Cayley-Hamilton Research Articles

An Approach to Isotropic Tensor Functions and Their Derivatives Via Omega Matrix Calculus

In this work we show how to obtain a closed form expression of any isotropic tensor function $F\left (\boldsymbol{A}\right )$ and their associated derivatives with $\boldsymbol{A}$ a second order tensor in a finite dimensional space. Our approach is based on a recent work of the author (SIAM Rev. 62(1):264–280, 2020) extending the Omega operator calculus, originally devised by MacMahon to describe partitions of natural numbers, to the realm of matrix analysis, namely, the Omega Matrix Calculus (OMC). The OMC is conceptually simple and useful in practice. Indeed, we show that the Cayley-Hamilton theorem and an improvement for low-rank second order tensors due to Segercrantz (Am. Math. Mon. 99(1):42–44, 1992), the representation of isotropic tensor functions and their first derivative of Itskov (Proc. R. Soc. Lond. Ser. A, Math. Phys. Eng. Sci. 459(2034):1449–1457, 2003), and Theorem 1.1 of Norris (Q. Appl. Math. 66(4):725–741, 2008) are all special cases of a general Omega expression introduced in this work.

Recurrences for the genus polynomials of linear sequences of graphs

Abstract Given a finite graph H, the n th member Gn of an H-linear sequence is obtained recursively by attaching a disjoint copy of H to the last copy of H in G n−1 by adding edges or identifying vertices, always in the same way. The genus polynomial Γ G (z) of a graph G is the generating function enumerating all orientable embeddings of G by genus. Over the past 30 years, most calculations of genus polynomials Γ Gn (z) for the graphs in a linear family have been obtained by partitioning the embeddings of Gn into types 1, 2, …, k with polynomials Γ G n j $\begin{array}{} \Gamma_{G_n}^j \end{array}$ (z), for j = 1, 2, …, k; from these polynomials, we form a column vector V n ( z ) = [ Γ G n 1 ( z ) , Γ G n 2 ( z ) , … , Γ G n k ( z ) ] t $\begin{array}{} V_n(z) = [\Gamma_{G_n}^1(z), \Gamma_{G_n}^2(z), \ldots, \Gamma_{G_n}^k(z)]^t \end{array}$ that satisfies a recursion Vn (z) = M(z)V n−1(z), where M(z) is a k × k matrix of polynomials in z. In this paper, the Cayley-Hamilton theorem is used to derive a k th degree linear recursion for Γ n (z), allowing us to avoid the partitioning, thereby yielding a reduction from k 2 multiplications of polynomials to k such multiplications. Moreover, that linear recursion can facilitate proofs of real-rootedness and log-concavity of the polynomials. We illustrate with examples.

Relation between the entropy and the linear entropy in the ion–laser interaction

It is shown that, in the interaction of a field and a qubit, there exists a relation between the linear entropy and the von Neumann entropy. The Cayley–Hamilton theorem is used to obtain such relation. In the case we study, the qubit is given by the internal degrees of freedom of an ion trapped in a Paul trap and the field given by its external (vibrational) degrees of freedom.

Low Complexity Signal Detection for Massive-MIMO Systems

Maximum likelihood detection is infeasible in uplink multiuser massive multiple-input and multiple-output (m-MIMO) systems due to the large dimension of the MIMO systems. Accordingly, suboptimal or near-optimal alternatives like linear minimum mean square error (LMMSE) detector and Zero Forcing (ZF) are used. However, the LMMSE and the ZF detectors need matrix inversion, which is computationally costly. We propose two detection schemes for massive MIMO, which compute an approximate inverse based on the Cayley-Hamilton theorem, and have quadratic complexity in the number of users. Simulation results exhibit the similarity of the BER performance of the proposed schemes to that of the ideal ZF or LMMSE.

On metric transformations with a U(1) gauge field

We study metric transformations including not just the field strength tensor of a $U(1)$ gauge field, but also its dual tensor. We first consider an arbitrary symmetric matrix built up with these two tensors in the metric transformation. It turns out the form of transformation reduces to a quite simple form on imposing the parity evenness of the transformed metric and by utilising the Cayley-Hamilton theorem as well as other useful identities. Interestingly, the same form for the transformation was recently argued in the process of seeking for a generic metric transformation but without the inclusion of the dual tensor.

Matrix zeros of polynomials

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 analogue 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.

T-Jordan Canonical Form and T-Drazin Inverse Based on the T-Product

In this paper, we introduce the tensor similar transforation and propose the T-Jordan canonical form and its properties. The concept of T-minimal polynomial and T-characteristic polynomial are raised. As a special case, we present properties when two tensors commutes via the tensor T-product. The Cayley-Hamilton theorem also holds for tensor cases. Then we focus on the tensor decomposition theory. T-polar, T-LU, T-QR and T-Schur decomposition of tensors are obtained. When a F-square tensor is not invertible via the T-product, we give the T-group inverse and T-Drazin inverse which can be viewed as the extension of matrix cases. The expression of T-group and T-Drazin inverse are given by the T-Jordan canonical form. The polynomial form of T-Drazin inverse is also proposed. As an application we raise the T-linear system and get its solution. In the last part, we give the T-core-nilpotent decomposition and show that the T-index and T-Drazin inverse can be given by a limitation process.

