In this paper, we shall deal with the solvability of interval systems of linear equations in max-plus algebra. Max-plus algebra is an algebraic structure in which classical addition and multiplication are replaced by $\oplus$ and $\otimes$, where $a\oplus b=max\{a,b\}$, $a\otimes b=a+b$. The notation $𝔸\otimes x=𝕓$ represents an interval system of linear equations, where $𝔸=[\overline{b},\overline{A}]$ and $𝕓=[\underline{b},\overline{b}]$ are given interval matrix and interval vector, respectively. We can define several types of solvability of interval systems. In this paper, we define the T4 solvability and give an algorithm...

For every positive definite quadratic form in $n$ variables the reciprocal of the square root of the discriminant is equal to the arithmetic mean of the values assumed by the form on the $n-1$ sphere centered at $0$ and with radius $1$ raised to the ($-\frac{n}{2}$)-th. power. Various consequences are deduced from this, in particular a simplification of some calculations from which one obtains the possibility of solving linear systems using spherical means rather than determinants.

From the fact that the unique solution of a homogeneous linear algebraic system is the trivial one we can obtain the existence of a solution of the nonhomogeneous system. Coefficients of the systems considered are elements of the Colombeau algebra $\overline{ℝ}$ of generalized real numbers. It is worth mentioning that the algebra $\overline{ℝ}$ is not a field.

Recently, Wang (2017) has introduced the $K$-nonnegative double splitting using the notion of matrices that leave a cone $K\subseteq {ℝ}^n$ invariant and studied its convergence theory by generalizing the corresponding results for the nonnegative double splitting by Song and Song (2011). However, the convergence theory for $K$-weak regular and $K$-nonnegative double splittings of type II is not yet studied. In this article, we first introduce this class of splittings and then discuss the convergence theory for these sub-classes...

In this paper explicit expressions for solutions of Cauchy problems and two-point boundary value problems concerned with the generalized Riccati matrix differential equation are given. These explicit expressions are computable in terms of the data and solutions of certain algebraic Riccati equations related to the problem. The interplay between the algebraic and the differential problems is used in order to obtain approximate solutions of the differential problem in terms of those of the algebraic...