7,774 research outputs found
Dynamics of piecewise linear maps and sets of nonnegative matrices
We consider functions and ,
where is a finite set of nonnegative matrices and by "min" and "max" we
mean coordinate-wise minimum and maximum. We transfer known results about
properties of to . In particular we show existence of nonnegative
generalized eigenvectors for , give necessary and sufficient conditions for
existence of strictly positive eigenvector for , study dynamics of on
the positive cone. We show the existence and construct matrices and ,
possibly not in , such that and for any
strictly positive vector .Comment: 20 page
Hypergraph conditions for the solvability of the ergodic equation for zero-sum games
The ergodic equation is a basic tool in the study of mean-payoff stochastic
games. Its solvability entails that the mean payoff is independent of the
initial state. Moreover, optimal stationary strategies are readily obtained
from its solution. In this paper, we give a general sufficient condition for
the solvability of the ergodic equation, for a game with finite state space but
arbitrary action spaces. This condition involves a pair of directed hypergraphs
depending only on the ``growth at infinity'' of the Shapley operator of the
game. This refines a recent result of the authors which only applied to games
with bounded payments, as well as earlier nonlinear fixed point results for
order preserving maps, involving graph conditions.Comment: 6 pages, 1 figure, to appear in Proc. 54th IEEE Conference on
Decision and Control (CDC 2015
Matrix Roots of Eventually Positive Matrices
Eventually positive matrices are real matrices whose powers become and remain
strictly positive. As such, eventually positive matrices are a fortiori matrix
roots of positive matrices, which motivates us to study the matrix roots of
primitive matrices. Using classical matrix function theory and Perron-Frobenius
theory, we characterize, classify, and describe in terms of the real Jordan
canonical form the th-roots of eventually positive matrices.Comment: Accepted for publication in Linear Algebra and its Application
Substitution Delone Sets
This paper addresses the problem of describing aperiodic discrete structures
that have a self-similar or self-affine structure. Substitution Delone set
families are families of Delone sets (X_1, ..., X_n) in R^d that satisfy an
inflation functional equation under the action of an expanding integer matrix
in R^d. This paper studies such functional equation in which each X_i is a
discrete multiset (a set whose elements are counted with a finite
multiplicity). It gives necessary conditions on the coefficients of the
functional equation for discrete solutions to exist. It treats the case where
the equation has Delone set solutions. Finally, it gives a large set of
examples showing limits to the results obtained.Comment: 34 pages, latex file; some results in Sect 5 rearranged and theorems
reformulate
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