71,692 research outputs found
Monochromatic sums and products
Suppose that is coloured with colours. Then there is some
colour class containing at least quadruples of the form .Comment: 48 pages, accepted for publication in Discrete Analysis. Second
version has minor changes arising from the referee report. Third version
updated to DAJ format. in Discrete Analysis 2016:
Duality of compact groups and Hilbert C*-systems for C*-algebras with a nontrivial center
In the present paper we prove a duality theory for compact groups in the case
when the C*-algebra A, the fixed point algebra of the corresponding Hilbert
C*-system (F,G), has a nontrivial center Z and the relative commutant satisfies
the minimality condition A.'\cap F = Z as well as a technical condition called
regularity. The abstract characterization of the mentioned Hilbert C*-system is
expressed by means of an inclusion of C*-categories T_\c < T, where T_\c{i}s a
suitable DR-category and T a full subcategory of the category of endomorphisms
of A. Both categories have the same objects and the arrows of T can be
generated from the arrows of T_\c{a}nd the center Z.
A crucial new element that appears in the present analysis is an abelian
group C(G), which we call the chain group of G, and that can be constructed
from certain equivalence relation defined on G^, the dual object of G. The
chain group, which is isomorphic to the character group of the center of G,
determines the action of irreducible endomorphisms of A when restricted to Z.
Moreover, C(G) encodes the possibility of defining a symmetry also
for the larger category T of the previous inclusion.Comment: Final version appeared in Int. J. Math. 15 (2004) 759-812. Minor
changes w.r.t. to the previous versio
Invariance of generalized wordlength patterns
The generalized wordlength pattern (GWLP) introduced by Xu and Wu (2001) for
an arbitrary fractional factorial design allows one to extend the use of the
minimum aberration criterion to such designs. Ai and Zhang (2004) defined the
-characteristics of a design and showed that they uniquely determine the
design. While both the GWLP and the -characteristics require indexing the
levels of each factor by a cyclic group, we see that the definitions carry over
with appropriate changes if instead one uses an arbitrary abelian group. This
means that the original definitions rest on an arbitrary choice of group
structure. We show that the GWLP of a design is independent of this choice, but
that the -characteristics are not. We briefly discuss some implications of
these results.Comment: To appear in: Journal of Statistical Planning and Inferenc
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