Search

Let $\mathcal X$ be a Banach space and let T be a bounded linear operator on $\mathcal {X}$ . We denote by S(T) the set of all complex $\lambda \in \mathcal {C}$ such that T does not have the single-valued extension property. In this paper it is shown that if MC is a 2 × 2 upper triangular operator matrix acting on the Banach space $\mathcal {X} \oplus \mathcal {Y}$ , then the passage from σLD(A) ∪ σLD(B) to σLD(MC) is accomplished by removing certain open subsets of σd(A) ∩ σLD(B) from the former, that is, there is the equality σLD(A) ∪ σLD(B) = σLD(MC) ∪ℵ, where ℵ is the union of certain of the holes in σLD(MC) which happen to be subsets of σd(A) ∩ σLD(B). Generalized Weyl’s theorem and generalized Browder’s theorem are liable to fail for 2 × 2 operator matrices. In this paper, we also explore how generalized Weyl’ theorem, generalized Browder’s theorem, generalized a-Weyl’s theorem and generalized a-Browder’s theorem survive for 2 × 2 upper triangular operator matrices on the Banach space.

In recent years, model order reduction (MOR) has been interested in more and more scientists. A lot of MOR algorithms have been introduced by many different approaches, among which preserving the dominant poles of the original system and Hankel singular values of the original system in order reduction system are appropriate approaches with many advantages. The article introduces a new MOR algorithm applied for stable and unstable linear systems, based on the idea of preserving the dominant poles of the original system during the order reduction. The algorithm will switch matrix-A of the original high-order system into the upper triangular matrix, then arrange the poles under the measure of dominance- H, H2, and mixed points on the main diagonal of upper triangular matrix-A, in order to attain a small error order reduction and preserve dominant poles simultaneously. The effectiveness of the new algorithm is illustrated through the order reduction of the high-order controller. Simulation results have proven the correctness of the algorithm.

Let H and K be infinite-dimensional separable Hilbert spaces, and denote by MC:=AC0B the upper triangular linear relation matrix with unknown C in H⊕K. For given relations A and B, some necessary a...

In this paper, we study the cotorsion dimensions and cotorsion triples in the recollements of abelian categories. The main results are that recollements induce new (resp. complete hereditary) cotorsion triples from the middle category and that the cotorsion dimensions are bounded under certain conditions. As an application, the cotorsion triples in the recollements of module categories with respect to triangular matrix algebras are recovered.

A ring R is called right Ikeda–Nakayama ring (right IN-ring) if for any two right ideals I, J of R, $$l(I)+l(J)=l(I \cap J)$$ . In this paper, we introduce the concept of Essential Ikeda–Nakayama rings (EIN-rings) as a generalization of right IN-rings. This class of rings includes semiprime rings. We prove that for a left nonsingular EIN-ring R, closed ideals of R are right annihilator in R. We show that the class of EIN-rings is closed under direct product and upper triangular matrix rings. Furthermore, a ring R is an Armendariz EIN-ring if and only if R[x] is an EIN-ring.

In this paper we establish eigenvector delocalization and bulk universality for L\'{e}vy matrices, which are real, symmetric, $N \times N$ random matrices $\textbf{H}$ whose upper triangular entries are independent, identically distributed $\alpha$-stable laws. First, if $\alpha \in (1, 2)$ and $E \in \mathbb{R}$ is any energy bounded away from $0$, we show that every eigenvector of $\textbf{H}$ corresponding to an eigenvalue near $E$ is completely delocalized and that the local spectral statistics of $\textbf{H}$ around $E$ converge to those of the Gaussian Orthogonal Ensemble (GOE) as $N$ tends to $\infty$. Second, we show for almost all $\alpha \in (0, 2)$, there exists a constant $c(\alpha) > 0$ such that the same statements hold if $|E| < c (\alpha)$., Comment: 76 pages, 1 figure. Version 2: Minor changes in the introduction; Version 3: More detailed exposition, updated references, and a new figure

