Sergei Sergeev

CSR expansions of matrix powers in max algebra

We study the behavior of max-algebraic powers of a reducible nonnegative n by n matrix A. We show that for t>3n^2, the powers A^t can be expanded in max-algebraic powers of the form CS^tR, where C and R are extracted from columns and rows of certain Kleene stars and S is diadonally similar to a Boolean matrix. We study the properties of individual terms and show that all terms, for a given t>3n^2, can be found in O(n^4 log n) operations. We show that the powers have a well-defined ultimate behavior, where certain terms are totally or partially suppressed, thus leading to ultimate CS^tR terms and the corresponding ultimate expansion. We apply this expansion to the question whether {A^ty, t>0} is ultimately linear periodic for each starting vector y, showing that this question can be also answered in O(n^4 log n) time. We give examples illustrating our main results.

Weak CSR expansions and transience bounds in max-plus algebra

Abstract This paper aims to unify and extend existing techniques for deriving upper bounds on the transient of max-plus matrix powers. To this aim, we introduce the concept of weak CSR expansions: A t = C S t R ⊕ B t . We observe that most of the known bounds (implicitly) take the maximum of (i) a bound for the weak CSR expansion to hold, which does not depend on the values of the entries of the matrix but only on its pattern, and (ii) a bound for the C S t R term to dominate. To improve and analyze (i), we consider various cycle replacement techniques and show that some of the known bounds for indices and exponents of digraphs apply here. We also show how to make use of various parameters of digraphs. To improve and analyze (ii), we introduce three different kinds of weak CSR expansions (named after Nachtigall, Hartman–Arguelles, and Cycle Threshold). As a result, we obtain a collection of bounds, in general incomparable to one another, but better than the bounds found in the literature.

Characterizing matrices with

A matrix

{\bf {X}}

A

-simple image eigenspace in max-min semiring

is said to have

Characterization of tropical hemispaces by (P,R)-decompositions

X

BASIC SOLUTIONS OF SYSTEMS WITH TWO MAX-LINEAR INEQUALITIES

-simple image eigenspace if any eigenvector

Reachability of eigenspaces for interval circulant matrices in max-algebra

x

ON THE MAX-ALGEBRAIC CORE OF A NONNEGATIVE MATRIX ∗

belonging to the interval

Multiorder, Kleene stars and cyclic projectors in the geometry of max cones

X=\{x\colon \underline{x}\leq x\leq\overline{x}\}