Topology

Milgram R. (ed.)'s Algebraic and Geometric Topology, Part 2 PDF

By Milgram R. (ed.)

ISBN-10: 0821814338

ISBN-13: 9780821814338

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Extra resources for Algebraic and Geometric Topology, Part 2

Example text

It is in particular true that μ, ν 2 Lp implies that σγ 2 Lp and that it is sufficient that one of the two measures is absolutely continuous in order to get the same on σγ . Models and applications of optimal transport 25 The simplest possible generalization of problem (B) is the following: min k(x)jv(x)jdx : r v D μ ν, which corresponds, by duality with the functions u such that jruj Ä k, to min dk (x, y)dγ : γ 2 (μ, ν), where dk (x, y) D inf ω(0)Dx, ω(1)Dy Lk (ω) :D 0 k(ω(t))jω0 (t)jdt is the distance associated with the Riemannian metric k.

An. (40), no. 1, 1–20, 2008. [11] C. Jimenez, Optimisation de probl`emes de transport, PhD thesis, Universit´e du Sud-Toulon-Var, 2005. [12] L. Kantorovich, On the transfer of masses. Dokl. Acad. Nauk. USSR, (37), 7–8, 1942. [13] R. J. McCann, A convexity principle for interacting gases. Adv. Math. (128), no. 1, 153–159, 1997. [14] G. Monge, M¨umoire sur la th¨uorie des d¨ublais et des remblais, Histoire de l’Acad´emie Royale des Sciences de Paris, 666–704, 1781. [15] C. Villani. Topics in Optimal Transportation.

R A wage function ψ : ! R, where ψ(x) stands for the salary that workers employed by the firm located at x receive from the firm (this is an unknown of the problem). r A revenue function φ : ! R (unknown) standing for the revenues net of commuting cost that residents earn; people living at x will solve maxy ψ(y) c(x, y) :D φ(x) so as to choose where to work according to this optimization problem and, conversely, firms located at y will decide whom to hire solving minx φ(x) C c(x, y) and getting again ψ(y) as a (minimal) wage to be assured at y so that there are workers who do accept to work at y.

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Algebraic and Geometric Topology, Part 2 by Milgram R. (ed.)


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