Overdetermined PDE Systems on Some Classes of Riemannian Manifolds
Maryam Samavaki, Jukka Tuomela
Department of Physics and Mathematics, University of Eastern Finland, P.O. Box 111, FI-80101, Joensuu, Finland.
Abstract: We study several classes of Riemannian manifolds which are dened by imposing a certain condition on the Ricci tensor. We consider the following cases: Ricci recurrent, Cotton, quasi Einstein, and pseudo Ricci symmetric condition. Such conditions can be interpreted as overdetermined PDE systems whose unknowns are the components of the Riemannian metric, and perhaps, in addition, some auxiliary functions. Hence even if the dimension of the manifold is small it is not easy to compute interesting examples by hand, and indeed very few examples appear in the literature. We will present large families of nontrivial examples of such manifolds. The relevant PDE systems are rst transformed into an involutive form. After that in many cases, one can actually solve the resulting system explicitly. However, the involutive form itself already gives a lot of information about the possible solutions to the given problem. We will also discuss some relationships between the relevant classes.
Keywords: Cotton tensor, conformally conservative manifold, pseudo Ricci symmetric manifold, quasi Einstein manifold, Ricci recurrent manifold, overdetermined PDE.
Pages: 1 – 17 | Full PDF Paper
The Inverse Problem Relative to the Shapley Value and the Convexity
University of Texas, Mathematics, Arlington, Texas 76019, USA.
Abstract: In an earlier work, for a given Shapley Value of a cooperative TU game we derived a procedure to find a new game in which the Shapley Value is the same, but it is coalitional rational. Recently, a similar problem has been solved for other two efficient values, the Egalitarian Allocation and the Egalitarian Nonseparable Contribution. In this paper, by using very similar ideas, we considered the problem: for a given Shapley Value of a cooperative TU game, find out a new game in which the Shapley Value is the same, but the game is convex. The procedure for finding such a game has been derived, depending on a parameter, and it was proved that in the new game, which is convex, the Shapley Value is unchanged and coalitional rational. As a corollary, it followed that in this new game the Egalitarian Nonseparable Contribution is unchanged and coalitional rational. Numerical examples illustrate the procedure.
Keywords: Inverse Set, Almost Null Family of Game, Convexity Threshold, Egalitarian Nonseparable Contribution.
Pages: 18 – 24 | Full PDF Paper