Resource theories of knowledge

How far can we take the resource theoretic approach to explore physics? Resource theories like LOCC, reference frames and quantum thermodynamics have proven a powerful tool to study how agents who are subject to certain constraints can act on physical systems. This approach has advanced our understanding of fundamental physical principles, such as the second law of thermodynamics, and provided operational measures to quantify resources such as entanglement or information content. In this work, we significantly extend the approach and range of applicability of resource theories. Firstly we generalize the notion of resource theories to include any description or knowledge that agents may have of a physical state, beyond the density operator formalism. We show how to relate theories that differ in the language used to describe resources, like micro and macroscopic thermodynamics. Finally, we take a top-down approach to locality, in which a subsystem structure is derived from a global theory rather than assumed. The extended framework introduced here enables us to formalize new tasks in the language of resource theories, ranging from tomography, cryptography, thermodynamics and foundational questions, both within and beyond quantum theory.Comment: 28 pages featuring figures, examples, map and neatly boxed theorems, plus appendi

The convex real projective orbifolds with radial or totally geodesic ends: a survey of some partial results

A real projective orbifold has a radial end if a neighborhood of the end is foliated by projective geodesics that develop into geodesics ending at a common point. It has a totally geodesic end if the end can be completed to have the totally geodesic boundary. The purpose of this paper is to announce some partial results. A real projective structure sometimes admits deformations to parameters of real projective structures. We will prove a homeomorphism between the deformation space of convex real projective structures on an orbifold $\mathcal{O}$ with radial or totally geodesic ends with various conditions with the union of open subspaces of strata of the corresponding subset of $$Hom(\pi_{1}(\mathcal{O}), PGL(n+1, \mathbb{R}))/PGL(n+1, \mathbb{R}).$$ Lastly, we will talk about the openness and closedness of the properly (resp. strictly) convex real projective structures on a class of orbifold with generalized admissible ends.Comment: 36 pages, 2 figure. Corrected a few mistakes including the condition (NA) on page 22, arXiv admin note: text overlap with arXiv:1011.106

Polytopal linear algebra

We investigate similarities between the category of vector spaces and that of polytopal algebras, containing the former as a full subcategory. In Section 2 we introduce the notion of a polytopal Picard group and show that it is trivial for fields. The coincidence of this group with the ordinary Picard group for general rings remains an open question. In Section 3 we survey some of the previous results on the automorphism groups and retractions. These results support a general conjecture proposed in Section 4 about the nature of arbitrary homomorphisms of polytopal algebras. Thereafter a further confirmation of this conjecture is presented by homomorphisms defined on Veronese singularities. This is a continuation of the project started in our papers "Polytopal linear groups" (J. Algebra 218 (1999), 715--737), "Polytopal linear retractions" preprint, math.AG/0001049) and "Polyhedral algebras, arrangements of toric varieties, and their groups" (preprint, http://www.mathematik.uni-osnabrueck.de/K-theory/0232/index.html). The higher $K$-theoretic aspects of polytopal linear objects will be treated in "Polyhedral $K$-theory" (in preparation).Comment: 21 pages, uses pstricks and P. Taylor's CD package. Beitr. Algebra Geom., to appea