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In the present paper we shall obtain the following result: Let n ≥ 2 and m ≥ 1 be integers. Let p (x 1 , ... , x m) be a polynomial with zero constant term in non-commutative variables over an algebraically closed field K. Suppose that 1 ≤ r ≤ n − 1 , where r is the order of p. Then p (T n (K)) is a dense subset of T n (K) (r − 1) (with respect to the Zariski topology). This is a solution of a problem presented by Gargate and de Mello and a supplement to a result obtained by Panja and Prasad in 2023. [ABSTRACT FROM AUTHOR]

Galois duals of Multi-twisted (MT) codes are considered in this study. We describe a MT code C as a module over a principal ideal domain. Hence, C has a generator polynomial matrix (GPM) G that satisfies an identical equation. We prove a GPM formula for the Euclidean dual C ⊥ using the identical equation of the Hermite normal form of G . Next, we aim to replace the Euclidean dual with the Galois dual. The Galois inner product is an asymmetric form, so we distinguish between the right and left Galois duals. We show that the right and left Galois duals of a MT code are MT as well but with possibly different shift constants. Some interconnected identities for the right and left Galois duals of a linear code are established and we also introduce the two-sided Galois dual. We use a condition that makes the two-sided Galois dual of a MT code MT, then we describe its GPM. Two special cases are also studied, one when the right and left Galois duals trivially intersect and the other when they coincide. A necessary and sufficient condition is established for the equality of the right and left Galois duals of any linear code. [ABSTRACT FROM AUTHOR]

The systems of nonlinear differential equations of certain types can be simplified to matrix forms. Two types of matrix differential equations will be considered in the paper, one is Fermat type matrix differential equation where n = 2 and n = 3, another is Malmquist type matrix differential equation , where α (≠ 0), β, γ are constants. By solving the systems of nonlinear differential equations, we obtain some properties on the meromorphic matrix solutions of the above matrix differential equations. In addition, we also consider two types of nonlinear differential equations, one of them is called Bi-Fermat differential equation. [ABSTRACT FROM AUTHOR]

In this paper we study the backward stability of running a backward stable eigenstructure solver on a pencil S (λ) that is a strong linearization of a rational matrix R (λ) expressed in the form R (λ) = D (λ) + C (λ I ℓ - A) - 1 B , where D (λ) is a polynomial matrix and C (λ I ℓ - A) - 1 B is a minimal state-space realization. We consider the family of block Kronecker linearizations of R (λ) , which have the following structure S (λ) : = M (λ) K ^ 2 T C K 2 T (λ) B K ^ 1 A - λ I ℓ 0 K 1 (λ) 0 0 , where the blocks have some specific structures. Backward stable eigenstructure solvers, such as the QZ or the staircase algorithms, applied to S (λ) will compute the exact eigenstructure of a perturbed pencil S ^ (λ) : = S (λ) + Δ S (λ) and the special structure of S (λ) will be lost, including the zero blocks below the anti-diagonal. In order to link this perturbed pencil with a nearby rational matrix, we construct in this paper a strictly equivalent pencil S ~ (λ) = (I - X) S ^ (λ) (I - Y) that restores the original structure, and hence is a block Kronecker linearization of a perturbed rational matrix R ~ (λ) = D ~ (λ) + C ~ (λ I ℓ - A ~) - 1 B ~ , where D ~ (λ) is a polynomial matrix with the same degree as D (λ) . Moreover, we bound appropriate norms of D ~ (λ) - D (λ) , C ~ - C , A ~ - A and B ~ - B in terms of an appropriate norm of Δ S (λ) . These bounds may be, in general, inadmissibly large, but we also introduce a scaling that allows us to make them satisfactorily tiny, by making the matrices appearing in both S (λ) and R (λ) have norms bounded by 1. Thus, for this scaled representation, we prove that the staircase and the QZ algorithms compute the exact eigenstructure of a rational matrix R ~ (λ) that can be expressed in exactly the same form as R (λ) with the parameters defining the representation very near to those of R (λ) . This shows that this approach is backward stable in a structured sense. Several numerical experiments confirm the obtained backward stability results. [ABSTRACT FROM AUTHOR]

