Calculus on the Sierpinski Gasket I: Polynomials, Exponentials and Power Series

We study the analog of power series expansions on the Sierpinski gasket, for analysis based on the Kigami Laplacian. The analog of polynomials are multiharmonic functions, which have previously been studied in connection with Taylor approximations and splines. Here the main technical result is an estimate of the size of the monomials analogous to x^n/n!. We propose a definition of entire analytic functions as functions represented by power series whose coefficients satisfy exponential growth conditions that are stronger than what is required to guarantee uniform convergence. We present a characterization of these functions in terms of exponential growth conditions on powers of the Laplacian of the function. These entire analytic functions enjoy properties, such as rearrangement and unique determination by infinite jets, that one would expect. However, not all exponential functions (eigenfunctions of the Laplacian) are entire analytic, and also many other natural candidates, such as the heat kernel, do not belong to this class. Nevertheless, we are able to use spectral decimation to study exponentials, and in particular to create exponentially decaying functions for negative eigenvalues

Estimation of pairwise sequence similarity of mammalian enhancers with word neighbourhood counts

Motivation: The identity of cells and tissues is to a large degree governed by transcriptional regulation. A major part is accomplished by the combinatorial binding of transcription factors at regulatory sequences, such as enhancers. Even though binding of transcription factors is sequence-specific, estimating the sequence similarity of two functionally similar enhancers is very difficult. However, a similarity measure for regulatory sequences is crucial to detect and understand functional similarities between two enhancers and will facilitate large-scale analyses like clustering, prediction and classification of genome-wide datasets

Monetary benefits of preventing childhood lead poisoning with lead-safe window replacement

Previous estimates of childhood lead poisoning prevention benefits have quantified the present value of some health benefits, but not the costs of lead paint hazard control or the benefits associated with housing and energy markets. Because older housing with lead paint constitutes the main exposure source today in the U.S., we quantify health benefits, costs, market value benefits, energy savings, and net economic benefits of lead-safe window replacement (which includes paint stabilization and other measures). The benefit per resident child from improved lifetime earnings alone is $21,195 in pre-1940 housing and$8,685 in 1940-59 housing (in 2005 dollars). Annual energy savings are $130 to$486 per housing unit, with or without young resident children, with an associated increase in housing market value of $5,900 to$14,300 per housing unit, depending on home size and number of windows replaced. Net benefits are $4,490 to$5,629 for each housing unit built before 1940, and $491 to$1,629 for each unit built from 1940-1959, depending on home size and number of windows replaced. Lead-safe window replacement in all pre-1960 U.S. housing would yield net benefits of at least $67 billion, which does not include many other benefits. These other benefits, which are shown in this paper, include avoided Attention Deficit Hyperactivity Disorder, other medical costs of childhood lead exposure, avoided special education, and reduced crime and juvenile delinquency in later life. In addition, such a window replacement effort would reduce peak demand for electricity, carbon emissions from power plants, and associated long-term costs of climate change

The PAZAR database of gene regulatory information coupled to the ORCA toolkit for the study of regulatory sequences

The PAZAR database unites independently created and maintained data collections of transcription factor and regulatory sequence annotation. The flexible PAZAR schema permits the representation of diverse information derived from experiments ranging from biochemical protein–DNA binding to cellular reporter gene assays. Data collections can be made available to the public, or restricted to specific system users. The data ‘boutiques’ within the shopping-mall-inspired system facilitate the analysis of genomics data and the creation of predictive models of gene regulation. Since its initial release, PAZAR has grown in terms of data, features and through the addition of an associated package of software tools called the ORCA toolkit (ORCAtk). ORCAtk allows users to rapidly develop analyses based on the information stored in the PAZAR system. PAZAR is available at http://www.pazar.info. ORCAtk can be accessed through convenient buttons located in the PAZAR pages or via our website at http://www.cisreg.ca/ORCAtk

The Effects of Alignment Quality, Distance Calculation Method, Sequence Filtering, and Region on the Analysis of 16S rRNA Gene-Based Studies

Pyrosequencing of PCR-amplified fragments that target variable regions within the 16S rRNA gene has quickly become a powerful method for analyzing the membership and structure of microbial communities. This approach has revealed and introduced questions that were not fully appreciated by those carrying out traditional Sanger sequencing-based methods. These include the effects of alignment quality, the best method of calculating pairwise genetic distances for 16S rRNA genes, whether it is appropriate to filter variable regions, and how the choice of variable region relates to the genetic diversity observed in full-length sequences. I used a diverse collection of 13,501 high-quality full-length sequences to assess each of these questions. First, alignment quality had a significant impact on distance values and downstream analyses. Specifically, the greengenes alignment, which does a poor job of aligning variable regions, predicted higher genetic diversity, richness, and phylogenetic diversity than the SILVA and RDP-based alignments. Second, the effect of different gap treatments in determining pairwise genetic distances was strongly affected by the variation in sequence length for a region; however, the effect of different calculation methods was subtle when determining the sample's richness or phylogenetic diversity for a region. Third, applying a sequence mask to remove variable positions had a profound impact on genetic distances by muting the observed richness and phylogenetic diversity. Finally, the genetic distances calculated for each of the variable regions did a poor job of correlating with the full-length gene. Thus, while it is tempting to apply traditional cutoff levels derived for full-length sequences to these shorter sequences, it is not advisable. Analysis of β-diversity metrics showed that each of these factors can have a significant impact on the comparison of community membership and structure. Taken together, these results urge caution in the design and interpretation of analyses using pyrosequencing data