We give a simple proof of Strassen’s theorem on stochastic dominance using linear programming duality, without requiring measure-theoretic arguments. The result extends to generalized inequalities using conic optimization duality and provides an additional, intuitive optimization formulation for stochastic dominance.

In this lecture, we show the applications of the strong duality theorem, and discuss how to obtain min-max theorems and combinatorial algorithms from linear programming. We first introduce the 2 player, zero-sum game and show that this can be solved by minimax theorem and we also prove the minimax theorem by the LP-duality theorem. After that, we introduce some applications of minimax theorem, ...

We prove stable versions of trace theorems on the sphere in L with optimal constants, thus obtaining rather precise information regarding near-extremisers. We also obtain stability for the trace theorem into L for q > 2, by combining a refined Hardy–Littlewood–Sobolev inequality on the sphere with a duality-stability result proved very recently by Carlen. Finally, we extend a local version of C...

We study Translation functors and Wall-Crossing functors on infinite dimensional representations of a complex semisimple Lie algebra using D -modules. This functorial machinery is then used to prove the Endomorphism-theorem and the Structure-theorem, two important results established earlier by W. Soergel in a totally different way. Other applications to the category O of Bernstein-GelfandGelfa...

In multiflow maximization problems, there are several combinatorial duality relations, such as Ford-Fulkerson’s max-flow min-cut theorem for single commodity flows, Hu’s max-biflow min-cut theorem for two-commodity flows, Lovász-Cherkassky duality theorem for free multiflows, and so on. In this paper, we provide a unified framework for such multiflow combinatorial dualities by using the notion ...

3 General results on summability and duality 4 3.1 Total algebras and duality . . . . . . . . . . . . . . . . . . . . . . 4 3.1.1 Series and infinite sums . . . . . . . . . . . . . . . . . . . 4 3.1.2 Summable families in Hom spaces. . . . . . . . . . . . . . 5 3.1.3 Substitutions . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2 Theorem of Cartier-Quillen-Milnor-Moore (analytic form) . ....

We present a Stone duality for bitopological spaces in analogy to the duality between topological spaces and frames, and discuss the resulting notions of sobriety and spatiality. Under the additional assumption of regularity, we prove a characterisation theorem for subsets of a bisober space that are compact in one and closed in the other topology. This is in analogy to the celebrated Hofmann-M...

1. Tate’s theorem 2 Exercises 5 2. Introduction to group schemes 5 Definition (as a functor) 6 Definition (as a group object) 6 Examples of group schemes 8 Rank and the augmentation ideal 9 Subgroup schemes, morphisms and kernels 11 Diagonalizable group schemes 13 Constant group schemes 14 Exercises 14 3. Duality and Deligne’s theorem 16 Cartier duality 16 Deligne’s theorem 19 Exercises 22 4. É...

We prove a general width duality theorem for combinatorial structures with well-defined notions of cohesion and separation. These might be graphs and matroids, but can be much more general or quite di↵erent. The theorem asserts a duality between the existence of high cohesiveness somewhere local and a global overall tree structure. We describe cohesive substructures in a unified way in the form...