Adapting Logic to Physics: The Quantum-Like Eigenlogic Program.

Considering links between logic and physics is important because of the fast development of quantum information technologies in our everyday life. This paper discusses a new method in logic inspired from quantum theory using operators, named Eigenlogic. It expresses logical propositions using linear algebra. Logical functions are represented by operators and logical truth tables correspond to the eigenvalue structure. It extends the possibilities of classical logic by changing the semantics from the Boolean binary alphabet using projection operators to the binary alphabet employing reversible involution operators. Also, many-valued logical operators are synthesized, for whatever alphabet, using operator methods based on Lagrange interpolation and on the Cayley–Hamilton theorem. Considering a superposition of logical input states one gets a fuzzy logic representation where the fuzzy membership function is the quantum probability given by the Born rule. Historical parallels from Boole, Post, Poincaré and Combinatory Logic are presented in relation to probability theory, non-commutative quaternion algebra and Turing machines. An extension to first order logic is proposed inspired by Grover’s algorithm. Eigenlogic is essentially a logic of operators and its truth-table logical semantics is provided by the eigenvalue structure which is shown to be related to the universality of logical quantum gates, a fundamental role being played by non-commutativity and entanglement.

Computational approach for transient behaviour of M/M (a, b)/1 bulk service queueing system with working vacation

A new computational technique is used to evaluate the Transient behaviour of Single Server Bulk Service Queueing System with Working Vacation with arrival rate λ which follows a Poisson process and the service will be in bulk. In this model the server provides two types of services namely normal service and lower service. The normal service time follows an exponential distribution with parameter μ1. The lower service rate follows an exponential distribution with parameter μ2. The vacation time follows an exponential distribution with parameter α. According to Neuts, the server begins service only when a minimum of ‘a’ customers in the waiting room and a maximum service capacity is ‘b’. An infinitesimal generator matrix is formed for all transitions. Time dependent solutions and Steady state solutions are acquired by using Cayley Hamilton theorem. Numerical studies have been done for Time dependent average number of customers in the queue, Transient probabilities of server in vacation and server busy for several values of t, λ, µ1, μ2, α, a and b.

Cayley--Hamilton theorem for $\alpha$-upper level partition of fuzzy square matrices

Cayley--Hamilton theorem for $\alpha$-upper level partition of fuzzy square matrices

On the [formula omitted]-dimensional Cayley–Hamilton theorem and its application to an algebraic decision problem inferred from the [formula omitted] norm analysis of delay systems

We consider a recursion formula for multi-dimensional powers of a finite set of matrices, which can be interpreted as a natural generalization of the celebrated Cayley–Hamilton theorem, and we show how it allows to solve an algebraic decision problem on a semigroup of matrices, which bears similarities to the observability problem of a switched linear system. This problem appears in the computation of the H2 norm of a stable system described by a class of linear time-invariant delay differential equations (DDAEs) with multiple delays. The H2 norm of a DDAE may not be finite even if there are seemingly no direct feedthrough terms. We show that necessary and sufficient conditions for a finite H2 norm consist of an infinite number of linear equations to be satisfied, inducing the algebraic decision problem, and that using the generalized Cayley–Hamilton theorem checking these conditions can be turned into a check of a finite number of equations. We conclude with some comments on the computation of the H2 norm whenever it is finite and by stating an open problem.

Some Counterexamples for Cayley–Hamilton Theorem for Doubly Infinite Matrices

We study the question about generalization of the Cayley–Hamilton theorem. Namely, the problem if for every banded periodic doubly infinite matrix A there exists polynomial Q such that Q(A) is a matrix with constant diagonals. We present a class of doubly infinite matrices for which such polynomial does not exist.

Controllability analysis of nonlinear fractional order differential systems with state delay and non-instantaneous impulsive effects

This manuscript prospects the controllability analysis of non-instantaneous impulsive Volterra type fractional differential systems with state delay. By enroling an appropriate Grammian matrix with the assistance of Laplace transform, the conditions to obtain the necessary and sufficiency for the controllability of non-instantaneous impulsive Volterra-type fractional differential equations are derived using algebraic approach and Cayley-Hamilton theorem. A distinctive approach presents in the manuscript, i have taken non-instantaneous impulses into the fractional order dynamical system with state delay and studied the controllability analysis, since this not exists in the available source of literature. Inclusively, i have provided two illustrative examples with the existence of non-instantaneous impulse into the fractional dynamical system. So this demonstrates the validity and efficacy of our obtained criteria of the main section.

Controllability criteria of fractional differential dynamical systems with non-instantaneous impulses

AbstractThis manuscript prospects the controllability criteria of non-instantaneous impulsive Volterra type fractional differential systems. By enroling an appropriate Gramian matrix that is often defined by the Mittag-Leffler function and with the assistance of Laplace transform, the necessary and sufficiency conditions for the controllability of non-instantaneous impulsive Volterra-type fractional differential equations are derived by using algebraic approach and Cayley–Hamilton theorem. An important feature present in our paper is that we have taken non-instantaneous impulses into the fractional order dynamical system and studied the controllability analysis, since this do not exist in the available source of literature. Inclusively, we have provided two illustrative examples with the existence of non-instantaneous impulse into the fractional dynamical system. So this demonstrates the validity and efficacy of our obtained criteria of the main section.