We introduce the notion of $$J$$ -Hermitianity of a matrix, as a generalization of Hermitianity, and, more generally, of closure by $$J$$ -Hermitianity of a set of matrices. Many well known algebras, like upper and lower triangular Toeplitz, Circulants and $$\tau $$ matrices, as well as certain algebras that have dimension higher than the matrix order, turn out to be closed by $$J$$ -Hermitianity. As an application, we generalize some theorems about displacement decompositions presented in [1, 2], by assuming the matrix algebras involved closed by $$J$$ -Hermitianity. Even if such hypothesis on the structure is not necessary in the case of algebras generated by one matrix, as it has been proved in [3], our result is relevant because it could yield new low complexity displacement formulas involving not one-matrix-generated commutative algebras.

Let R be an associative and commutative ring with unity 1 and consider k∈N such that 1+1+⋯+1=k is invertible. Let UT∞(k)(R) be the group of upper triangular infinite matrices whose diagonal entries...

Let [Formula: see text] and [Formula: see text] be artin algebras and [Formula: see text] the triangular matrix algebra with [Formula: see text] a finitely generated ([Formula: see text])-bimodule. We construct support [Formula: see text]-tilting modules and ([Formula: see text]-)tilting modules in [Formula: see text] from that in [Formula: see text] and [Formula: see text], and give the converse constructions under some condition.

In this paper, we show that a matrix algebra $$\mathcal {LM}_{I}^{p}(\mathbb {C})$$ is pseudo-contractible if and only if I is finite. In addition, for each non-empty set I, we show that $$\mathcal {LM}_{I}^{p}(\mathbb {C})$$ is always pseudo-amenable. Amenability, approximate biprojectivity and approximate biflatness of upper triangular algebras with respect to $$\mathcal {LM}_{I}^{p}(\mathbb {C})$$ are discussed here, where $$1\le p\le 2$$ .

This paper explores how to construct an adaptive iteration learning control (AILC) mechanism for a class of discrete-time nonrepetitive systems subject to iteration-varying unknown parameters and unidentical initial condition. Firstly, for the linear discrete-time nonrepetitive systems, by minimizing the discrepancy of the real system outputs from the estimated system outputs, a gradient-type adaptation law is designed to estimate the system lower triangular parameter matrix and the system initial state. Especially, the current parametric estimation is updated by virtue of the input-output data and the previous estimation. Secondly, an AILC mechanism is constructed based on the estimated system lower triangular parameter matrix, where the control input algorithm and the adaptation law are scheduled in an interactive mode. Thirdly, the boundedness of the estimation error between the real system matrix and the estimation one is derived by means of vector norm theory. Based on the boundedness of the estimation error, the robust condition of the proposed AILC is given. Finally, the proposed AILC is investigated for a class of nonlinear affine systems and the corresponding results are captured. Simulation results illustrate the validity and effectiveness of the proposed AILC schemes.

In probabilistic state inference, we seek to estimate the state of an (autonomous) agent from noisy observations. It can be shown that, under certain assumptions, finding the estimate is equivalent to solving a linear least squares problem. Solving such a problem is done by calculating the upper triangular matrix $\boldsymbol R$ from the coefficient matrix $\boldsymbol A$ , using the QR or Cholesky factorizations; this matrix is commonly referred to as the “square root matrix”. In sequential estimation problems, we are often interested in periodic optimization of the state variable order, e.g., to reduce fill-in, or to apply a predictive variable ordering tactic; however, changing the variable order implies expensive re-factorization of the system. Thus, we address the problem of modifying an existing square root matrix $\boldsymbol R$ , to convey reordering of the variables. To this end, we identify several conclusions regarding the effect of column permutation on the factorization, to allow efficient modification of $\boldsymbol R$ , without accessing $\boldsymbol A$ at all, or with minimal re-factorization. The proposed parallelizable algorithm achieves a significant improvement in performance over the state-of-the-art incremental Smoothing And Mapping (iSAM2) algorithm, which utilizes incremental factorization to update $\boldsymbol R$ .

