The congruence subgroup problem

This is a short survey of the progress on the congruence subgroup problem since the sixties when the first major results on the integral unimodular groups appeared. It is aimed at the non-specialists and avoids technical details.Comment: 10 page

A topological realization of the congruence subgroup Kernel A

A number of years ago, Kumar Murty pointed out to me that the computation of the fundamental group of a Hilbert modular surface ([7],IV,${\S}$6), and the computation of the congruence subgroup kernel of SL(2) ([6]) were surprisingly similar. We puzzled over this, in particular over the role of elementary matrices in both computations. We formulated a very general result on the fundamental group of a Satake compactification of a locally symmetric space. This lead to our joint paper [1] with Lizhen Ji and Les Saper on these fundamental groups. Although the results in it were intriguingly similar to the corresponding calculations of the congruence subgroup kernel of the underlying algebraic group in [5], we were not able to demonstrate a direct connection (cf. [1], ${\S}$7). The purpose of this note is to explain such a connection. A covering space is constructed from inverse limits of reductive Borel-Serre compactifications. The congruence subgroup kernel then appears as the group of deck transformations of this covering. The key to this is the computation of the fundamental group in [1]

Crystal structure of 11-(2,3-dimethoxy-phenyl)-14-methyl-12-oxa-8,14-diaza-tetracyclo[8.3.3.01,10.02,7]hexadeca-2(7),3,5-triene-9,13-dione

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Crystal structure of ethyl (2S,2′ R)-1′-benzyl-3-oxo-3H-dispiro[1-benzothio-phene-2,3′-pyrrolidine-2′,11″-indeno-[1,2-b]quinoxaline]-4′-carboxylate

The authors thank the TBI X-ray facility, CAS in Crystallography and BioPhysics, University of Madras, Chennai, India, for the data collection.Peer reviewedPublisher PD

Synthetic sequence generator for recommender systems - memory biased random walk on sequence multilayer network

Personalized recommender systems rely on each user's personal usage data in the system, in order to assist in decision making. However, privacy policies protecting users' rights prevent these highly personal data from being publicly available to a wider researcher audience. In this work, we propose a memory biased random walk model on multilayer sequence network, as a generator of synthetic sequential data for recommender systems. We demonstrate the applicability of the synthetic data in training recommender system models for cases when privacy policies restrict clickstream publishing.Comment: The new updated version of the pape

Ultraviolet Fe II Emission in Fainter Quasars: Luminosity Dependences, and the Influence of Environments

We investigate the strength of ultraviolet Fe II emission in fainter quasars com- pared with brighter quasars for 1.0 :( z :( 1.8, using the SDSS (Sloan Digital Sky Survey) DR7QSO catalogue and spectra of Schneider et al., and the SFQS (SDSS Faint Quasar Survey) catalogue and spectra of Jiang et al. We quantify the strength of the UV Fe II emission using the W 2400 equivalent width of Weymann et al., which is defined between two rest-frame continuum windows at 2240–2255 and 2665–2695 ˚A. The main results are the following. (1) We find that for W 2400 2: 25 ˚A there is a universal (i.e. for quasars in general) strengthening of W 2400 with decreasing intrinsic luminosity, L3000. (2) In conjunction with previous work by Clowes et al., we find that there is a further, differential, strengthening of W 2400 with decreasing L3000 for those quasars that are members of Large Quasar Groups (LQGs). (3) We find that increasingly strong W 2400 tends to be associated with decreasing FWHM of the neighbouring Mg II λ2798 broad emission line. (4) We suggest that the dependence of W 2400 on L3000 arises from Lyα fluorescence. (5) We find that stronger W 2400 tends to be associated with smaller virial estimates from Shen et al. of the mass of the central black hole, by a factor ∼ 2 between the ultrastrong emitters and the weak. Stronger W 2400 emission would correspond to smaller black holes that are still growing. The differential effect for LQG members might then arise from preferentially younger quasars in the LQG environments