The Theory of the Interleaving Distance on Multidimensional Persistence Modules

In 2009, Chazal et al. introduced $\epsilon$-interleavings of persistence modules. $\epsilon$-interleavings induce a pseudometric $d_I$ on (isomorphism classes of) persistence modules, the interleaving distance. The definitions of $\epsilon$-interleavings and $d_I$ generalize readily to multidimensional persistence modules. In this paper, we develop the theory of multidimensional interleavings, with a view towards applications to topological data analysis. We present four main results. First, we show that on 1-D persistence modules, $d_I$ is equal to the bottleneck distance $d_B$. This result, which first appeared in an earlier preprint of this paper, has since appeared in several other places, and is now known as the isometry theorem. Second, we present a characterization of the $\epsilon$-interleaving relation on multidimensional persistence modules. This expresses transparently the sense in which two $\epsilon$-interleaved modules are algebraically similar. Third, using this characterization, we show that when we define our persistence modules over a prime field, $d_I$ satisfies a universality property. This universality result is the central result of the paper. It says that $d_I$ satisfies a stability property generalizing one which $d_B$ is known to satisfy, and that in addition, if $d$ is any other pseudometric on multidimensional persistence modules satisfying the same stability property, then $d\leq d_I$. We also show that a variant of this universality result holds for $d_B$, over arbitrary fields. Finally, we show that $d_I$ restricts to a metric on isomorphism classes of finitely presented multidimensional persistence modules.Comment: Major revision; exposition improved throughout. To appear in Foundations of Computational Mathematics. 36 page

The persistence landscape and some of its properties

Persistence landscapes map persistence diagrams into a function space, which may often be taken to be a Banach space or even a Hilbert space. In the latter case, it is a feature map and there is an associated kernel. The main advantage of this summary is that it allows one to apply tools from statistics and machine learning. Furthermore, the mapping from persistence diagrams to persistence landscapes is stable and invertible. We introduce a weighted version of the persistence landscape and define a one-parameter family of Poisson-weighted persistence landscape kernels that may be useful for learning. We also demonstrate some additional properties of the persistence landscape. First, the persistence landscape may be viewed as a tropical rational function. Second, in many cases it is possible to exactly reconstruct all of the component persistence diagrams from an average persistence landscape. It follows that the persistence landscape kernel is characteristic for certain generic empirical measures. Finally, the persistence landscape distance may be arbitrarily small compared to the interleaving distance.Comment: 18 pages, to appear in the Proceedings of the 2018 Abel Symposiu

Sociodemographic Factors Associated with AIDS Knowledge in a Random Sample of University Students

A telephone survey was used to assess knowledge of the transmission, prevalence, and infectivity of acquired immunodeficiency syndrome (AIDS), and the safety of casual contact among 214 randomly selected university students. Males were more knowledgeable than females overall (odds ratio [OR], men/women = 4.8). Although most students understood the dangers of unprotected sex and intravenous needle sharing, up to 30% believed some kinds of casual contact (e.g., shared eating utensils) can transmit AIDS. Older students (≥ 23 yrs) were more knowledgeable than those 17 to 19 years old about the safety of casual contact (OR = 3.8). Students are in need of education programs that stress the ways AIDS is not transmitted. Since most students identified newspapers and television as their main sources of information, these may be effective vehicles for education efforts.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/73095/1/j.1525-1446.1991.tb00654.x.pd

'Education, education, education' : legal, moral and clinical

This article brings together Professor Donald Nicolson's intellectual interest in professional legal ethics and his long-standing involvement with law clinics both as an advisor at the University of Cape Town and Director of the University of Bristol Law Clinic and the University of Strathclyde Law Clinic. In this article he looks at how legal education may help start this process of character development, arguing that the best means is through student involvement in voluntary law clinics. And here he builds upon his recent article which argues for voluntary, community service oriented law clinics over those which emphasise the education of students

Pre-neoplastic alterations define CLL DNA methylome and persist through disease progression and therapy

