5,024 research outputs found

    Tropical totally positive matrices

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    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

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    Given an n×n×pn\times n\times p cost array CC we consider the problem pp-P3AP which consists in finding pp pairwise disjoint permutations φ1,φ2,…,φp\varphi_1,\varphi_2,\ldots,\varphi_p of {1,…,n}\{1,\ldots,n\} such that ∑k=1p∑i=1nciφk(i)k\sum_{k=1}^{p}\sum_{i=1}^nc_{i\varphi_k(i)k} is minimized. For the case p=np=n the planar 3-dimensional assignment problem P3AP results. Our main result concerns the pp-P3AP on cost arrays CC that are layered Monge arrays. In a layered Monge array all n×nn\times n matrices that result from fixing the third index kk are Monge matrices. We prove that the pp-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 pp-3PAP which can be represented as matrix with bandwidth ≤4p−3\le 4p-3. This structural result allows us to provide a dynamic programming algorithm that solves the pp-P3AP in polynomial time on layered Monge arrays when pp is fixed.Comment: 16 pages, appendix will follow in v

    Monge Distance between Quantum States

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    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

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    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 W2,pW^{2,p} estimates for solutions to the linearized Monge--Amp\`ere equations

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    In this paper, we establish global W2,pW^{2,p} 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 W2,pW^{2,p} estimates of Winter for fully nonlinear, uniformly elliptic equations, and also linearized counterparts of Savin's global W2,pW^{2,p} estimates for the Monge-Amp\`ere equations.Comment: v2: presentation improve
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