A generalized view of Duality is offered as a bridge between physical sciences and the more abstract philosophical dimensions bordering on mysticism. To that end several examples of duality are first cited from from conventional physics sectors to illustrate the obvious powers of this principle. These include items from reciprocity in Newtonian mechanics to the problem of measurement duality th...

We present a duality principle for the Legendre transform that yields the shortest path between the graphs of functions and embodies the underlying Nash equilibrium. A useful feature of the algorithm for the shortest path obtained in this way is that its implementation has a local character in the sense that it is applicable at any point in the domain with no reference to calculations made earl...

In this paper we relate two generalisations of the finite monoid recognisers of automata theory for the study of circuit complexity classes: Boolean spaces with internal monoids and typed monoids. Using the setting of stamps, this allows us to generalise a number of results from algebraic automata theory as it relates to Büchi’s logic on words. We obtain an Eilenberg theorem, a substitution pri...

The scale and diversity of networked sources of data and computer programs is rapidly swamping human abilities to digest and even locate relevant information. The high speed of computing has compounded this problem by the generation of even larger amounts of data, derived in ways that are generally opaque to human users. The result is an increasing gulf between human and computer abilities. Soc...

A dinner table seats k guests and holds n discrete morsels of food. Guests select morsels in turn until all are consumed. Each guest has a ranking of the morsels according to how much he would enjoy eating them; these rankings are commonly known. A gallant knight always prefers one food division over another if it provides strictly more enjoyable collections of food to one or more other players...

The quantum duality principle (QDP) for homogeneous spaces gives four recipes to obtain, from a quantum homogeneous space, a dual one, in the sense of Poisson duality. One of these recipes fails (for lack of the initial ingredient) when the homogeneous space we start from is not a quasi-affine variety. In this work we solve this problem for the quantum Grassmannian, a key example of quantum pro...

Let E a C( AT) be a subspace of continuous functions defined on a compact Hausdorff space K. We characterize those spaces for which the rational functions with denominators and numerators from E are dense. Despite the non-linear structure of rational functions, this characterization uses only methods from linear functional analysis. As special cases, we recover various results on the density of...

We present a new duality principle, in which we do not suppose that the range of the functions that are to be optimized is a subset of a linear space. Therefore, unlike the classical optimization theory, our main result can be applied for example to functions with ranges in Heisenberg groups.

The theory of duality for linear programs is well-developed and has been successful in advancing both the theory and practice of linear programming. In principle, much of this broad framework can be extended to mixed integer linear programs, but this has proven difficult, in part because duality theory does not integrate well with current computational practice. This paper surveys what is known...