In this paper we present a class of convergent infinite products ∏ k = 1 ∞ H k = lim k → ∞ (H k H k − 1 ⋯ H 1) of 3 × 3 hollow matrices of the form H k := [ 0 a k a k b k 0 b k c k c k 0 ] , (0 < a k , b k , c k < 1 / 2). These sequences of hollow matrices are obtained from the Ando-Li-Mathias matrix geometric mean of 2 × 2 positive definite matrices. We further provide a three dimensional analytic manifold and an analytic self-map realizing such sequences. [ABSTRACT FROM AUTHOR]

We introduce a new family of linearizations of rational matrices, which we call block full rank linearizations. The theory of block full rank linearizations is useful as it establishes very simple criteria to determine if a pencil is a linearization of a rational matrix in a target set or in the whole underlying field, by using rank conditions. Block full rank linearizations allow us to recover locally information about zeros and poles. To recover the pole-zero information at infinity, we will define the grade of the new block full rank linearizations as linearizations at infinity and the notion of degree of a rational matrix will be used. Moreover, the eigenvectors of a rational matrix associated with its eigenvalues in a target set can be obtained from the eigenvectors of its block full rank linearizations in that set. This new family of linearizations generalizes and includes the structures appearing in most of the linearizations for rational matrices constructed in the literature. As example, we study the structure and properties of the linearizations in [P. Lietaert et al., Automatic rational approximation and linearization of nonlinear eigenvalue problems, 2021]. [ABSTRACT FROM AUTHOR]

Let A 1 , ... , A s be unitary commutative rings which do not have non-trivial idempotents and let A = A 1 ⊕ ⋯ ⊕ A s be their direct sum. We describe all idempotents in the 2 × 2 matrix ring M 2 (A [ [ X ] ]) over the ring A [ [ X ] ] of formal power series with coefficients in A and in an arbitrary set of variables X. We apply this result to the matrix ring M 2 ( Z n [ [ X ] ]) over the ring Z n [ [ X ] ] where Z n ≅ Z / n Z for an arbitrary positive integer n greater than 1. Our proof is elementary and uses only the Cayley-Hamilton theorem (for 2 × 2 matrices only) and, in the special case A = Z n , the Chinese remainder theorem and the Euler-Fermat theorem. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to initiate the study of the images of non-multilinear polynomials on upper triangular matrix algebras. We shall give a complete description of the images of completely homogenous polynomials on 2 × 2 upper triangular matrix algebras. As a consequence, we show that the image of some special completely homogenous polynomials on 2 × 2 upper triangular matrix algebras are not vector spaces. [ABSTRACT FROM AUTHOR]

In this paper we establish some Positivstellensätze for polynomial matrices, applying the Scherer–Hol theorem. Firstly, we give a representation for polynomial matrices positive definite on subsets of compact polyhedra. Then we establish a Putinar-Vasilescu Positivstellensatz for polynomial matrices. Next we propose a matrix version of the Dickinson–Povh Positivstellensatz. Finally, we establish a version of Marshall's theorem for polynomial matrices, approximating positive semi-definite polynomial matrices using sums of squares. [ABSTRACT FROM AUTHOR]

Let D be the unit disk. Kutzschebauch and Studer (Bull. Lond. Math. Soc. 51 (2019) 995–1004) recently proved that, for each continuous map A:D¯→SL(2,C), which is holomorphic in D, there exist continuous maps E,F:D¯→sl(2,C), which are holomorphic in D, such that A=eEeF. Also they asked if this extends to arbitrary compact bordered Riemann surfaces. We prove that this is possible. [ABSTRACT FROM AUTHOR]

We show that any element of the special linear group SL2(R) is a product of two exponentials if the ring R is either the ring of holomorphic functions on an open Riemann surface or the disc algebra. This is sharp: one exponential factor is not enough since the exponential map corresponding to SL2(C) is not surjective. Our result extends to the linear group GL2(R), where it is sharp as well. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to describe the images of non-commutative multilinear polynomials on upper triangular matrix algebras. [ABSTRACT FROM AUTHOR]

