I Basic Results on Normal Form Games 3 I.1 The Minmax Theorem 4 I.1.a Definitions and Notations 4 I.1.b A Basic Theorem 5 I.1.c Convexity 6 I.1.d Mixed Strategies 6 I.1.e Note on the Separation Theorem 9 Exercises 10 I.2 Complements to the Minmax Theorem 15 I.2.a The Topology on S 16 I.2.b Lack of Continuity: Regularization 16 I.2.c Lack of Compactness: Approximation 19 I.2.d Measurability: Sym...

Convexity is important in several elds, and we have some theories on it. In this thesis, we discuss a kind of combinatorial convexity, in particular, antimatroids and convex geometries. An antimatroid is a combinatorial abstraction of convexity. It has some di erent origins; by Dilworth in lattice theory, by Edelman and Jamison in the notions of convexity, by Korte{Lov asz who were motivated by...

A set K of nodes of a graph G is geodesically convex (respectively, monophonically convex) if K contains every node on every shortest (respectively, chordless) path joining nodes in K. We investigate the classes of graphs which are characterized by certain local convexity conditions with respect to geodesic convexity, in particular, those graphs in which balls around nodes are convex, and those...

Based on a classic paper by Ginibre [Commun. Math. Phys. 8, 26 (1968)] it is shown that, whenever Bogoliubov's approximation, that is, the replacement of a(0) and a*(0) by complex numbers in the Hamiltonian, asymptotically yields the right pressure, it also implies the asymptotic equality of |a(0)|(2)/V and a*(0)a(0)/V in symmetry breaking fields, irrespective of the existence or absence of Bos...

In this article we introduce a notion of normalized angle for Lorentzian pre-length spaces. This concept allows us to prove some equivalences the definition timelike curvature bounds from below Specifically, establish comparison theorems known as local version Toponogov theorem and Alexandrov convexity property. Finally, an application obtain first variation Formula non-negatively curved global...

Gaussian comparison theorems are useful tools in probability theory; they are essential ingredients in the classical proofs of many results in empirical processes and extreme value theory. More recently, they have been used extensively in the analysis of underdetermined linear inverse problems. A prominent role in the study of those problems is played by Gordon’s Gaussian min-max theorem. It ha...

It is proved that if 0 < λ ≤ f(x) ≤ Λ for x ∈ Ω, where Ω ⊂ R is a bounded convex domain, and f is L-Dini continuous on Ω, then there exist infinitely many biLipschitz maps F : Ω → R such that detDF (x) = f(x) for a.e. x ∈ Ω. Moreover, these mappings can be chosen to have convex potentials. We relate our result to a classical theorem by H. M. Reimann; however, the emphasis is on the novel use of...