Proof. First we prove the “if” direction. Fix agent 1 and preferences ✓1. Suppose that S1 is not obviously dominant in G = hH, , A,A, P, c, (Ii)i2N , gi. We need to demonstrate that there exists G̃ that is i-indistinguishable from G, such that G,G̃(S1) is not weakly dominant in G̃ . We proceed by construction. Let (S0 1, I1, h sup, S 1 , d sup c , hinf , Sinf 1, d inf c ) be such that I1 2 ↵(S1, S...

Proof. Let us first assume that c > 0. To prove that sup(cA) = c(sup A), we claim that sup(cA) ≤ c(sup A) and sup(cA) ≥ c(sup A). For the first inequality, let a′ ∈ cA be arbitrary. Then a′ = ca for some a ∈ A. But since a ≤ sup A, we have a′ = ca ≤ c(sup A). This shows that c(sup A) is an upper bound of cA, hence we have sup(cA) ≤ c(sup A). The reverse inequality also follows in a similar mann...

We develop a mixed finite-element approximation scheme for Kirchhoff plate theory based on the reformulation of Kirchhoff plate theory of Ortiz and Morris [1]. In this reformulation the moment-equilibrium problem for the rotations is in direct analogy to the problem of incompressible two-dimensional elasticity. This analogy in turn opens the way for the application of diamond approximation sche...

This paper is concerned with the stability of Petrov-Galerkin discretizations with application to parabolic evolution problems in space-time weak form. We will prove that the discrete inf-sup condition for an a priori fixed Petrov-Galerkin discretization is satisfied uniformly under standard approximation and smoothness conditions without any further coupling between the discrete trialand test ...

Motivated by the work of Fleming [5], we shall provide the general framework to associate inf-sup type values with the Isaacs equations. We shall show that upper and lower bounds for the generators of inf-sup type are upper and lower Hamiltonian in differential games respectively. In particular, lower (resp.upper) bound corresponds to progressive (resp. strictly progressive) strategy. Under Dyn...

Using an axiomatic formulation, we devise a diiusion equation on movies that preserves the 3D structure of objects, an interesting property for depth recovery tasks. Since this equation presents a strong singularity and is not handled by the classical theory of viscosity solutions, we study the existence and unicity of weak solutions, and point out some properties, including a variational inter...

and Applied Analysis 3 Our main result is the following theorem. Theorem 1.1. Suppose that F t, x satisfies assumptions (A) and 1.7 . Assume that lim sup r→ ∞ inf x∈RN,|x| r |x|−2α ∫T 0 F t, x dt ∞, 1.8 lim inf R→ ∞ sup x∈RN,|x| R |x|−2α ∫T 0 F t, x dt −∞. 1.9

In this paper we extend the so called inf-sup conditions to non linear problems having regular solution branches. The inf-sup conditions for non linear problems are introduced for instance in Caloz and Rappaz 1994]. Here we present a more general approach to include turning points on a solution branch. Our abstract results are applied to model examples.

and rðtÞ :1⁄4 sup{t , t : t [ T}; for all t [ T, where inf Y: 1⁄4 sup T and sup Y: 1⁄4 inf T, where Y denotes the empty set. We assume throughout that T has the topology that it inherits from the standard topology on the real numbers R. If s(t) . t, we say t is right-scattered, while if r(t) , t we say t is leftscattered. If s(t) 1⁄4 t and t , sup T we say t is right-dense, while if r(t) 1⁄4 t ...