Let R be a commutative unital ring. A well-known factorization problem is whether any matrix in SL n (R) is a product of elementary matrices with entries in R. To solve the problem, we use two approaches based on the notion of the Bass stable rank and on construction of a null-homotopy. Special attention is given to the case, where R is a ring or Banach algebra of holomorphic functions. Also, we consider a related problem on representation of a matrix in GL n (R) as a product of exponentials. [ABSTRACT FROM AUTHOR]

In the paper we study probability measure on real-orthogonal projections acting on complex Euclidean space. We proof an analog of Gleason's theorem. We study Hermitian measure and some special class of measures on real-orthogonal projections. [ABSTRACT FROM AUTHOR]

The richly developed theory of complex manifolds plays important roles in our understanding of holomorphic functions in several complex variables. It is natural to consider manifolds that will play similar roles in the theory of holomorphic functions in several non‐commuting variables. In this paper we introduce the class of nc‐manifolds, the mathematical objects that at each point possess a neighborhood that has the structure of an nc‐domain in the d‐dimensional nc‐universeMd. We illustrate the use of such manifolds in free analysis through the construction of the non‐commutative Riemann surface for the matricial square root function. A second illustration is the construction of a non‐commutative analog of the elementary symmetric functions in two variables. For any symmetric domain in M2 we construct a two‐dimensional non‐commutative manifold such that the symmetric holomorphic functions on the domain are in bijective correspondence with the holomorphic functions on the manifold. We also derive a version of the classical Newton–Girard formulae for power sums of two non‐commuting variables. [ABSTRACT FROM AUTHOR]

We introduce a new family of strong linearizations of matrix polynomials—which we call “block Kronecker pencils”—and perform a backward stability analysis of complete polynomial eigenproblems. These problems are solved by applying any backward stable algorithm to a block Kronecker pencil, such as the staircase algorithm for singular pencils or the QZ algorithm for regular pencils. This stability analysis allows us to identify those block Kronecker pencils that yield a computed complete eigenstructure which is exactly that of a slightly perturbed matrix polynomial. The global backward error analysis in this work presents for the first time the following key properties: it is a rigorous analysis valid for finite perturbations (i.e., it is not a first order analysis), it provides precise bounds, it is valid simultaneously for a large class of linearizations, and it establishes a framework that may be generalized to other classes of linearizations. These features are related to the fact that block Kronecker pencils are a particular case of the new family of “strong block minimal bases pencils”, which are robust under certain perturbations and, so, include certain perturbations of block Kronecker pencils. [ABSTRACT FROM AUTHOR]

Livernet and Loday constructed a polarization of the nonsymmetric associative operadwith one operation into a symmetric operadwith two operations (the Lie bracket and Jordan product), and defined a one-parameter deformation of, which includes Poisson algebras. We combine this with the dendriform splitting of an associative operation into the sum of two nonassociative operations, and use Koszul duality for quadratic operads, to construct one-parameter deformations of the nonsymmetric dendriform and diassociative operads into the category of symmetric operads. [ABSTRACT FROM PUBLISHER]

An eigenvalue perturbation theory under rank-one perturbations is developed for classes of real matrices that are symmetric with respect to a non-degenerate bilinear form, or Hamiltonian with respect to a non-degenerate skew-symmetric form. In contrast to the case of complex matrices, the sign characteristic is a crucial feature of matrices in these classes. The behaviour of the sign characteristic under generic rank-one perturbations is analyzed in each of these two classes of matrices. Partial results are presented, but some questions remain open. Applications include boundedness and robust boundedness for solutions of structured systems of linear differential equations with respect to general perturbations as well as with respect to structured rank perturbations of the coefficients. [ABSTRACT FROM PUBLISHER]

The purpose of this paper is to give a correct proof of a result on the images of non-commutative multilinear polynomials on upper triangular matrix algebra. [ABSTRACT FROM AUTHOR]