We present characterizations and representations for the CMP inverse. Also, we explore the CMP inverse of a block triangular matrix and its sign pattern, propose a successive matrix squaring algori...

Let $N_\infty(F)$ be the ring of infinite strictly upper triangular matrices with entries in an infinite field. The description of the commuting maps defined on $N_\infty(F)$, i.e. the maps $f\colon N_\infty(F)\rightarrow N_\infty(F)$ such that $[f(X),X]=0$ for every $X\in N_\infty(F)$, is presented. With the use of this result, the form of $m$-commuting maps defined on $T_\infty(F)$ -- the ring of infinite upper triangular matrices, i.e. the maps $f\colon T_\infty(F)\rightarrow T_\infty(F)$ such that $[f(X),X^m]=0$ for every $X\in T_\infty(F)$, is found.

In this work, we explore an extension of the Omega calculus in the context of matrix analysis introduced recently by Neto [Matrix analysis and Omega calculus. SIAM Rev. 2020;62(1):264–280]. We obta...

In this paper, the active disturbance rejection control approach is addressed for a class of lower triangular uncertain stochastic nonlinear systems driven by coloured noises. An extended state obs...

Let N be the algebra of all n×n dominant block upper triangular matrices over a field. In this paper, we explicitly describe all linear Lie centralizers of N. We also describe linear Lie centralize...

This paper proposed a new alternative approach of the implicit diagonal block backward differentiation formula (BBDF) to solve linear and nonlinear first-order stiff ordinary differential equations (ODEs). We generate the solver by manipulating the numbers of back values to achieve a higher-order possible using the interpolation procedure. The algorithm is developed and implemented in C ++ medium. The numerical integrator approximates few solution points concurrently with off-step points in a block scheme over a non-overlapping solution interval at a single iteration. The lower triangular matrix form of the implicit diagonal causes fewer differentiation coefficients and ultimately reduces the execution time during running codes. We choose two intermediate points as off-step points appropriately, which are proven to guarantee the method's zero stability. The off-step points help to increase the accuracy by optimizing the local truncation error. The proposed solver satisfied theoretical consistency and zero-stable requirements, leading to a convergent multistep method with third algebraic order. We used the well-known and standard linear and nonlinear stiff IVP problems used in literature for validation to measure the algorithm's accuracy and processor time efficiency. The performance metrics are validated by comparing them with a proven solver, and the output shows that the alternative method is better than the existing one.

This paper investigates the stabilization problem for upper-triangular nonlinear systems subject to time-delay by using sampled-data control. In order to make the design process more feasible, the ...

In the present paper it is proved that a variety of associative PI-superalgebras with graded involution of finite basic rank over a field of characteristic zero is minimal of fixed *-graded exponent if, and only if, it is generated by a subalgebra of an upper block triangular matrix algebra equipped with a suitable elementary ℤ2-grading and graded involution.

The goal of the paper is to give a complete description of the image of polynomials with zero constant term on 2 × 2 upper triangular matrix algebras over an algebraically closed field.

We determine some Nichols algebras that admit a non-trivial quadratic relation associated to some families of upper triangular solutions of the Yang-Baxter equation of dimension 3., 28 pages

We study the differential polynomial identities of the algebra UTm(F) under the derivation action of the two dimensional metabelian Lie algebra, obtaining a generating set of the TL-ideal they constitute. Then we determine the Sn-structure of their proper multilinear spaces and, for the minimal cases m = 2, 3, their exact differential codimension sequence.

For any positive integer [Formula: see text], we clearly describe all finite-dimensional algebras [Formula: see text] such that the upper triangular matrix algebras [Formula: see text] are piecewise hereditary. Consequently, we describe all finite-dimensional algebras [Formula: see text] such that their derived categories of [Formula: see text]-complexes are triangulated equivalent to derived categories of hereditary abelian categories, and we describe the tensor algebras [Formula: see text] for which their singularity categories are triangulated orbit categories of the derived categories of hereditary abelian categories.

