Triangular Matrix Research Articles

On Galois representations with large image

For every prime number p ≥ 3 p\geq 3 and every integer m ≥ 1 m\geq 1 , we prove the existence of a continuous Galois representation ρ : G Q → G l m ( Z p ) \rho : G_\mathbb {Q} \rightarrow Gl_m(\mathbb {Z}_p) which has open image and is unramified outside { p , ∞ } \{p,\infty \} if p ≡ 3 p\equiv 3 mod 4 4 and is unramified outside { 2 , p , ∞ } \{2,p,\infty \} if p ≡ 1 p \equiv 1 mod 4 4 . We also revisit the question of the lifting of residual Galois representations in terms of embedding problems; that allows us to produce Galois representations with open image in the group of upper triangular matrices with diagonal entries equal to 1 1 , unramified outside { p , ∞ } \{p,\infty \} , for m m “small” comparing to p p .

A nonlinear optimal control approach for voltage source inverter-fed three-phase PMSMs

A nonlinear optimal control approach for voltage source inverter-fed three-phase PMSMs

Analysis of Projectile Impact Resisting Characteristics of Hollow Triangular Matrix Structure

The numerical simulation of engineering structural elements and spatial combination form of the polarizing body using common engineering materials is carried out, and the triangular hole matrix space combination structure is proposed. The penetration depth and deflection angle of the projectile under different projectile velocities and incident positions are obtained by numerical simulation of the projectile impacting the composite structure under various working conditions. At the same time, through the comparative study of ordinary solid shell shielding structures, it is found that the higher the projectile velocity, the higher the shell shielding effectiveness of triangular hole matrix space composite structure.

Derimorphisms over Algebras and Applications

The new concept “derimorphism” generalizing both derivation and homomorphism is defined. When a derimorphism is invertible, its inverse is a Rota–Baxter operator. The general theory of derimorphism is established. The classification of all derimorphisms over an associative unital algebra is obtained. Contrary to the nonexistence of nontrivial positive derivations, it is shown that nontrivial positive derimorphisms do exist over any pair of opposite orderings over [Formula: see text], the lattice-ordered full matrix algebra and upper triangular matrix algebra over a totally ordered field.

Involutions in Algebras of Upper-Triangular Matrices

Involutions in Algebras of Upper-Triangular Matrices

Formal matrix rings: Automorphisms

Automorphism groups of formal matrix algebras with zero trace ideals are studied. Such an algebra is represented as a splitting extension of some ring by some nilpotent ideal. Using this extension, the study of the structure of the automorphism group of an algebra in a certain sense is reduced to the study of the structure of some its subgroups and quotient groups. Then the structure of these subgroups and factor groups is found. The case of formal triangular matrix rings is specially considered. At the end of the paper we study automorphisms of ordinary matrix rings. In this case, a well-known homomorphism from the group of outer automorphisms to the Picard group of a certain ring is used.

Two new feather mite species (Acarina: Psoroptidia) from the Chestnut Bulbul, Hemixos castanonotus (Passeriformes: Pycnonotidae), in China

Two new species of feather mites from the bird host Hemixos castanonotus Swinhoe, 1870 (Pycnonotidae) collected in China are described: Trouessartia bulbuli sp. nov. (Trouessartiidae) and Pteroherpus guangdongensis sp. nov. (Pteronyssidae). Trouessartia bulbuli sp. nov. is close to T. latisetata Gaud, 1952 and differs in having the following features: in males of T. bulbuli tibia IV has apical thorn-like process at base of solenidia φ, epimerites IVa are wide and with a small extension of irregular form on the anterior end; in females, the copulatory opening is on the small wide-angular projection (external copulatory tube) on the margin of interlobar membrane and h1 setae are filiform. Pteroherpus guangdongensis sp. nov. belongs to the hoplophorus species group and is most close to P. meghalayensis Constantinescu, 2014.The new species has the following distinctive characters: in males, a very long genital apparatus and the epiandrum with long branches, extending beyond the base of setae 4a, a pair of small additional sclerites posterior to prodorsal shield is present, and adanal shield is shaped as a small longitudinal sclerite of irregular form; in females, the hysterosomal shield is split into unpaired anterior hysteronotal sclerite shaped as shaped as dumb-bell, central sclerite of triangular form, paired lateral opisthosomal sclerites and pygidial sclerites (similar to those in other species of hoplophorus group), and the hysteronotal gland openings gl are situated on the striated tegument.

