相关论文: On the max-algebraic core of a nonnegative matrix
We associate to any given finite set of valuations on the polynomial ring in two variables over an algebraically closed field a numerical invariant whose positivity characterizes the case when the intersection of their valuation rings has…
We make some remarks on the existence of a geodesically complete core for any compact non-positively curved space.
Several spectral radii formulas for infinite bounded nonnegative matrices in max algebra are obtained. We also prove some Perron-Frobenius type results for such matrices. In particular, we obtain results on block triangular forms, which are…
The max-plus algebra $\mathbb{R}\cup \{-\infty \}$ is a semiring with the two operations: addition $a \oplus b := \max(a,b)$ and multiplication $a \otimes b := a + b$. Roots of the characteristic polynomial of a max-plus matrix are called…
The core of an ideal is defined as the intersection of all of its reductions. In this paper we provide an explicit description for the core of a monomial ideal $I$ satisfying certain residual conditions, showing that ${\rm core}(I)$…
In this paper, we mainly study the trace norm of the adjacency matrix of a graph, also known as the energy of graph. We give the maximum trace norms for the graph and its complement. In fact, the above problem is stated and solved in a more…
Given a generic semidefinite program, specified by matrices with rational entries, each coordinate of its optimal solution is an algebraic number. We study the degree of the minimal polynomials of these algebraic numbers. Geometrically,…
The core of an ideal is the intersection of all its reductions. We describe the core of a zero-dimensional monomial ideal I as the largest monomial ideal contained in a general reduction of I. This provides a new interpretation of the core…
Suppose $\mathcal{X}$ is an $n$-correct set of nodes in the plane, that is, it admits a unisolvent interpolation with bivariate polynomials of total degree less than or equal to $n.$ Then an algebraic curve $q$ of degree $k\le n$ can pass…
We study the multiplication operation of square matrices over lattices. If the underlying lattice is distributive, then matrices form a semigroup; we investigate idempotent and nilpotent elements and the maximal subgroups of this matrix…
The numerical radius of a matrix is a scalar quantity that has many applications in the study of matrix analysis. Due to the difficulty in computing the numerical radius, inequalities bounding it have received a considerable attention in…
Semipositive matrices (matrices that map at least one nonnegative vector to a positive vector) and minimally semipositive matrices (semipositive matrices whose no column-deleted submatrix is semipositive) are well studied in matrix theory.…
In this paper we study the concept of algebraic core for convex sets in general vector spaces without any topological structure and then present its applications to problems of convex analysis and optimization. Deriving the equivalence…
The purpose of this paper is to investigate the interplay arising between max algebra, convexity and scaling problems. The latter, which have been studied in nonnegative matrix theory, are strongly related to max algebra. One problem is…
The square root rank of a nonnegative matrix $A$ is the minimum rank of a matrix $B$ such that $A=B \circ B$, where $\circ$ denotes entrywise product. We show that the square root rank of the slack matrix of the correlation polytope is…
Our aim in this paper is to initiate the study of exponent semigroups for rational matrices. We prove that every numerical semigroup is the exponent semigroup of some rational matrix. We also obtain lower bounds on the size of such matrices…
Our aim is to study matrix polynomials over max-algebras and their growth in terms of a max-induced semi-norm. We investigate the relationship between the asymptotic growth of polynomial products and the joint spectral radius of the…
Results on matrix canonical forms are used to give a complete description of the higher rank numerical range of matrices arising from the study of quantum error correction. It is shown that the set can be obtained as the intersection of…
We formulate conjectures regarding the maximum value and maximizing matrices of the permanent and of diagonal products on the set of stochastic matrices with bounded rank. We formulate equivalent conjectures on upper bounds for these…
Optimal matrices for problems involving the matrix numerical radius often have fields of values that are disks, a phenomenon associated with partial smoothness. Such matrices are highly structured: we experiment in particular with the…