We want to construct a homological link invariant whose Euler characteristic is MOY polynomial as Khovanov and Rozansky constructed a categorification of HOMFLY polynomial. The present paper gives the first step to construct a categorification of MOY polynomial. For the essential colored planar diagrams with additional data which is a sequence naturally induced by coloring, we define matrix fac...

This report presents a new analytic method for checking the Schur stability of interval polynomial matrix (polynomials whose coefficients are interval matrices). In the suggested method, the interval Markov matrices of the interval polynomial matrix are used to assess its stability. The validity of the suggested method is illustrated through numerical examples. The proposed technique is algebra...

A method for enclosing matrix polynomials is proposed. This method is applicable when the matrix is diagonalizable and supplies an interval matrix including the matrix polynomial. The computational cost of this method does not depend on the degree of the polynomial. Hence the proposed method is faster than previous approaches when the degree is large. Numerical results show the properties of th...

The energy of signed graph is the sum of the absolute values of the eigenvalues of its adjacency matrix. Two signed graphs are said to be equienergetic if they have same energy. In the literature the construction of equienergetic signed graphs are reported. In this paper we obtain the characteristic polynomial and energy of the join of two signed graphs and thereby we give another construction ...

Lyapunov’s direct method. Two matrix equations have been derived: matrix polynomial equation and continuous (discrete) Lyapunov matrix equation. Also, modifications of the existing sufficient conditions of convergence of Traub and Bernoilli algorithms for computing the dominant solvent of the matrix polynomial equation are derived. These results have been extended to large scale systems as well...

A standard way of dealing with a matrix polynomial P (λ) is to convert it into an equivalent matrix pencil – a process known as linearization. For any regular matrix polynomial, a new family of linearizations generalizing the classical first and second Frobenius companion forms has recently been introduced by Antoniou and Vologiannidis, extending some linearizations previously defined by Fiedle...

No polynomial algorithms are known for finding the coefficients of the characteristic polynomial and characteristic equation of a matrix in max-algebra. The following are proved: (1) The task of finding the max-algebraic characteristic polynomial for permutation matrices encoded using the lengths of their constituent cycles is IVP-complete. (2) The task of finding the lowest order finite term o...

This paper discusses a relationship between the prime factorizability of a normal full rank n-D (n > 2) polynomial matrix and its reduced minors. Two conjectures regarding the n-D polynomial matrix prime factorization problem are posed, and a partial solution to one of the conjectures is provided. Another related open problem of factorizing an n-D polynomial matrix that is not of normal full ra...

An algorithm is described for computing the structure at innnity and extracting the innnite zeros of a given polynomial matrix. The algorithm relies on numerically reliable operations only. Applications include computation of the subspace of impulsive solution of a set of linear diierential equations, derivation of the Smith form at innnity of a polynomial matrix, or also enhanced computation o...