Most human cancers converge to a deregulated methylome with reduced global levels and elevated methylation at select CpG islands. To investigate the emergence and dynamics of the cancer methylome, we characterized genome-wide DNA methylation in preneoplastic monoclonal B-cell lymphocytosis (MBL) and chronic lymphocytic leukemia (CLL), including serial samples collected across disease course. We detected the aberrant tumor-associated methylation landscape at CLL diagnosis and found no significant differentially methylated regions in the high-count MBL-to-CLL transition. Patient methylomes showed remarkable stability with natural disease and posttherapy progression. Single CLL cells were consistently aberrantly methylated, indicating a homogeneous transition to the altered epigenetic state and a distinct expression profile together with MBL cells compared with normal B cells. Our longitudinal analysis reveals the cancer methylome to emerge early, which may provide a platform for subsequent genetically driven growth dynamics, and, together with its persistent presence, suggests a central role in disease onset. SigNifiCANCe: DNA methylation data from a large cohort of patients with MBL and CLL show that epigenetic transformation emerges early and persists throughout disease stages with limited subsequent changes. Our results indicate an early role for this aberrant landscape in the normal-to-preneoplastic transition that may reflect a pan-cancer mechanism

An Open Access Database of Genome-wide Association Results

<p>Abstract</p> <p>Background</p> <p>The number of genome-wide association studies (GWAS) is growing rapidly leading to the discovery and replication of many new disease loci. Combining results from multiple GWAS datasets may potentially strengthen previous conclusions and suggest new disease loci, pathways or pleiotropic genes. However, no database or centralized resource currently exists that contains anywhere near the full scope of GWAS results.</p> <p>Methods</p> <p>We collected available results from 118 GWAS articles into a database of 56,411 significant SNP-phenotype associations and accompanying information, making this database freely available here. In doing so, we met and describe here a number of challenges to creating an open access database of GWAS results. Through preliminary analyses and characterization of available GWAS, we demonstrate the potential to gain new insights by querying a database across GWAS.</p> <p>Results</p> <p>Using a genomic bin-based density analysis to search for highly associated regions of the genome, positive control loci (e.g., MHC loci) were detected with high sensitivity. Likewise, an analysis of highly repeated SNPs across GWAS identified replicated loci (e.g., <it>APOE</it>, <it>LPL</it>). At the same time we identified novel, highly suggestive loci for a variety of traits that did not meet genome-wide significant thresholds in prior analyses, in some cases with strong support from the primary medical genetics literature (<it>SLC16A7, CSMD1, OAS1</it>), suggesting these genes merit further study. Additional adjustment for linkage disequilibrium within most regions with a high density of GWAS associations did not materially alter our findings. Having a centralized database with standardized gene annotation also allowed us to examine the representation of functional gene categories (gene ontologies) containing one or more associations among top GWAS results. Genes relating to cell adhesion functions were highly over-represented among significant associations (p < 4.6 × 10^-14), a finding which was not perturbed by a sensitivity analysis.</p> <p>Conclusion</p> <p>We provide access to a full gene-annotated GWAS database which could be used for further querying, analyses or integration with other genomic information. We make a number of general observations. Of reported associated SNPs, 40% lie within the boundaries of a RefSeq gene and 68% are within 60 kb of one, indicating a bias toward gene-centricity in the findings. We found considerable heterogeneity in information available from GWAS suggesting the wider community could benefit from standardization and centralization of results reporting.</p

Inverted Expression Profiles of Sex-Biased Genes in Response to Toxicant Perturbations and Diseases

10.1371/journal.pone.0056668PLoS ONE82

Trophic factors differentiate dopamine neurons vulnerable to Parkinson's disease

Recent studies suggest a variety of factors characterize substantia nigra neurons vulnerable to Parkinson's disease, including the transcription factors pituitary homeobox 3 (Pitx3) and orthodenticle homeobox 2 (Otx2) and the trophic factor receptor deleted in colorectal cancer (DCC), but there is limited information on their expression and localization in adult humans. Pitx3, Otx2, and DCC were immunohistochemically localized in the upper brainstem of adult humans and mice and protein expression assessed using relative intensity measures and online microarray data. Pitx3 was present and highly expressed in most dopamine neurons. Surprisingly, in our elderly subjects no Otx2 immunoreactivity was detected in dopamine neurons, although Otx2 gene expression was found in younger cases. Enhanced DCC gene expression occurred in the substantia nigra, and higher amounts of DCC protein characterized vulnerable ventral nigral dopamine neurons. Our data show that, at the age when Parkinson's disease typically occurs, there are no significant differences in the expression of transcription factors in brainstem dopamine neurons, but those most vulnerable to Parkinson's disease rely more on the trophic factor receptor DCC than other brainstem dopamine neurons