Spectral Measures for Sp(2)Sp(2)

Spectral measures provide invariants for braided subfactors via fusion modules. In this paper we study joint spectral measures associated to the compact connected rank two Lie group $SO(5)$ and its double cover the compact connected, simply-connected rank two Lie group $Sp(2)$, including the McKay graphs for the irreducible representations of $Sp(2)$ and $SO(5)$ and their maximal tori, and fusion modules associated to the $Sp(2)$ modular invariants.Comment: 41 pages, 45 figures. Title changed and notation corrected. arXiv admin note: substantial text overlap with arXiv:1404.186

Braided Subfactors, Spectral Measures, Planar algebras and Calabi-Yau algebras associated to SU(3) modular invariants

Braided subfactors of von Neumann algebras provide a framework for studying two dimensional conformal field theories and their modular invariants. We review this in the context of SU(3) conformal field theories through corresponding SU(3) braided subfactors and various subfactor invariants including spectral measures for the nimrep graphs, A_2-planar algebras and almost Calabi-Yau algebras.Comment: 45 pages, 25 figures. v3: minor correction to Figure 14; v2: figures of 0-1 parts of graphs included, some minor correction

Spectral Measures for G2G_2 II: finite subgroups

Joint spectral measures associated to the rank two Lie group $G_2$, including the representation graphs for the irreducible representations of $G_2$ and its maximal torus, nimrep graphs associated to the $G_2$ modular invariants have been studied. In this paper we study the joint spectral measures for the McKay graphs (or representation graphs) of finite subgroups of $G_2$. Using character theoretic methods we classify all non-conjugate embeddings of each subgroup into the fundamental representation of $G_2$ and present their McKay graphs, some of which are new.Comment: 33 pages, 20 figures; minor improvements to exposition. Accepted for publication in Reviews in Mathematical Physic

Spectral Measures for G2G_2

Spectral measures provide invariants for braided subfactors via fusion modules. In this paper we study joint spectral measures associated to the rank two Lie group $G_2$, including the McKay graphs for the irreducible representations of $G_2$ and its maximal torus, and fusion modules associated to all known $G_2$ modular invariants.Comment: 36 pages, 40 figures; correction to Sections 5.4 and 5.5, minor improvements to expositio

Limit theory for point processes in manifolds

Let $Y_i,i\geq1$, be i.i.d. random variables having values in an $m$-dimensional manifold $\mathcal {M}\subset \mathbb{R}^d$ and consider sums $\sum_{i=1}^n\xi(n^{1/m}Y_i,\{n^{1/m}Y_j\}_{j=1}^n)$, where $\xi$ is a real valued function defined on pairs $(y,\mathcal {Y})$, with $y\in \mathbb{R}^d$ and $\mathcal {Y}\subset \mathbb{R}^d$ locally finite. Subject to $\xi$ satisfying a weak spatial dependence and continuity condition, we show that such sums satisfy weak laws of large numbers, variance asymptotics and central limit theorems. We show that the limit behavior is controlled by the value of $\xi$ on homogeneous Poisson point processes on $m$-dimensional hyperplanes tangent to $\mathcal {M}$. We apply the general results to establish the limit theory of dimension and volume content estimators, R\'{e}nyi and Shannon entropy estimators and clique counts in the Vietoris-Rips complex on $\{Y_i\}_{i=1}^n$.Comment: Published in at http://dx.doi.org/10.1214/12-AAP897 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Tribo-Corrosion behaviour of TiCxOy thin films in bio fluids

In recent years, the development of thin film systems for decorative applications has attracted significant attention in scientific research. These decorative coatings require, not only an attractive appearance for market applications, but also an ability to protect the surface underneath. Because of this, corrosion, wear and their combined effects (termed tribo-corrosion) are particularly important for lifetime prediction. The tribo-corrosion behaviour of a range of single layered titanium oxycarbide, TiCxOy,coatings, produced by DC reactive magnetron sputtering, has been studied and reported as a function of electrode potential (-0.9 V, -0.5 V, 0.0 V and +0.5 V) and applied load (3, 6 and 9 N). The study was conducted in a reciprocating sliding tribosystem (Plint TE 67/E) in a bio fluid (an artificial perspiration solution) at room temperature. During the wear tests, both the open-circuit potential and the corrosion current were monitored. The results showed that electrode potential and load have a significant influence on the total material loss. The variations in Rp (polarization resistance) and Cf(capacitance) before and after sliding, obtained by Electrochemical Impedance Spectroscopy (EIS) were evaluated in order to provide an understanding of the resistance of the film in such conditions. Tribo-corrosion maps were generated, based on the results, indicating the change in mechanisms of the tribological and corrosion parameters for such coatings