Vol.10, Issue 7, July 2024

1-Convex Transferable Utility Games A Reappraisal

Pierre Dehez

Center for Operations Research and Econometrics (CORE), University of Louvain, 34 Voie du Roman Pays, 1348 Louvain-la-Neuve, Belgium

Abstract:

1-convex games have been introduced by Theo Driessen in his 1985 PhD dissertation. They form an interesting class of games for at least one reason: the core of a 1-convex n-player game is a regular simplex of dimension n – 1 or a single point. As a consequence, its nucleolus is the center of gravity of the core. We recall and extend the results obtained by Driessen and provide examples and applications.

Keywords: transferable utility games, core, nucleolus, Shapley value.

Pages: 97 – 118 | Full PDF Paper

The Strongly Solutions of Nonlinear Parabolic Partial Differential Equations Problems in Sobolev Spaces

Omar Mohammed Abdullah Al-haj, Mohammed Al-Hawmi

Department of Mathematics, Faculty of Education and Science, University of Saba Region, Marib, Yemen.

Abstract:

In this paper, we study the existence of a strong solutions for the initial-boundary value problems of the nonlinear degenerated parabolic equation

∂u/∂t + A(u) + g(x, t, u, ∇u) = f in Q

where A(u) = -div a(x, t, u, ∇u), is a Leary lions operator acted from the weighted Sobolev Space L^P(0, T, W₀^{1, p}(Ω, ω)) in to its dual L^P’(0, T, W₀^{-1, p’}(Ω, ω^*)) and g(x, t, u, ∇u) is a nonlinear term with critical growth condition with respect to u. The source term f is assumed to belong to L^p’(0, T, W₀^{-1, p’}(Ω, ω^*)).

Keywords: Weighted Sobolev Spaces, Boundary Value problems, parabolic problems, nonlinear equation, Compactness, a time mollification sequence, measurable, Compact set, Continuous, a symmetric bilinear, weak convergence, Holders inequality, strongly solution, integral, continuity, weak solution, converges strongly, positive constant.

Pages: 119 – 151 | Full PDF Paper