Triangular Matrix Research Articles

Nonlinear multi-image optical authentication based on QR decomposition and Kramer-Kronig relations

Abstract In this paper, a new nonlinear optical multi-image authentication scheme is proposed based on Kramers-Kronig digital holography and orthogonal triangular decomposition or QR decomposition. Here, the complex light field carrying the information of multiple images is modulated by random phase masks and propagated at certain distance. Afterwards, the QR decomposition is applied to the complex wavefront to generate the private keys and to add the non-linearity in the scheme. Next, the product of orthogonal matrix and upper triangular matrix is processed further. The obtained output is modulated by different phase masks and interfered with reference beam to record the encrypted image. For decryption, the Kramer-Kronig relation is utilized to extract the plaintext images directly with only the positive frequency part. A series of numerical simulations are conducted to validate the efficacy and robustness of proposed image authentication scheme.

Design and implementation of a nonlinear observer for a novel Quadrotor UAV: the HELIPLANE

This study presents the design and dynamic modeling of a new type of quadrotor UAV, referred to as the Heliplane, which integrates the vertical take-off and landing (VTOL) capabilities of a multirotor with the horizontal flight efficiency of a fixed-wing aircraft. The modeling process involved creating a dynamic representation of the Heliplane, with special attention to the determination of its aerodynamic coefficients, which were calculated using SolidWorks Flow Simulation. The aerodynamic forces, including lift and drag, were analyzed based on the flow characteristics around the Heliplane's structure. To further enhance the system's performance, a nonlinear observer based on a sliding mode approach was synthesized. This observer was designed in a triangular form to ensure accurate state estimation and robust control, even in the presence of model uncertainties. The observer's key advantage lies in its finite-time convergence, which guarantees quick and precise tracking of the Heliplane's state variables. Simulation results demonstrated the effectiveness of the proposed observer in estimating the rotational and translational velocities with minimal deviation. The sliding mode observer's performance suggests its suitability for control and fault detection tasks in UAV applications, particularly in challenging operational environments.

Finite orbits of the braid group actions

We study the finite orbits of the braid group Bn action on the space of n×n upper-triangular matrices with 1's along the diagonal. On one hand, we give a necessary condition for a matrix M to be in a finite orbit; on the other hand, we classify and provide lengths of finite orbits in low-dimensional matrices and some other important cases. As the finite orbits on 3×3 matrix were crucial to finding the algebraic solutions of the sixth Painlevé equation, we hope the finite orbits on generic n×n matrices to be useful to finding solutions of higher order Painleve type differential equations.

Randomized methods for computing joint eigenvalues, with applications to multiparameter eigenvalue problems and root finding

AbstractIt is well known that a family of $${n}\times {n}$$ n × n commuting matrices can be simultaneously triangularized by a unitary similarity transformation. The diagonal entries of the triangular matrices define the n joint eigenvalues of the family. In this work, we consider the task of numerically computing approximations to such joint eigenvalues for a family of (nearly) commuting matrices. This task arises, for example, in solvers for multiparameter eigenvalue problems and systems of multivariate polynomials, which are our main motivations. We propose and analyze a simple approach that computes eigenvalues as one-sided or two-sided Rayleigh quotients from eigenvectors of a random linear combination of the matrices in the family. We provide some analysis and numerous numerical examples, showing that such randomized approaches can compute semisimple joint eigenvalues accurately and lead to improved performance of existing solvers.

Rings Whose Invertible Elements Are Weakly Nil-Clean

Westudythoseringsinwhichallinvertible elements are weakly nil-clean, calling them UWNC rings. This somewhat extends results due to Karimi-Mansoub et al. in Contemp. Math. (2018), where rings in which all invertible elements are nil-clean were considered abbreviating them as UNC rings. Specifically, our main achievements are that the triangular matrix ring Tn(R) over a ring R is UWNC precisely when R is UNC.Besides, the notions UWNC and UNCdocoincide when2 ∈ J(R). Wealso describe UWNC2-primal rings R by proving that R is a ring with J(R) = Nil(R) such that U(R) = ±1+Nil(R). In particular, the polynomial ring R[x] over some arbitrary variable x is UWNC exactly when R is UWNC. Likewise, we furthermore apply the obtained results to group rings showing that if G is a locally finite p-group and R is a UWNC ring such that the prime p is a nilpotent in R, then RG is too a UWNC ring. Some other relevant assertions are proved in the present direction as well.

A user selection algorithm for downlink MU-MIMO systems using product of singular values of effective channels

User selection plays a crucial role when aiming to improve channel throughput in multiuser multiple- input multiple-output downlink systems with block diagonalization linear precoding. While an optimal group of users can be found by exhaustively searching all possible combinations of users, the process becomes prohibitively complex when there is a large number of users. In this paper, we propose a greedy suboptimal user selection algorithm, which iteratively selects a user to maximize the product of the singular values of the effective channels from a set of unselected users. The proposed algorithm is based on the iterative precoder design method, whereby interuser interference is perfectly cancelled with lower complexity than with the singular value decomposition-based precoding technique. Moreover, to avoid frequent use of singular value decomposition, the proposed algorithm applies QR decomposition to the selected channel matrices in order to obtain singular values, which are equivalent to the product of diagonal elements in the upper triangular matrix. Simulation results show that, under a high SNR regime, the proposed algorithm outperforms several other greedy algorithms and can provide the same performance as a capacity-based algorithm for both correlated and uncorrelated channels with significantly reduced complexity.

