71,691 research outputs found

    L-Series, modular imbeddings, and signatures

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    Monochromatic sums and products

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    Suppose that Fp\mathbb{F}_p is coloured with rr colours. Then there is some colour class containing at least crp2c_r p^2 quadruples of the form (x,y,x+y,xy)(x, y , x + y, xy).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

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    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 ϵ\epsilon 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

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    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 JJ-characteristics of a design and showed that they uniquely determine the design. While both the GWLP and the JJ-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 JJ-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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