Four different algorithms are designed to obtain the null-space of a polynomial matrix. In this first part we present two algorithms. These algorithms are based on classical methods of numerical linear algebra, namely the reduction into the column echelon form and the LQ factorization. Both algorithms take advantage of the block Toeplitz structure of the Sylvester matrix associated with the pol...

Abstract In this paper, we study the images of multilinear graded polynomials on algebra upper triangular matrices $UT_n$ . For positive integers $q\leq n$ , classify these $UT_{n}$ endowed with a particular elementary ${\mathbb {Z}}_{q}$ -grading. As consequence, obtain natural {Z}}_{n}$ We apply classification in order to give new condition for polynomial terms identities so that traceless it...

We formulate the Hopf algebra underlying the su(2|2) worldsheet S-matrix of the AdS 5 × S 5 string in the AdS/CFT correspondence. For this we extend the previous construction in the su(1|2) subsector due to Janik to the full algebra by specifying the action of the coproduct and the antipode on the remaining generators. The nontriviality of the coproduct is determined by length-changing effects ...

Existing software implementations for solving Linear Programming (LP) models are all based on full matrix inversion operations involving every constraint in the model in every step. This linear algebra component in these systems makes it difficult to solve dense models even with moderate size, and it is also the source of accumulating roundoff errors affecting the accuracy of the output. We pre...

In recent years a Hopf algebraic structure underlying the process of renormalization in quantum field theory was found. It led to a Birkhoff factorization for (regularized) Hopf algebra characters, i.e. for Feynman rules. In this work we would like to show that this Birkhoff factorization finds its natural formulation in terms of a classical r-matrix, coming from a Rota-Baxter structure underly...

It is an open problem whether an infinite-dimensional amenable Banach algebra exists whose underlying Banach space is reflexive. We give sufficient conditions for a reflexive, amenable Banach algebra to be finite-dimensional (and thus a finite direct sum of full matrix algebras). If A is a reflexive, amenable Banach algebra such that for each maximal left ideal L of A (i) the quotient A/L has t...

In this paper, we give the factorizations of Lucas and inverse matrices. We also investigate Cholesky factorization symmetric matrix. Moreover, obtain upper lower bounds for eigenvalues matrix by using some majorization techniques.

In recent years a Hopf algebraic structure underlying the process of renormalization in quantum field theory was found. It led to a Birkhoff factorization for (regularized) Hopf algebra characters, i.e. for Feynman rules. In this work we would like to show that this Birkhoff factorization finds its natural formulation in terms of a classical r-matrix, coming from a Rota-Baxter structure underly...