power of the transfer matrix, avoiding the matrix product of the N matrices. Furthermore, in the symmetric case of free oscillations in finite periodic Timoshenko-Ehrenfest beams, closed-form expressions for the solutions to the recursive relation have recently been derived, which we use to write the transfer matrix of a finite beam composed of N cells in a closed form. We find a good agreement between the natural frequencies calculated with the obtained expressions and finite element simulations. These expressions are very useful for studying the interaction of evanescent oscillations and find application, for instance, in the development of phononic topological insulators. Our formalism can be applied to waves propagating in finite periodic layers described by a 4 × 4-transfer matrix, such as electromagnetic waves in anisotropic optical media.

Controllability of Fractional Complex Networks

Controllability is a fundamental issue in the field of fractional complex network control, yet it has not received adequate attention in the past. This paper is dedicated to exploring the controllability of complex networks involving the Caputo fractional derivative. By utilizing the Cayley–Hamilton theorem and Laplace transformation, a concise proof is given to determine the controllability of linear fractional complex networks. Subsequently, leveraging the Schauder Fixed-Point theorem, controllability Gramian matrix, and fractional calculus theory, we derive controllability conditions for nonlinear fractional complex networks with a weighted adjacency matrix and Laplacian matrix, respectively. Finally, a numerical method for the controllability of fractional complex networks is obtained using Matlab (2021a)/Simulink (2021a). Three examples are provided to illustrate the theoretical results.

Extending nonstandard finite difference scheme rules to systems of nonlinear ODEs with constant coefficients

In this paper, we present a reformulation of Mickens' rules for nonstandard finite difference (NSFD) schemes to adapt them to systems of ODEs. We want to compare several methods within the class of NSFD schemes. These methods are expressed as exponential integrators which treat exactly the linear part of the equations. While the classical methods address each equation of a system separately, we propose here to address them as a whole, which involves a reformulation of Mickens' rules. This leads to enhance the results compared to the classical case, even if approximations of the matrix exponentials are used. Precisely, using Cayley–Hamilton theorem we obtain two correction factors, one of which is linked to the size of the system and the other to the nonlinear term, we are thus able to use test cases which show the impact of one or another corrector. We prove the consistency and the zero-stability of the method in the nonlinear case to conclude to the convergence of the method.

Some applications of trace of powers of matrices to graph theory, algebra and combinatorics

In this paper, using Cayley Hamilton Theorem, we exhibit a combinatorial formula for Tr(An), (n ∈ ℤ, A ∈M3 (K)), over the field K(= ℝorℂ)using Tr (A), det(A) and sum of principal minors of A. Further, a formula for ∑nk=0 Tr(Ak) is derived. For specific values of Tr(A) and det(A), the formula of Tr(An) is highly simplified. We exhibit some applications of our results in Graph Theory, Algebra and Combinatorics. We have verified our results on various matrices using Scilab and found that Scilabfails tremendously because of the truncation. But our formulae give exact result.

Family of right conoid hypersurfaces with light-like axis in Minkowski four-space

<abstract><p>In the realm of the four-dimensional Minkowski space $ \mathbb{L}^{4} $, the focus is on hypersurfaces classified as right conoids and defined by light-like axes. Matrices associated with the fundamental form, Gauss map, and shape operator, all specifically tailored for these hypersurfaces, are currently undergoing computation. The intrinsic curvatures of these hypersurfaces are determined using the Cayley-Hamilton theorem. The conditions of minimality are addressed by the analysis. The Laplace-Beltrami operator for such hypersurfaces is computed, accompanied by illustrative examples aimed at fostering a more profound understanding of the involved mathematical principles. Additionally, scrutiny is applied to the umbilical condition, and the introduction of the Willmore functional for these hypersurfaces is presented.</p></abstract>

On Certain Algebraic Properties of Symbolic 3-Plithogenic Real Square Matrices

The main objective of this article is to study the inverse of invertible symbolic 3-plithogenic real square matrices using the concept of adjoints and characteristic polynomials. Also, the symbolic 3-plithogenic version of Cayley-Hamilton theorem was proved and provided enough examples to enhance understanding.

Constructing Multidimensional Difference Equations From a State-Space Representation Using the Generalized Cayley-Hamilton Theorem

Constructing Multidimensional Difference Equations From a State-Space Representation Using the Generalized Cayley-Hamilton Theorem

RIGHT CONOID HYPERSURFACES IN FOUR-SPACE

The right conoid hypersurfaces in the four-dimensional Euclidean space $\mathbb{E}^{4}$ are introduced. The matrices corresponding to the fundamental form, Gauss map, and shape operator of these hypersurfaces are calculated. By utilizing the Cayley--Hamilton theorem, the curvatures of these specific hypersurfaces are determined. Furthermore, the conditions for minimality are presented. Additionally, the Laplace--Beltrami operator of this family is computed, and some examples are provided.

