Of McKay Correspondence, Non-linear Sigma-model and Conformal Field Theory

The ubiquitous ADE classification has induced many proposals of often mysterious correspondences both in mathematics and physics. The mathematics side includes quiver theory and the McKay Correspondence which relates finite group representation theory to Lie algebras as well as crepant resolutions of Gorenstein singularities. On the physics side, we have the graph-theoretic classification of the modular invariants of WZW models, as well as the relation between the string theory nonlinear $\sigma$-models and Landau-Ginzburg orbifolds. We here propose a unification scheme which naturally incorporates all these correspondences of the ADE type in two complex dimensions. An intricate web of inter-relations is constructed, providing a possible guideline to establish new directions of research or alternate pathways to the standing problems in higher dimensions.Comment: 35 pages, 4 figures; minor corrections, comments on toric geometry and references adde

The structure of TeV-bright shell-type supernova remnants

Aims. Two-dimensional MHD simulations are used to model the emission properties of TeV-bright shell-type supernova remnants (SNRs) and to explore their nature. Methods. In the leptonic scenario for the TeV emission, the $\gamma$-ray emission is produced via Inverse Compton scattering of background soft photons by high-energy electrons accelerated by the shocks of the SNRs. The TeV emissivity is proportional to the magnetic field energy density and MHD simulations can be used to model the TeV structure of such remnants directly. 2D MHD simulations for SNRs are then performed under the assumption that the ambient interstellar medium is turbulent with the magnetic field and density fluctuations following a Kolmogorov-like power-law spectrum. Results. (1) As expected, these simulations confirm early 1D and 2D modelings of these sources, namely the hydrodynamical evolution of the shock waves and amplification of magnetic field by Rayleigh-Taylor convective flows and by shocks propagating in a turbulent medium; (2) We reproduce rather complex morphological structure for $\gamma$-rays, suggesting intrinsic variations of the source morphology not related to the structure of the progenitor and environment; (3)The observed radial profile of several remnants are well reproduced with an ambient medium density of $0.1-1$ cm$^{-3}$. An even lower ambient density leads to a sharper drop of the TeV brightness with radius than what is observed near the outer edge of these remnants. Conclusions. In a turbulent background medium, we can reproduce the observed characteristics of several shell-type TeV SNRs with reasonable parameters except for a higher ambient density than that inferred from X-ray observations.Comment: 7pages,12figures,Accepted for publication in A&A. arXiv admin note: text overlap with arXiv:1306.439

Inverse Projection Representation and Category Contribution Rate for Robust Tumor Recognition

Sparse representation based classification (SRC) methods have achieved remarkable results. SRC, however, still suffer from requiring enough training samples, insufficient use of test samples and instability of representation. In this paper, a stable inverse projection representation based classification (IPRC) is presented to tackle these problems by effectively using test samples. An IPR is firstly proposed and its feasibility and stability are analyzed. A classification criterion named category contribution rate is constructed to match the IPR and complete classification. Moreover, a statistical measure is introduced to quantify the stability of representation-based classification methods. Based on the IPRC technique, a robust tumor recognition framework is presented by interpreting microarray gene expression data, where a two-stage hybrid gene selection method is introduced to select informative genes. Finally, the functional analysis of candidate's pathogenicity-related genes is given. Extensive experiments on six public tumor microarray gene expression datasets demonstrate the proposed technique is competitive with state-of-the-art methods.Comment: 14 pages, 19 figures, 10 table