A Study on Matrix Sequence with Generalized Guglielmo Numbers Components

We know that the diverse applications of matrix sequences in fields such as physics, engineering, architecture, nature, and art. Numerous authors have delved into the study of these matrix sequences in existing literature. In this study, we define and investigate the generalized Guglielmo matrix sequence. For this aim we explore four specific cases of that sequence that are called triangular matrix sequences, Lucas-triangular matrix sequences, oblong matrix sequences, and pentagonal matrix sequences. Next, we present Binet's formulas, generating functions, the summation formulas and some elementary identities for these sequences. Moreover, we give some identities and matrices related with these sequences. Furthermore, we show that there always exist some relationship between generalized triangular, Lucas-triangular, oblong and pentagonal matrix sequences.

Sobre matrizes pentadiagonais não estritamente diagonais dominantes

Based on Crout's method, we will present, in this work, new non singularity criteria and sufficient conditions for existence of the LU factorization, for non strictly diagonally dominant pentadiagonal matrices. Crout's method is a recursive process of n stages that obtains the factorization A = LU of a pentadiagonal matrix of order n. In this recursive process of obtaining both the lower triangular matrix L and the upper triangular matrix U, the parameters alpha_i, 1 <= i <= n, must be non-zero to ensure that det(A) neq 0 and A = LU. Crout's recursive method is replaced by the analysis of sufficient conditions that can be verified simultaneously with low computational cost.

Rota–Baxter Operators of Weight Zero on Upper-Triangular Matrices of Order Three

Rota–Baxter Operators of Weight Zero on Upper-Triangular Matrices of Order Three

Generalised Lat-Igusa-Todorov Algebras and Morita Contexts

AbstractIn this paper we define (special) GLIT classes and (special) GLIT algebras. We prove that GLIT algebras, which generalise Lat-Igusa-Todorov algebras, satisfy the finitistic dimension conjecture and give several properties and examples. In addition we show that special GLIT algebras are exactly those that have finite finitistic dimension. Lastly we study Morita algebras arising from a Morita context and give conditions for them to be (special) GLIT in terms of the algebras and bimodules used in their definition. As a consequence we obtain simple conditions for a triangular matrix algebra to be (special) GLIT and also prove that the tensor product of a GLIT $$\mathbb {K}$$ K -algebra with a path algebra of a finite quiver without oriented cycles is GLIT.

Dimension of Planar Non-conformal Attractors with Triangular Derivative Matrices

We study the dimension of the attractor and quasi-Bernoulli measures of parametrized families of iterated function systems of non-conformal and non-affine maps. We introduce a transversality condition under which, relying on a weak Ledrappier-Young formula, we show that the dimensions equal to the root of the subadditive pressure and the Lyapunov dimension, respectively, for almost every choice of parameters. We also exhibit concrete examples satisfying the transversality condition with respect to the translation parameters.

Computing the Network’s Equilibrium Point at the Fault Clearing Instant in Transient Stability Studies

This paper proposes an approach for computing the network’s equilibrium point related to the fault clearing time in transient stability studies. The computation of this point is not a trivial task, particularly when the algebraic network’s equations are expressed in the power balance form. A natural attempt to solve this problem is using Newton’s method. However, convergence issues are found because of the lack of a general strategy for initializing nodal voltages at the clearing time. This problem has not been widely discussed in the existing literature and, therefore, is comprehensively analyzed in this paper. Furthermore, the paper proposes the use of a network’s model based on current injections and an extended admittance matrix to overcome the problem. This model is efficiently solved via the fixed-point iteration method, which involves factorization of the extended admittance matrix into the product of a lower triangular matrix [L] and an upper triangular matrix [U]. This solution executes a just once and only forward–backward substitution during the iterative solution process. Case studies clearly demonstrate the proposal’s effectiveness in computing the equilibrium point in operating conditions where Newton’s method fails to converge.

Gradings on the algebra of triangular matrices as a Lie algebra: Revisited

We investigate the group gradings on the algebras of upper triangular matrices over an arbitrary field, viewed as Lie algebras. Classification results were obtained in 2017 by the same authors when the base field has characteristic different from 2. In this paper we provide streamlined proofs of these results. Moreover we present a complete classification of isomorphism classes of the group gradings on these algebras over an arbitrary field. Recall that two graded Lie algebras L1 and L2 are practically-isomorphic if there exists an (ungraded) algebra isomorphism L1→L2 that induces a graded-algebra isomorphism L1/z(L1)→L2/z(L2). We provide a classification of the practically-isomorphism classes of the group gradings on the Lie algebra of upper triangular matrices. The latter classification is a better alternative way to consider these gradings up to being essentially the same object. Finally, we investigate in details the case where the characteristic of the base field is 2, a topic that was neglected in previous works.