The Cayley–Hamilton Theorem and Resolvent Representation

The Cayley–Hamilton Theorem and Resolvent Representation

Twisted Hypersurfaces in Euclidean 5-Space

The twisted hypersurfaces x with the (0,0,0,0,1) rotating axis in five-dimensional Euclidean space E5 is considered. The fundamental forms, the Gauss map, and the shape operator of x are calculated. In E5, describing the curvatures by using the Cayley–Hamilton theorem, the curvatures of hypersurfaces x are obtained. The solutions of differential equations of the curvatures of the hypersurfaces are open problems. The umbilically and minimality conditions to the curvatures of x are determined. Additionally, the Laplace–Beltrami operator relation of x is given.

Dynamical evolution of three-state spin-1 non-Hermitian system with Cayley–Hamilton theorem

In this paper, we study the time evolution of three-states spin-1 non-Hermitian systems. The state vector of the wave function for Schrödinger equation is obtained analytically by using the Cayley–Hamilton theorem. We note that the behavior of the non-Hermitian three-states system corresponds to the Hermitian three-state system. Furthermore, the fluctuation between three-state spin-1 is periodically oscillating when the eigenvalues of the non-Hermitian system are real. Moreover, if the eigenvalues are imaginary, we explore that from our analysis and analytical solution that the probabilities’ amplitudes always grow with time for all physical parameters in the three-state, i.e. including the [Formula: see text]-symmetry non-Hermitian system. It means that the Schrödinger equation becomes the growth exponential.

A Hypersurfaces of Revolution Family in the Five-Dimensional Pseudo-Euclidean Space E25

We present a family of hypersurfaces of revolution distinguished by four parameters in the five-dimensional pseudo-Euclidean space E25. The matrices corresponding to the fundamental form, Gauss map, and shape operator of this family are computed. By utilizing the Cayley–Hamilton theorem, we determine the curvatures of the specific family. Furthermore, we establish the criteria for maximality within this framework. Additionally, we reveal the relationship between the Laplace–Beltrami operator of the family and a 5×5 matrix.

Differential Geometry of the Family of Helical Hypersurfaces with a Light-like Axis in Minkowski Spacetime L4

We investigate the class of helical hypersurfaces parametrized by x=x(u,v,w), characterized by a light-like axis in Minkowski spacetime L4. We determine the matrices that represent the fundamental forms, Gauss map, and shape operator of x. Furthermore, employing the Cayley–Hamilton theorem, we compute the curvatures associated with x. We explore the conditions under which the curvatures of x possess the property of being umbilical. Moreover, we provide the Laplace–Beltrami operator for the family of helical hypersurfaces with a light-like axis in L4.

Minimally modified gravity with auxiliary constraints formalism

We investigate the possibility of reducing the number of degrees of freedom (d.o.f.) starting from generic metric theories of gravity by introducing multiple auxiliary constraints (ACs), under the restriction of retaining spatial covariance as a gauge symmetry. Arbitrary numbers of scalar-, vector- and tensor-type ACs are considered a priori, yet we find that no vector- and tensor-type constraints should be introduced, and that scalar-type ACs should be no more than four for the purpose of constructing minimally modified gravity (MMG) theories which propagate only two tensorial d.o.f., like general relativity (GR). Through a detailed Hamiltonian analysis, we exhaust all the possible classifications of ACs and find out the corresponding minimalizing and symmetrizing conditions for obtaining the MMG theories. In particular, no condition is required in the case of four ACs, hence in this case the theory can couple with matter consistently and naturally. To illustrate our formalism, we build a concrete model for this specific case by using the Cayley-Hamilton theorem and derive the dispersion relation of the gravitational waves, which is subject to constraints from the observations.

Adjoint Majorana QCD2 at finite N

The mass spectrum of 1 + 1-dimensional SU(N) gauge theory coupled to a Majorana fermion in the adjoint representation has been studied in the large N limit using Light-Cone Quantization. Here we extend this approach to theories with small values of N, exhibiting explicit results for N = 2, 3, and 4. In the context of Discretized Light-Cone Quantization, we develop a procedure based on the Cayley-Hamilton theorem for determining which states of the large N theory become null at finite N. For the low-lying bound states, we find that the squared masses divided by g2N, where g is the gauge coupling, have very weak dependence on N. The coefficients of the 1/N2 corrections to their large N values are surprisingly small. When the adjoint fermion is massless, we observe exact degeneracies that we explain in terms of a Kac-Moody algebra construction and charge conjugation symmetry. When the squared mass of the adjoint fermion is tuned to g2N/π, we find evidence that the spectrum exhibits boson-fermion degeneracies, in agreement with the supersymmetry of the model at any value of N.

Generalized Helical Hypersurface with Space-like Axis in Minkowski 5-Space

We introduce the generalized helical hypersurface having a space-like axis in five-dimensional Minkowski space. We compute the first and second fundamental form matrices, Gauss map, and shape operator matrix of the hypersurface. Additionally, we compute the curvatures of the hypersurface by using the Cayley–Hamilton theorem. Moreover, we give some relations for the mean and the Gauss–Kronecker curvatures of the hypersurface. Finally, we obtain the Laplace–Beltrami operator of the hypersurface.