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

Decrease in gyrase A protein expression in _E. coli_ cells inhibited by antisense ribozymes

RNase P complexed with external guide sequence (EGS) represents a novel nucleic-acid-based gene interference approach to modulate gene expression. Nucleic acid-based gene interference technologies represent promising strategies for specific inhibition of mRNA sequences of choice. Recently, small interfering RNAs have been implicated in inducing endogenous RNase of the RNA-induced silencing complex in the RNA interference pathway to inhibit gene expression and growth of several human viruses. We report down regulation of protein expression of _E. coli_ gyrase A, an essential gene for DNA supercoiling and antibiotic susceptibility in BL21 (DE3) strain of _E. coli_, using Ribonuclease P based external guide sequence (EGS) technique. EGS directed against gyrase A gene that was cloned into pUC vector, which contains the ampicillin (Amp) resistance gene. The recombinant plasmid pT7EGyrA was transformed into BL21 (DE3) and inductions were performed using IPTG. Western blot was done to investigate the downregulation of gyrase A protein. The results showed a significant decrease of gyrase A suggesting the utility of EGS RNAs in gene therapy applications, by inhibiting the expression of essential proteins

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]

Many Molecular Properties from One Kernel in Chemical Space

We introduce property-independent kernels for machine learning modeling of arbitrarily many molecular properties. The kernels encode molecular structures for training sets of varying size, as well as similarity measures sufficiently diffuse in chemical space to sample over all training molecules. Corresponding molecular reference properties provided, they enable the instantaneous generation of ML models which can systematically be improved through the addition of more data. This idea is exemplified for single kernel based modeling of internal energy, enthalpy, free energy, heat capacity, polarizability, electronic spread, zero-point vibrational energy, energies of frontier orbitals, HOMO-LUMO gap, and the highest fundamental vibrational wavenumber. Models of these properties are trained and tested using 112 kilo organic molecules of similar size. Resulting models are discussed as well as the kernels' use for generating and using other property models

Clustering at high redshift: The connection between Lyman Alpha emitters and Lyman break galaxies

We present a physically motivated semi-analytic model to understand the clustering of high redshift Lyman Alpha Emitters (LAEs). We show that the model parameters constrained by the observed luminosity functions, can be used to predict large scale bias and angular correlation function of LAEs. These predictions are shown to reproduce the observations remarkably well. We find that average masses of dark matter halos hosting LAEs brighter than threshold narrow band magnitude ~ 25 are ~ 10^11 M_\odot. These are smaller than that of typical Lyman Break Galaxies (LBGs) brighter than similar threshold continuum magnitude by a factor ~ 10. This results in a smaller clustering strength of LAEs compared to LBGs. However, using the observed relationship between UV continuum and Lyman-alpha luminosity of LAEs, we show that both LAEs and LBGs belong to the same parent galaxy population with narrow band techniques having greater efficiency in picking up galaxies with low UV luminosity. We also show that the lack of evidence for the presence of the one halo term in the observed LAE angular correlation functions can be attributed to sub-Poisson distribution of LAEs in dark matter halos as a result of their low halo occupations.Comment: Accepted for publication in MNRAS. arXiv admin note: text overlap with arXiv:1208.209