It is proved that for matrices [Formula: see text], [Formula: see text] in the [Formula: see text] by [Formula: see text] upper triangular matrix ring [Formula: see text] over a domain [Formula: see text], if [Formula: see text] is nonzero and central in [Formula: see text] then [Formula: see text]. The [Formula: see text] by [Formula: see text] full matrix rings over right Noetherian domains are also shown to have this property. In this article we treat a ring property that is a generalization of this result, and a ring with such a property is said to be weakly reversible-over-center. The class of weakly reversible-over-center rings contains both full matrix rings over right Noetherian domains and upper triangular matrix rings over domains. The structure of various sorts of weakly reversible-over-center rings is studied in relation to the questions raised in the process naturally. We also consider the connection between the property of being weakly reversible-over-center and the related ring properties.

Relations between Laplace’s spheroidal harmonics associated with different spheroidal coordinates are derived. The transition matrices for the functions of the 1st kind are lower triangular and are related by inversion. The matrices for the functions of the 2nd kind are the transposed ones for the functions of the 1st kind. The series for the functions of the 1st kind are finite, and those for the 2nd kind are infinite. In the latter case the region of convergence is considered. Using the derived relations, the rigid solution to the electrostatic problem for the multi-layered scatterers with nonconfocal spheroidal boundaries of the layers is obtained and the Rayleigh approximation is constructed, as well as an approximate approach to a similar light scattering problem, which provides reliable results far beyond the range of applicability of the Rayleigh approximation, is suggested.

Our contribution is the development of novel representations and investigations of main properties of the MPCEP inverse. Precisely, we present representations of the MPCEP inverse which involve appropriate Moore–Penrose inverses, projections and full-rank decompositions, as well as limit and integral representations. Determinantal representations for the MPCEP inverse are also established. We study perturbation formulae with perturbation bounds of MPCEP inverse. An equivalent condition for the continuity of the MPCEP inverse is proposed. The MPCEP inverse of a suitable upper block triangular matrix is given. The successive matrix squaring algorithm and splitting method for computing the MPCEP inverse are presented. Some appropriate constrained systems of linear equations are solved applying the MPCEP inverse.

We (completely) determine those natural numbers n for which the full matrix ring $\mathbb{M}_n(\mathbb{F}_2)$ and the triangular matrix ring $\mathbb{T}_n(\mathbb{F}_2)$ over the two elements field $\mathbb{F}_2$ are either n-torsion clean or are almost n-torsion clean, respectively. These results somewhat address and settle a question, recently posed by Danchev–Matczuk in Contemp. Math. (2019) as well as they supply in a more precise aspect the nil-cleanness property of the full matrix $n\times n$ ring $\mathbb{M}_n(\mathbb{F}_2)$ for all naturals $n\geq 1$ , established in Linear Algebra & Appl. (2013) by Breaz–Calugareanu–Danchev–Micu and again in Linear Algebra & Appl. (2018) by Ster as well as in Indag. Math. (2019) by Shitov.

Given an n × n real symmetric matrix A, let λ n ( A ) denote its smallest eigenvalue. Let K n denote the set of all n × n invertible, lower triangular ( 0 , 1 ) matrices, and c n : = m i n { λ n ( Y Y t ) : Y ∈ K n } . Then, c n is the smallest singular value in K n . Hong and Loewy introduced c n as a mean to obtain inequalities involving eigenvalues of certain GCD (greatest common divisor) and LCM (least common multiple) matrices. Since then, c n has been used in many papers to obtain additional spectral inequalities for GCD and LCM matrices, and their generalizations. Due to its wide spread, it became important to obtain good bounds for c n . In this paper we obtain such bounds, and consequently determine the asymptotic behavior of c n , proving a conjecture of Kaarnioja. Moreover, we prove the uniqueness of the matrix Y ∈ K n for which c n is attained, proving a conjecture of Altinisik, Keskin, Yildiz and Demirbuken.

