30,630 research outputs found
On higher holonomy invariants in higher gauge theory II
This is the second of a series of two technical papers devoted to the
analysis of holonomy invariants in strict higher gauge theory with end
applications in higher Chern--Simons theory. We provide a definition of trace
over a crossed module such to yield surface knot invariants upon application to
2-holonomies. We show further that the properties of the trace are best
described using the theory quandle crossed modules.Comment: Latex, 34 pages, no figure
Crystal isomorphisms in Fock spaces and Schensted correspondence in affine type A
We are interested in the structure of the crystal graph of level Fock
spaces representations of . Since
the work of Shan [26], we know that this graph encodes the modular branching
rule for a corresponding cyclotomic rational Cherednik algebra. Besides, it
appears to be closely related to the Harish-Chandra branching graph for the
appropriate finite unitary group, according to [8]. In this paper, we make
explicit a particular isomorphism between connected components of the crystal
graphs of Fock spaces. This so-called "canonical" crystal isomorphism turns out
to be expressible only in terms of: - Schensted's classic bumping procedure, -
the cyclage isomorphism defined in [13], - a new crystal isomorphism, easy to
describe, acting on cylindric multipartitions. We explain how this can be seen
as an analogue of the bumping algorithm for affine type . Moreover, it
yields a combinatorial characterisation of the vertices of any connected
component of the crystal of the Fock space
Linear Connections in Non-Commutative Geometry
A construction is proposed for linear connections on non-commutative
algebras. The construction relies on a generalisation of the Leibnitz rules of
commutative geometry and uses the bimodule structure of . A special
role is played by the extension to the framework of non-commutative geometry of
the permutation of two copies of . The construction of the linear
connection as well as the definition of torsion and curvature is first proposed
in the setting of the derivations based differential calculus of Dubois-
Violette and then a generalisation to the framework proposed by Connes as well
as other non-commutative differential calculi is suggested. The covariant
derivative obtained admits an extension to the tensor product of several copies
of . These constructions are illustrated with the example of the
algebra of matrices.Comment: 15 pages, LMPM ../94 (uses phyzzx
Towards correct-by-construction product variants of a software product line: GFML, a formal language for feature modules
Software Product Line Engineering (SPLE) is a software engineering paradigm
that focuses on reuse and variability. Although feature-oriented programming
(FOP) can implement software product line efficiently, we still need a method
to generate and prove correctness of all product variants more efficiently and
automatically. In this context, we propose to manipulate feature modules which
contain three kinds of artifacts: specification, code and correctness proof. We
depict a methodology and a platform that help the user to automatically produce
correct-by-construction product variants from the related feature modules. As a
first step of this project, we begin by proposing a language, GFML, allowing
the developer to write such feature modules. This language is designed so that
the artifacts can be easily reused and composed. GFML files contain the
different artifacts mentioned above.The idea is to compile them into FoCaLiZe,
a language for specification, implementation and formal proof with some
object-oriented flavor. In this paper, we define and illustrate this language.
We also introduce a way to compose the feature modules on some examples.Comment: In Proceedings FMSPLE 2015, arXiv:1504.0301
Explaining Gabriel-Zisman localization to the computer
This explains a computer formulation of Gabriel-Zisman localization of
categories in the proof assistant Coq. It includes both the general
localization construction with the proof of GZ's Lemma 1.2, as well as the
construction using calculus of fractions. The proof files are bundled with the
other preprint "Files for GZ localization" posted simultaneously
- …