Fredholm properties of upper triangular matrices of relations

Fredholm properties of upper triangular matrices of relations

Backström algebras

We introduce Backstr\"om pairs and Backstr\"om rings, study their derived categories and construct for them a sort of categorical resolutions. For the latter we define the global dimension, construct a sort of semi-orthogonal decomposition of the derived category and deduce that the derived dimension of a Backstr\"om ring is at most $2$. Using this semi-orthogonal decomposition, we define a description of the module category as the category of elements of a special bimodule. We also construct a partial tilting for a Backstr\"om pair to a ring of triangular matrices and define the global dimension of the latter.

Complexes with the derived double centralizer property

In representation theory, the double centralizer property is an important property for a module (bimodule). In this paper, we extend this property to complexes in derived categories of algebras, under the name derived double centralizer property. Let A be a finite dimensional algebra over an algebraically closed field. Characterizations for complexes of finitely generated A-modules with the derived double centralizer property and for tilting complexes in the bounded derived category with the endomorphism algebras hereditary are given. In particular, when A is a upper triangular matrix algebra, all complexes of A-A-bimodules with this property are classified.

Perturbations of Tensor-Schur decomposition and its applications to multilinear control systems and facial recognitions

Perturbation analysis has been primarily considered to be one of the main issues in many fields. The Schur decomposition can factor a square matrix as the product of a unitary matrix and an upper triangular matrix, which contains eigenvalues of the square matrix. In view of the importance of T-eigenvalue problems, this paper discusses a tensor Schur decomposition (T-Schur), the T-Schur decomposition =U∗T∗U∗, which is based on T-product multiplication of third-order tensors. We present the normwise and componentwise perturbation analysis for the unitary tensor U, the upper triangular tensor T and T-eigenvalues of the tensor A. We explore some applications of the T-Schur decomposition and perform the T-Schur form to solve the tensor pole assignment. We give an algorithm to solve the tensor based T-Sylvester equation and present its perturbation bound and the backward error. The T-Sylvester equation can also be used to estimate the condition of the T-sign function. We apply the T-Schur decomposition to facial recognation and compare it with other types of tensor decompositions.

IFP for Rings with Involution

In this note, we introduce the concept of semi-*-IFP, the involutive version of semi-IFP, which is a generalization of quasi-*-IFP and *-reducedness of *-rings. We study the basic structure and properties of *-rings having semi-*-IFP and give results for IFPs in rings with involution. Several results and counterexamples are stated to connect the involutive versions of IFP. We discuss the conditions for the involutive IFPs to be extended into *-subrings of the ring of upper triangular matrices. In *-rings with quasi-*-IFP, it is shown that Köthe’s conjecture has a strong affirmative solution. We investigate its related properties and the relationship between *-rings with quasi-*-IFP and *-Armendariz properties.

Tensor Eigenvalue and SVD from the Viewpoint of Linear Transformation

A linear transformation from vector space to another vector space can be represented as a matrix. This close relationship between the matrix and the linear transformation is helpful for the study of matrices. In this paper, the tensor is regarded as a generalization of the matrix from the viewpoint of the linear transformation instead of the quadratic form in matrix theory; we discuss some operations and present some definitions and theorems related to tensors. For example, we provide the definitions of the triangular form and the eigenvalue of a tensor, and the theorems of the tensor QR decomposition and the tensor singular value decomposition. Furthermore, we explain the significance of our definitions and their differences from existing definitions.

