65,912 research outputs found
Point-Free, Set-Free Concrete Linear Algebra
International audienceWe show how a simple variant of Gaussian elimination can be used to model abstract linear algebra directly, using matrices only to represent all categories of objects, with operations such as subspace intersection and sum. We can even provide effective support for direct sums and subalgebras. We have formalized this work in Coq, and used it to develop all of the group representation theory required for the proof of the Odd Order Theorem, including results such as the Jacobson Density Theorem, Clifford's Theorem, the Jordan-Holder Theorem for modules, the Wedderburn Structure Theorem for semisimple rings (the basis for character theory).On présente une formalisation en Coq de l'algèbre linéaire où tous les objets sont représentés par des matrices, y compris les sous-espaces. Ce développement a été utilisé pour élaborer la formalisation des éléments de théorie de la représentation nécessaires à la prévue du théorème de Feit-Thompson
An Algebraic Framework for Compositional Program Analysis
The purpose of a program analysis is to compute an abstract meaning for a
program which approximates its dynamic behaviour. A compositional program
analysis accomplishes this task with a divide-and-conquer strategy: the meaning
of a program is computed by dividing it into sub-programs, computing their
meaning, and then combining the results. Compositional program analyses are
desirable because they can yield scalable (and easily parallelizable) program
analyses.
This paper presents algebraic framework for designing, implementing, and
proving the correctness of compositional program analyses. A program analysis
in our framework defined by an algebraic structure equipped with sequencing,
choice, and iteration operations. From the analysis design perspective, a
particularly interesting consequence of this is that the meaning of a loop is
computed by applying the iteration operator to the loop body. This style of
compositional loop analysis can yield interesting ways of computing loop
invariants that cannot be defined iteratively. We identify a class of
algorithms, the so-called path-expression algorithms [Tarjan1981,Scholz2007],
which can be used to efficiently implement analyses in our framework. Lastly,
we develop a theory for proving the correctness of an analysis by establishing
an approximation relationship between an algebra defining a concrete semantics
and an algebra defining an analysis.Comment: 15 page
Constructing cell data for diagram algebras
We show how the treatment of cellularity in families of algebras arising from
diagram calculi, such as Jones' Temperley--Lieb wreaths, variants on Brauer's
centralizer algebras, and the contour algebras of Cox et al (of which many
algebras are special cases), may be unified using the theory of tabular
algebras. This improves an earlier result of the first author (whose hypotheses
covered only the Brauer algebra from among these families).Comment: Approximately 38 pages, AMSTeX. Revised in light of referee comments.
To appear in the Journal of Pure and Applied Algebr
Unitary Easy Quantum Groups: the free case and the group case
Easy quantum groups have been studied intensively since the time they were
introduced by Banica and Speicher in 2009. They arise as a subclass of
(-algebraic) compact matrix quantum groups in the sense of Woronowicz. Due
to some Tannaka-Krein type result, they are completely determined by the
combinatorics of categories of (set theoretical) partitions. So far, only
orthogonal easy quantum groups have been considered in order to understand
quantum subgroups of the free orthogonal quantum group .
We now give a definition of unitary easy quantum groups using colored
partitions to tackle the problem of finding quantum subgroups of . In
the free case (i.e. restricting to noncrossing partitions), the corresponding
categories of partitions have recently been classified by the authors by purely
combinatorial means. There are ten series showing up each indexed by one or two
discrete parameters, plus two additional quantum groups. We now present the
quantum group picture of it and investigate them in detail. We show how they
can be constructed from other known examples using generalizations of Banica's
free complexification. For doing so, we introduce new kinds of products between
quantum groups.
We also study the notion of easy groups.Comment: 39 page
Fixed Point Algebras for Easy Quantum Groups
Compact matrix quantum groups act naturally on Cuntz algebras. The first
author isolated certain conditions under which the fixed point algebras under
this action are Kirchberg algebras. Hence they are completely determined by
their -groups. Building on prior work by the second author, we prove that
free easy quantum groups satisfy these conditions and we compute the -groups
of their fixed point algebras in a general form. We then turn to examples such
as the quantum permutation group , the free orthogonal quantum group
and the quantum reflection groups . Our fixed point-algebra
construction provides concrete examples of free actions of free orthogonal easy
quantum groups, which are related to Hopf-Galois extensions
Reeh-Schlieder Defeats Newton-Wigner: On alternative localization schemes in relativistic quantum field theory
Many of the "counterintuitive" features of relativistic quantum field theory
have their formal root in the Reeh-Schlieder theorem, which in particular
entails that local operations applied to the vacuum state can produce any state
of the entire field. It is of great interest, then, that I.E. Segal and, more
recently, G. Fleming (in a paper entitled "Reeh-Schlieder Meets Newton-Wigner")
have proposed an alternative "Newton-Wigner" localization scheme that avoids
the Reeh-Schlieder theorem. In this paper, I reconstruct the Newton-Wigner
localization scheme and clarify the limited extent to which it avoids the
counterintuitive consequences of the Reeh-Schlieder theorem. I also argue that
neither Segal nor Fleming has provided a coherent account of the physical
meaning of Newton-Wigner localization.Comment: 25 pages, LaTe
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