The normal map has proven to be a powerful tool for solving generalized equations of the form: find ; £ C, with 0 e F{z) + NiAz). where C is a convex set and Nr(z) is the normal cone to C at ;. In this paper, we use the 7"-map, a generalization of the normal map, to solve equations of the more general form: find : E dom(r), with 0 ^ E(z) + T(z). where T is a maximal monotone multifunction. We p...

The co-elementary hierarchy is a nested ordinalindexed sequence of classes of mappings between compacta, with each successor level being defined inductively from the previous one using the topological ultracopower construction. The lowest level is the class of continuous surjections; and the next level up, the co-existential maps, is already a much more restricted class. Co-existential maps are...

In this note, we provide fixed-point characterizations for two solution concepts in finite games: bestresponse sets (BRS’s) and self-admissible sets (SAS’s). The BRS concept is due to Pearce [7, 1984]. The SAS concept is a weak-dominance analog to a BRS, and is due to BrandenburgerFriedenberg-Keisler [4, 2006]. BRS’s are important because they characterize the epistemic condition of rationality...

We consider foundational questions related to the de nition of functions by corecursion. This method is especially suited to functions into the greatest xed point of some monotone operator, and it is most applicable in the context of non-wellfounded sets. We review the work on the Special Final Coalgebra Theorem of Aczel [NWF] and the Corecursion Theorem of Barwise and Moss [VC]. We prove some ...

We present a new approach to the infinitesimal structure of Lipschitz maps into L, and as an application, we give an alternative proof of the main theorem of [CK06], that the Heisenberg group does not admit a bi-Lipschitz embedding in L. The proof uses the metric differentiation theorem of [Pau01] and the cut metric description in [CK06] to reduce the nonembedding argument to a classification o...

The co-elementary hierarchy is a nested ordinal-indexed sequence of classes of mappings between compacta, with each successor level being defined inductively from the previous one using the topological ultracopower construction. The lowest level is the class of continuous surjections; and the next level up, the coexistential maps, is already a much more restricted class. Co-existential maps are...

In this paper, we consider probability measures μ and ν on a d–dimensional sphere in R, d ≥ 1, and cost functions of the form c(x,y) = l( |x−y| 2 2 ) that generalize those arising in geometric optics where l(t) = − log t. We prove that if μ and ν vanish on (d− 1)–rectifiable sets, if |l′(t)| > 0, limt→0+ l(t) = +∞, and g(t) := t(2 − t)(l′(t))2 is monotone then there exists a unique optimal map ...

We say that a continuous map / of a compact interval to itself is linear Markov if it is piecewise linear, and the set of all fk(x), where k > 0 and x is an endpoint of a linear piece, is finite. We provide an effective classification, up to topological conjugacy, for linear Markov maps and an effective procedure for determining whether such a map is transitive. We also consider expanding Marko...

(1) Ord: objects are (pre)ordered sets (= sets with a reflexive and transitive relation, no antisymmetry condition), with monotone maps. Formally: (X, a) with 1. a(x, x), 2. a(x, y) ∧ a(y, z) a(x, z), f : (X, a) − → (Y, b) with a(x, y) b(f (x), f (y)). (2) Met: objects are (generalized) metric spaces (=sets with a function a : X ×X − → [0, ∞] that is 0 on the diagonal and satisfies the triangle...

Abstract. Sticky particle solutions to the one-dimensional pressureless gas dynamics equations can be constructed by a suitable metric projection onto the cone of monotone maps, as was shown in recent work by Natile and Savaré. Their proof uses a discrete particle approximation and stability properties for first order differential inclusions. Here we give a more direct proof that relies on a re...