H∞ Filtering for a Class of Singular Time-delay Systems

This paper describes the H∞ filtering problem for a class of nonlinear singular time-delay systems. A sufficient condition for the existence of H∞ filter is given. Based on the Cayley-Hamilton Theorem and Lyapunov stability theory, the H∞ filter is designed such that the filtering error is asymptotically stable and satisfies the prescribed H∞ performance. In terms of linear matrix inequality, a sufficient condition for the solvability of the addressed problem is obtained. Finally, some numerical examples are employed to demonstrate the effectiveness of the proposed approaches.

An Extension of the Cayley–Hamilton Theorem to Different Orders Fractional Linear Systems and Its Application to Electrical Circuits

The classical Cayley-Hamilton theorem is extended to different orders fractional continuous-time linear systems and to fractional electrical circuits.

Modeling Crane-Induced Ship Motion Using the Moving Frame Method

A decline in oil-related revenues challenges Norway to focus on new types of offshore installations. Often, ship-mounted crane systems transfer cargo or crew onto offshore installations such as floating windmills. This project analyzes the motion of a ship induced by an onboard crane in operation using a new theoretical approach to dynamics: the moving frame method (MFM). The MFM draws upon Lie group theory and Cartan's moving frames. This, together with a compact notation from geometrical physics, makes it possible to extract the equations of motion, expeditiously. While others have applied aspects of these mathematical tools, the notation presented here brings these methods together; it is accessible, programmable, and simple. In the MFM, the notation for multibody dynamics and single body dynamics is the same; for two-dimensional (2D) and three-dimensional (3D), the same. Most importantly, this paper presents a restricted variation of the angular velocity to use in Hamilton's principle. This work accounts for the masses and geometry of all components, interactive motor couples and prepares for buoyancy forces and added mass. This research solves the equations numerically using a relatively simple numerical integration scheme. Then, the Cayley–Hamilton theorem and Rodriguez's formula reconstruct the rotation matrix for the ship. Furthermore, this work displays the rotating ship in 3D, viewable on mobile devices. This paper presents the results qualitatively as a 3D simulation. This research demonstrates that the MFM is suitable for the analysis of “smart ships,” as the next step in this work.

Schwarzschild-Tangherlini black holes in five-dimensional massive (bi-)gravity

We will present main results of our recent investigations on the existence of the Schwarzschild-Tangherlini black holes in a five-dimensional (nonlinear) massive gravity as well as in its dynamical extension, a five-dimensional massive bi-gravity. In particular, we will show how to use the well-known Cayley-Hamilton theorem to construct five-and higher dimensional massive graviton terms. Then, we will present the proof of the existence of the Schwarzschild-Tangherlini black holes in the five-dimensional massive (bi-)gravity.

On three-loop RGE for the Higgs sector of 2HDM

We discuss renormalization group equations (RGE) for the parameters of the Higgs sector in general Two-Higgs-Doublet Model (2HDM). We present the three-loop results but consider only contributions due to self-couplings of the Higgs doublets. We study the structure of RGE and express beta-functions in terms of reparametrization invariants with respect to higgs-basis rotations. The Cayley-Hamilton theorem is utilized to reduce both the number of independent tensor structures in matrix RGE and the number of invariants to a minimal set. As a by-product of our calculation we discovered that two-loop RGE of the scalar sector in general QFT with multiple higgses were not properly implemented in a number of public packages. The latter give rise to a wrong result when mixing in the scalar sector is allowed.

Modelling Subsea ROV robotics using the moving frame method

This paper introduces a new method (notation and theory) in engineering dynamics. The goal of this paper is to apply this new method to the analysis of multi-body systems, specifically for the analysis of the motion of a Remotely Operated Vehicle (ROV). Norway conducts operations on a variety of structures in the North Sea. Research that can improve the efficiency of these operations are of high interest. This paper researches the motion of an ROV induced by the motion of the robotic manipulators, motor torques, and buoyancy. This research also introduces a new method in engineering dynamics: the Moving Frame Method (MFM). The MFM draws upon Lie group theory and Cartan’s Moving Frames. This, together with a compact notation from geometrical physics, makes it possible to extract the equations of motion, expeditiously. While others have applied pieces and aspects of these mathematical tools, the notation presented here brings them together; it is accessible, programmable and simple. The notation for multi-body dynamics and single rigid body dynamics is the same. Most important, this paper presents a restricted variation of the angular velocity to use in Hamilton’s Principle to extract the equations of motion. This research solves the equations using a relatively simple numerical integration scheme. The Cayley-Hamilton theorem and Rodriguez’s formula reconstructs the rotation matrix for the ROV. This work displays the rotating ship in 3D, viewable on mobile devices. This paper presents the results qualitatively as a 3D simulation. This research demonstrates that the MFM is suitable for the analysis of “smart ROVs” as the next step in this work.