Regular Objects, Multiplicative Unitaries and Conjugation

The notion of left (resp. right) regular object of a tensor C*-category equipped with a faithful tensor functor into the category of Hilbert spaces is introduced. If such a category has a left (resp. right) regular object, it can be interpreted as a category of corepresentations (resp. representations) of some multiplicative unitary. A regular object is an object of the category which is at the same time left and right regular in a coherent way. A category with a regular object is endowed with an associated standard braided symmetry. Conjugation is discussed in the context of multiplicative unitaries and their associated Hopf C*-algebras. It is shown that the conjugate of a left regular object is a right regular object in the same category. Furthermore the representation category of a locally compact quantum group has a conjugation. The associated multiplicative unitary is a regular object in that category.Comment: 48 pages, Late

Ergodic actions of S_\mu U(2) on C*-algebras from II_1 subfactors

To a proper inclusion N\subset M of II_1 factors of finite Jones index [M:N], we associate an ergodic C*-action of the quantum group S_\mu U(2). The deformation parameter is determined by -1<\mu<0 and [M:N]=|\mu+\mu^{-1}|. The higher relative commutants can be identified with the spectral spaces of the tensor powers of the defining representation of the quantum group. This ergodic action may be thought of as a virtual subgroup of S_\mu U(2) in the sense of Mackey arising from the tensor category generated by M regarded as a bimodule over N. \mu is negative as M is a real bimodule.Comment: 25 pages, references adde

A rigidity result for extensions of braided tensor C*-categories derived from compact matrix quantum groups

Let G be a classical compact Lie group and G_\mu the associated compact matrix quantum group deformed by a positive parameter \mu (or a nonzero and real \mu in the type A case). It is well known that the category Rep(G_\mu) of unitary f.d. representations of G_\mu is a braided tensor C*-category. We show that any braided tensor *-functor from Rep(G_\mu) to another braided tensor C*-category with irreducible tensor unit is full if |\mu|\neq 1. In particular, the functor of restriction to the representation category of a proper compact quantum subgroup, cannot be made into a braided functor. Our result also shows that the Temperley--Lieb category generated by an object of dimension >2 can not be embedded properly into a larger category with the same objects as a braided tensor C*-subcategory.Comment: 19 pages; published version, to appear in CMP; for a more detailed exposition see v

The radial plot in meta-analysis : approximations and applications

Fixed effects meta-analysis can be thought of as least squares analysis of the radial plot, the plot of standardized treatment effect against precision (reciprocal of the standard deviation) for the studies in a systematic review. For example, the least squares slope through the origin estimates the treatment effect, and a widely used test for publication bias is equivalent to testing the significance of the regression intercept. However, the usual theory assumes that the within-study variances are known, whereas in practice they are estimated. This leads to extra variability in the points of the radial plot which can lead to a marked distortion in inferences that are derived from these regression calculations. This is illustrated by a clinical trials example from the Cochrane database. We derive approximations to the sampling properties of the radial plot and suggest bias corrections to some of the commonly used methods of meta-analysis. A simulation study suggests that these bias corrections are effective in controlling levels of significance of tests and coverage of confidence intervals

Thinning out redundant empirical data

Given a set $X$ of "empirical" points, whose coordinates are perturbed by errors, we analyze whether it contains redundant information, that is whether some of its elements could be represented by a single equivalent point. If this is the case, the empirical information associated to $X$ could be described by fewer points, chosen in a suitable way. We present two different methods to reduce the cardinality of $X$ which compute a new set of points equivalent to the original one, that is representing the same empirical information. Though our algorithms use some basic notions of Cluster Analysis they are specifically designed for "thinning out" redundant data. We include some experimental results which illustrate the practical effectiveness of our methods.Comment: 14 pages; 3 figure

Stable Border Bases for Ideals of Points

Let $X$ be a set of points whose coordinates are known with limited accuracy; our aim is to give a characterization of the vanishing ideal $I(X)$ independent of the data uncertainty. We present a method to compute a polynomial basis $B$ of $I(X)$ which exhibits structural stability, that is, if $\widetilde X$ is any set of points differing only slightly from $X$, there exists a polynomial set $\widetilde B$ structurally similar to $B$, which is a basis of the perturbed ideal $I(\widetilde X)$.Comment: This is an update version of "Notes on stable Border Bases" and it is submitted to JSC. 16 pages, 0 figure