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- G. L. Litvinov, P. Del Moral, W. H. Fleming, J. S. Golan, M. Gondran, V. N. Kolokoltsov +20 others
- 2006

This paper is a brief introduction to idempotent and tropical mathematics. Tropical mathematics can be treated as the result of the so-called Maslov dequantization of the traditional mathematics over numerical fields as the Planck constant tends to zero taking imaginary values. Bibliography: 187 titles. To Anatoly Vershik with admiration and gratitude This… (More)

- G. Cohen, S. Gaubert, J. P. Quadrat
- 1995

We show that Continuous Timed Petri Nets (CTPN) can be modeled by generalized polynomial recurrent equations in the (min,+) semiring. We establish a correspondence between CTPN and Markov decision processes. We survey the basic system theoretical results available: behavioral (input-output) properties, algebraic representations, asymptotic regime. A… (More)

- G. Cohen, S. Gaubert, J.-P. Quadrat
- 1997

In general semimodules, we say that the image of a linear operator B and the kernel of a linear operator C are direct factors if every equivalence class modulo C crosses the image of B at a unique point. For linear maps represented by matrices over certain idempotent semifields such as the (max, +)-semiring, we give necessary and sufficient conditions for… (More)

- J. P. QUADRAT
- 1996

We revisit some results obtained recently in min-plus algebra following the ideas of statistical mechanics. Computation of geodes-ics in a graph can be done by min-plus matrix products. A min-plus matrix is seen as a kind of finite states mechanical system. The energy of this system is the eigenvalue of its min-plus matrix. The graph interpretation of the… (More)

- G. Cohen, S. Gaubert, J.-P. Quadrat, Peter Kokotovic
- 2004

In maxplus algebra, linear projectors on an image of a morphism B parallel to the kernel of another morphism C can be built under transversality conditions of the two morphisms. The existence of a transverse to an image or a kernel of a morphism is obtained under some regularity conditions. We show that those regularity and transversality conditions can be… (More)

- G. Cohen, S. Gaubert, J. P. Quadrat
- 1997

| We study uid analogues of a subclass of Petri nets, called Fluid Timed Event Graphs with Multipliers, which are a timed extension of weighted T-Systems studied in the Petri Net literature. These event graphs can be studied naturally, with a new algebra, analogous to the min-plus algebra, but deened on piecewise linear concave increasing functions, endowed… (More)

The Traffic Assignment problem consists in determining the routes used by sets of network users taking into account the link congestions. In deterministic modelling, Wardrop Equilibriums are computed. They can be reduced to huge nonlinear multiflow problems in the simplest cases. In stochastic modelling, Logit Assignments are used. They can be obtained,… (More)

- Bibliography, B D O Anderson, M Deistler, L Farina, L Benvenuti, Nonnegative +8 others
- 1996

realizations of a linear system with nonnegative impulse response.

- J. P. Quadrat
- 1998

We study here the duality appearing between probability calculus and optimization by substituting the semifield (R ∪ {+∞}, min, +) for the standard semiring (R + , +, ×).

- O Fall, J P Quadrat
- 1998

We study here the min-plus analogues of Jackson networks of queues and show that the corresponding geodesic problems on Z m can be reduced to minimal cost flow problems on complete graphs having m nodes. In some particular cases, these flow problems can be solved explicitly giving formulae analogue , in the min-plus algebra, to the standard product forms.… (More)