Characterizing Jordan embeddings between block upper-triangular subalgebras via preserving properties

Let Mn be the algebra of n×n complex matrices. We consider arbitrary subalgebras A of Mn which contain the algebra of all upper-triangular matrices, and their Jordan embeddings. We first describe Jordan embeddings ϕ:A→Mn as maps of the formϕ(X)=TXT−1orϕ(X)=TXtT−1, where T∈Mn is an invertible matrix, and then we obtain a simple criterion of when one block upper-triangular subalgebra Jordan-embeds into another (and in that case we describe the form of such embeddings). As a main result, we characterize Jordan embeddings ϕ:A→Mn (when n≥3) as continuous injective maps which preserve commutativity and spectrum. We show by counterexamples that all these assumptions are indispensable (unless A=Mn when injectivity is superfluous).

A uniformly convergent analysis for multiple scale parabolic singularly perturbed convection-diffusion coupled systems: Optimal accuracy with less computational time

This study addresses time-dependent multiple-scale reaction-convection-diffusion initial boundary value systems characterized by strong coupling in the reaction matrix and weak coupling in the convection terms for a locally optimal accurate solution. The discrete problem, which typically loses its tridiagonal structure, expands its bandwidth to four in such coupled systems, resulting in a substantial computational load. Our objective is to mitigate this computational burden through a splitting approach that transforms the non-tridiagonal matrix into a tridiagonal form while maintaining the consistency, local optimal accuracy in space, and stability of the numerical scheme. We employ equidistributed non-uniform grids, guided by a carefully chosen monitor function, to approximate the continuous space domain. The discretization strategy targets local optimal linear accuracy across space and time on the domain's interior points. In addition, we have also provided the global convergence analysis of the present splitting approach, mathematically. The mathematical evidence is also obtained from the numerical experiments by comparing the splitting approach (either diagonal or triangular forms) of the reaction matrix to its coupled form. The results strongly confirm the effectiveness of this approach in delivering uniform linear accuracy, based on the present problem discretizations while significantly reducing the computational costs.

Some properties of *-weak rings with involution.

Throughout this paper, we introduced the concept of *-weak (*-IFP, quasi-*-IFP, and *-reversible) *-rings also study properties and the basic structure of *-weak *-rings, giving some of the results. Moreover, we will clarify the conditions for the *-weak *rings to extend into subrings with the involution of the ring, at the upper triangular matrices , with the same diagonal).

Strong commutativity preserving maps on rank k strictly triangular matrices

Let n ⩾ 2 be an integer and let N n ( F ) be the algebra of n × n strictly upper triangular matrices over a field F with center Z ( N n ( F ) ) . In this paper, we obtain a structural characterization of additive maps ψ : N n ( F ) → N n ( F ) satisfying [ ψ ( A ) , ψ ( B ) ] − [ A , B ] ∈ Z ( N n ( F ) ) for all A , B ∈ N n ( F ) . We deduce from this result a characterization of strong commutativity preserving additive maps on rank k strictly upper triangular matrices over F , where 1 ⩽ k ⩽ n − 1 is a fixed integer such that k ≠ n − 1 when F is the Galois field of two elements.

The role of semicentral idempotents in triangular matrix rings

The circle composition e 1 ° e 2 = e 1 + e 2 − e 1 e 2 is well-known from the seminal book of Jacobson. A part of our motivation aims to find non-orthogonal idempotents e 1 and e 2 such that e 1 ° e 2 is an idempotent. Such idempotents are the products r l where r is a right semicentral and l is a left semicentral idempotent of the ring of upper triangular matrices over a ring. We prove that in the semigroup of upper triangular matrices over ring with only trivial idempotents every idempotent matrix can be represented as a circle composition of products of the type r l .

On P-Stable Functions

Let h_1, h_2 be two analytic functions defined in the open unit disc Δ:={z∈C:|z|<1} which are normalized by the condition h_1 (0)=1=h_2 (0). Then h_1 is P-stable with respect to h_2, whenever (P_n (h_1,z))/(h_1 (z))≺1/(h_2 (z)) (z∈Δ),holds for all n∈N. Here ’≺’ stands for subordination and P_n (h,z)=P_n (z)*h(z) where P_n (z) denote the n-degree polynomial induced by the (n+1)th row entities in an admissible lower triangular matrix. The main purpose of this article is to prove that the function ((Az+1)/(Bz+1))^δ is P-stable with respect to (Bz+1)^(-δ), for δ∈(0,1] and -1≤B<A≤0 but not P-stable with respect to itself, when -1≤B<A<0 and δ∈(0,1]. As an application, considered different admissible lower triangular matrices to derive various results related on stability.

Certain homological properties of triangular matrix rings

In this paper, we study the (Gorenstein) flat dimension of modules over triangular matrix rings, and give estimations of the (Gorenstein) weak global dimension and the supremum of flat lengths of injective modules over triangular matrix rings. As applications, some new Ding–Chen rings are constructed.