Bounded Adaptive Function Activated Recurrent Neural Network for Solving the Dynamic QR Factorization

The orthogonal triangular factorization (QRF) method is a widespread tool to calculate eigenvalues and has been used for many practical applications. However, as an emerging topic, only a few works have been devoted to handling dynamic QR factorization (DQRF). Moreover, the traditional methods for dynamic problems suffer from lagging errors and are susceptible to noise, thereby being unable to satisfy the requirements of the real-time solution. In this paper, a bounded adaptive function activated recurrent neural network (BAFARNN) is proposed to solve the DQRF with a faster convergence speed and enhance existing solution methods’ robustness. Theoretical analysis shows that the model can achieve global convergence in different environments. The results of the systematic experiment show that the BAFARNN model outperforms both the original ZNN (OZNN) model and the noise-tolerant zeroing neural network (NTZNN) model in terms of accuracy and convergence speed. This is true for both single constants and time-varying noise disturbances.

A unified approach to generalized Pascal-like matrices: q-analysis

In this paper, we present a general method to construct q-analogues and other generalizations of Pascal-like matrices. Our matrices are obtained as functions of strictly lower triangular matrices and include several types of generalized Pascal-like matrices and matrices related with modified Hermite polynomials of two variables and other polynomial sequences. We find explicit expressions for products, powers, and inverses of the matrices and also some factorization formulas using this method.

The image of polynomials and Waring type problems on upper triangular matrix algebras

Let p be a polynomial in non-commutative variables x1,x2,…,xn with constant term zero over an algebraically closed field K. The object of study in this paper is the image of this kind of polynomial over the algebra of upper triangular matrices Tm(K). We introduce a family of polynomials called multi-index p-inductive polynomials for a given polynomial p. Using this family we will show that, if p is a polynomial identity of Tt(K) but not of Tt+1(K), then p(Tm(K))⊆Tm(K)(t−1). Equality is achieved in the case t=1,m−1 and an example has been provided to show that equality does not hold in general. We further prove existence of d such that each element of Tm(K)(t−1) can be written as sum of d many elements of p(Tm(K)). It has also been shown that the image of Tm(K)× under a word map is Zariski dense in Tm(K)×.

Invertibility properties of operator matrices on Hilbert spaces

Denote by $$T_n^d(A)$$ an upper triangular operator matrix of dimension n whose diagonal entries are given and the others are unknown. In this article, we provide necessary and sufficient conditions for various types of Fredholm and Weyl completions of $$T_n^d(A)$$ . As consequences, we recover many known results of Zhang and Wu (J Math Anal Appl 392(2):103–110, 2012) for the case $$n=2$$ already existing in the literature, as well as some perturbation results of Wu and Huang (Acta Math Sin (Engl Ser) 36(7):783–796, 2020; Ann Funct Anal 11(3):780–798, 2020) given for the case of arbitrary $$n\ge 2$$ .

Properties and characterizations of dual sharp orders

In this paper, we extend the sharp order to dual ring. To be specific, we introduce D-sharp, C-sharp, G- and G-sharp binary relations of dual matrices, and discuss relevant properties of these dual binary relations by using dual group generalized inverse(DGGI for short), group dual generalized inverse(GDGI for short) and matrix decomposition. We conclude that D-sharp and G-sharp binary relations are partial orders. The dual binary relations are connected with the sharp order by using the upper triangular block matrix. Furthermore, we discuss relationships among dual binary relations mentioned above. When dual parts of dual matrix pairs are all zero matrices, dual binary relations(D-sharp, C-sharp, G- and G-sharp binary relations) are equivalent to the sharp order.

Riesz ideal convergence in neutrosophic normed spaces

In this paper, we define concept of ideal(I) convergent sequences as a domain of regular Riesz triangular matrix and also using neutrosophic normed space. In neutrosophic normed spaces, we describe the idea of a generalization of the Riesz ideal convergence and the Riesz Cauchy sequence, and we also show certain aspects of these ideas. Finally, we define the concept of I∗-convergence in neutrosophic normed space and show that it is related to Riesz I-convergence in neutrosophic normed space. Moreover, we clearly show some results related to these theories and provide counterexamples.

On real forms of Dubrovin-Ugaglia Poisson brackets

Dubrovin and Ugaglia introduced a holomorphic Poisson structure on the space of semisimple Frobenius manifolds, which is identified with the entries of an upper triangular matrix with ones on the diagonal. In this paper, we compute explicitly the real forms of the Dubrovin-Ugaglia Poisson structures, arising from the real forms of the Lie algebra son(C).