Given a hyperplane arrangement in an affine space equipped with a linear functional, we define two finite-dimensional, noncommutative algebras, both of which are motivated by the geometry of hypertoric varieties. We show that these algebras are Koszul dual to each other, and that the roles of the two algebras are reversed by Gale duality. We also study the centers and representation categories ...

2 Algebraic preliminaries 4 2.1 Notations and conventions . . . . . . . . . . . . . . . . . . . . 4 2.2 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2.1 Local Duality . . . . . . . . . . . . . . . . . . . . . . . 5 2.2.2 Global Duality . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4 Prelimin...

The zero duality gap that underpins the duality theory is one of the central ingredients in optimisation. In convex programming, it means that the optimal values of a given convex program and its associated dual program are equal. It allows, in particular, the development of efficient numerical schemes. However, the zero duality gap property does not always hold even for finite dimensional prob...

In this letter we implement a recently proposed spacetime duality approach to dualize a two dimensional, Abelian, gauge field theory, which has no dual version under p–duality. Our result suggests that spacetime duality spans a new, wider, class of dual theories, which cannot be related one to another by p–duality transformations.

Duality is an important notion for nonlinear programming (NLP). It provides a theoretical foundation for many optimization algorithms. Duality can be used to directly solve NLPs as well as to derive lower bounds of the solution quality which have wide use in other high-level search techniques such as branch and bound. However, the conventional duality theory has the fundamental limit that it le...

we study the notion of harmonicity in the sense of symplectic geometry, and investigate the geometric properties of harmonic thom forms and distributional thom currents, dual to different types of submanifolds. we show that the harmonic thom form associated to a symplectic submanifold is nowhere vanishing. we also construct symplectic smoothing operators which preserve the harmonicity of distri...