Introduction to linear logic and ludics, part II

This paper is the second part of an introduction to linear logic and ludics, both due to Girard. It is devoted to proof nets, in the limited, yet central, framework of multiplicative linear logic and to ludics, which has been recently developped in an aim of further unveiling the fundamental interactive nature of computation and logic. We hope to offer a few computer science insights into this new theory

Computational Tools for Cohomology of Toric Varieties

In this review, novel non-standard techniques for the computation of cohomology classes on toric varieties are summarized. After an introduction of the basic definitions and properties of toric geometry, we discuss a specific computational algorithm for the determination of the dimension of line-bundle valued cohomology groups on toric varieties. Applications to the computation of chiral massless matter spectra in string compactifications are discussed and, using the software package cohomCalg, its utility is highlighted on a new target space dual pair of (0,2) heterotic string models.Comment: 17 pages, 4 tables; prepared for the special issue "Computational Algebraic Geometry in String and Gauge Theory" of Advances in High Energy Physics, cohomCalg implementation available at http://wwwth.mppmu.mpg.de/members/blumenha/cohomcalg

The Lifting Properties of A-Homotopy Theory

In classical homotopy theory, two spaces are homotopy equivalent if one space can be continuously deformed into the other. This theory, however, does not respect the discrete nature of graphs. For this reason, a discrete homotopy theory that recognizes the difference between the vertices and edges of a graph was invented, called A-homotopy theory [1-5]. In classical homotopy theory, covering spaces and lifting properties are often used to compute the fundamental group of the circle. In this paper, we develop the lifting properties for A-homotopy theory. Using a covering graph and these lifting properties, we compute the fundamental group of the 5-cycle , giving an alternate approach to [4].Comment: 27 pages, 3 figures, updated version. Minor changes to the introduction and clarification that the computation of the fundamental group of the 5-cycle originally appeared in [4]. Title changed from "Computing A-Homotopy Groups Using Coverings and Lifting Properties" to "The Lifting Properties of A-Homotopy Theory

Covariance approach to the free photon field

We introduce photon theory following the same principles as for introduction of the quantum theory of a single particle, using a C*-algebraic approach based on covariance systems. The basic symmetries are additivity of the fields and additivity of test functions. We write down in explicit form a state of this covariance system. It turns out to reproduce the traditional Fock representation of the free photon field, with a Lorentz invariant vacuum. Properties of smeared-out photons are discussed.Comment: Latex, 24 pages, to appear in: "Probing the structure of Quantum Mechanics: nonlinearity, nonlocality, computation and axiomatics", eds. D. Aerts, M. Czachor, and T. Durt (World Scientific, 2002