5,024 research outputs found
Tropical totally positive matrices
We investigate the tropical analogues of totally positive and totally
nonnegative matrices. These arise when considering the images by the
nonarchimedean valuation of the corresponding classes of matrices over a real
nonarchimedean valued field, like the field of real Puiseux series. We show
that the nonarchimedean valuation sends the totally positive matrices precisely
to the Monge matrices. This leads to explicit polyhedral representations of the
tropical analogues of totally positive and totally nonnegative matrices. We
also show that tropical totally nonnegative matrices with a finite permanent
can be factorized in terms of elementary matrices. We finally determine the
eigenvalues of tropical totally nonnegative matrices, and relate them with the
eigenvalues of totally nonnegative matrices over nonarchimedean fields.Comment: The first author has been partially supported by the PGMO Program of
FMJH and EDF, and by the MALTHY Project of the ANR Program. The second author
is sported by the French Chateaubriand grant and INRIA postdoctoral
fellowshi
Planar 3-dimensional assignment problems with Monge-like cost arrays
Given an cost array we consider the problem -P3AP
which consists in finding pairwise disjoint permutations
of such that
is minimized. For the case
the planar 3-dimensional assignment problem P3AP results.
Our main result concerns the -P3AP on cost arrays that are layered
Monge arrays. In a layered Monge array all matrices that result
from fixing the third index are Monge matrices. We prove that the -P3AP
and the P3AP remain NP-hard for layered Monge arrays. Furthermore, we show that
in the layered Monge case there always exists an optimal solution of the
-3PAP which can be represented as matrix with bandwidth . This
structural result allows us to provide a dynamic programming algorithm that
solves the -P3AP in polynomial time on layered Monge arrays when is
fixed.Comment: 16 pages, appendix will follow in v
Monge Distance between Quantum States
We define a metric in the space of quantum states taking the Monge distance
between corresponding Husimi distributions (Q--functions). This quantity
fulfills the axioms of a metric and satisfies the following semiclassical
property: the distance between two coherent states is equal to the Euclidean
distance between corresponding points in the classical phase space. We compute
analytically distances between certain states (coherent, squeezed, Fock and
thermal) and discuss a scheme for numerical computation of Monge distance for
two arbitrary quantum states.Comment: 9 pages in LaTex - RevTex + 2 figures in ps. submitted to Phys. Rev.
Some recent results in the analysis of greedy algorithms for assignment problems
We survey some recent developments in the analysis of greedy algorithms for assignment and transportation problems. We focus on the linear programming model for matroids and linear assignment problems with Monge property, on general linear programs, probabilistic analysis for linear assignment and makespan minimization, and on-line algorithms for linear and non-linear assignment problems
Global estimates for solutions to the linearized Monge--Amp\`ere equations
In this paper, we establish global estimates for solutions to the
linearized Monge-Amp\`ere equations under natural assumptions on the domain,
Monge-Amp\`ere measures and boundary data. Our estimates are affine invariant
analogues of the global estimates of Winter for fully nonlinear,
uniformly elliptic equations, and also linearized counterparts of Savin's
global estimates for the Monge-Amp\`ere equations.Comment: v2: presentation improve
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