This paper develops a novel equivalent input disturbance (EID)-based adaptive sliding mode control (SMC) method for singularly perturbed systems (SPSs). Firstly, the block diagonalization approach is introduced to decompose the full-order SPSs exactly, and slow and fast subsystems are obtained by solving the upper and lower triangular matrices individually. Secondly, an EID is constructed to estimate the unknown disturbances with the observer gain and error system convergence analyzed. Then, depending on the decoupled reduced-order system models, a Lyapunov equation-based solution is adopted to construct a composite sliding surface. Finally, combined with the EID estimation, an adaptive SMC law is proposed to compensate the adverse effect of disturbances and the reachability condition is proven. The presented control strategy is free of any priori disturbances information while the satisfactory system performance can be guaranteed. Simulation results on two examples illustrate its superiority over the existing methods.

This paper proposes a hybrid iteration method for solving symmetric saddle point problem arising in CFD. It is an implicit alternative direction iteration method and named as the ULT-HSS (upper and lower triangular, Hermitian and skew- Hermitian splitting) method. The convergence analysis is provided, and the necessary and sufficient conditions are given for the convergence of the method. Some practical approaches are formulated for setting the optimal parameter of the method. Numerical experiments are given to show its efficiency.

This paper studies error analysis and asymptotic variance of a closed-loop subspace model identification method for a system described with the output-error state-space representation. For details, since the procedure of the identification method includes the QR factorization of stacked data Hankel matrices, this study investigates asymptotic properties of block elements of the triangular matrix obtained from the QR factorization. The set of the block elements is separated into two components, namely, the signal-based component and the noise-based component. The contributions are to derive asymptotic properties of both components and to obtain the asymptotic covariance matrix of the vectorization of the noise-based component.

In this study, semi-global practical asymptotic stability of a class of nonlinear cascade systems with upper triangular configuration has been investigated. In particular, using general results presented on stabilization of the discrete-time systems, a semi-global practical asymptotic stabilizing controller has been designed and the essential conditions for the semi-global asymptotic stability of this class of nonlinear cascade systems have been presented. The controller-design framework is based on the approximate discrete-time model of the system and the corresponding cross term constructed Lyapunov function. To illustrate the effectiveness of the proposed scheme, it has been applied to some examples and also been compared with other counterpart results in this context.

Let F be a field and let k, n1, . . . , nk be positive integers with n1 + · · · + nk = n > 2. We denote by Tn1,...,nk a block triangular matrix algebra over F with unity In and center Z(Tn1,...,nk ). Fixing an integer 1 1 upper triangular matrices over an arbitrary field is addressed.

Explicit time advancement for continuous finite elements requires the inversion of a global mass matrix. For spectral element simulations on quadrilaterals and hexahedra, there is an accurate appro...

In this paper, the problem of fuzzy adaptive control of unknown nonlinear fractional-order systems with external disturbances and unknown control directions is studied. We exploit a decomposition of the control gain matrix into a symmetric positive-definite matrix, a diagonal matrix with diagonal entries + 1 or 1 , and a unity upper triangular matrix. Fuzzy logic systems are used for estimating the unknown nonlinear functions. Based on the fractional Lyapunov direct method and some proposed lemmas, a novel fuzzy adaptive controller is designed. The proposed method can guarantee that all the signals in the closed-loop systems remain bounded and the tracking errors converge to an arbitrary small region of the origin. In addition, for updating the parameters of the fuzzy system, fractional-order adaptations laws are proposed. Lastly, an illustrative example is given to demonstrate the effectiveness of the proposed results.

We obtain the two new variants of an explicit parametrization for the general Lorentz group. Formulas are given for the direct and inverse four-dimensional Lorentz transformations. These formulas use the orthogonal three- or four-dimensional matrices. We find the infinitesimal operators of the proper Lorentz group and the multiplication formulas (commutators) of the infinitesimal operators. The orthogonal three- and four-dimensional matrices are parameterized by lower triangular matrices containing three or six independent parameters.