A Numerical Analyst Looks at the "Cutoff Phenomenon" in Card Shuffling and Other Markov Chains

Diaconis and others have shown that certain Markov chains exhibit a "cutoff phenomenon" in which, after an initial period of seemingly little progress, convergence to the steady state occurs suddenly. Since Markov chains are just powers of matrices, how can such effects be explained in the language of applied linear algebra? We attempt to do this, focusing on two examples: random walk on a hypercube, which is essentially the same as the problem of Ehrenfest urns, and the celebrated case of riffle shuffling of a deck of cards. As is typical with transient phenomena in matrix processes, the reason for the cutoff is not readily apparent from an examination of eigenvalues or eigenvectors, but it is reflected strongly in pseudosprectra - provided they are measured in the 1-norm, not the 2-norm. We illustrate and explain the cutoff phenomenon with Matlab computations based in part on a new explicit formula for the entries of the $n \times n$ "riffle shuffle matrix", and note that while the normwise cutoff may occur at one point, such as $\frac{3}{2} \log_{2} n$ for the riffle shuffle, weak convergence may occur at an equally precise earlier point such as $\log_{2} n$

Attention-deficit hyperactivity disorder shares copy number variant risk with schizophrenia and autism spectrum disorder.

To access publisher's full text version of this article, please click on the hyperlink in Additional Links field or click on the hyperlink at the top of the page marked DownloadAttention-deficit/hyperactivity disorder (ADHD) is a highly heritable common childhood-onset neurodevelopmental disorder. Some rare copy number variations (CNVs) affect multiple neurodevelopmental disorders such as intellectual disability, autism spectrum disorders (ASD), schizophrenia and ADHD. The aim of this study is to determine to what extent ADHD shares high risk CNV alleles with schizophrenia and ASD. We compiled 19 neuropsychiatric CNVs and test 14, with sufficient power, for association with ADHD in Icelandic and Norwegian samples. Eight associate with ADHD; deletions at 2p16.3 (NRXN1), 15q11.2, 15q13.3 (BP4 & BP4.5-BP5) and 22q11.21, and duplications at 1q21.1 distal, 16p11.2 proximal, 16p13.11 and 22q11.21. Six of the CNVs have not been associated with ADHD before. As a group, the 19 CNVs associate with ADHD (OR = 2.43, P = 1.6 × 10-21), even when comorbid ASD and schizophrenia are excluded from the sample. These results highlight the pleiotropic effect of the neuropsychiatric CNVs and add evidence for ADHD, ASD and schizophrenia being related neurodevelopmental disorders rather than distinct entities.Innovative Medicines Initiative Joint Undertaking from the European Union's Seventh Framework Programme (EU-FP7/2007-2013) European Union (EU) EU-FP7-People-2011-IAPP grant Research Council of Norway KG Jebsen Stiftelsen South-East Norway Health Authorit

Efficient computation of structured gradients using automatic differentiation

The advent of robust automatic differentiation tools is an exciting and important development in scientific computing. It is particularily noteworthy that the gradient of a scalar-valued function of many variables can be computed with essentially the same time complexity as required to evaluate the function itself. This is true, in theory, when the "reverse mode" of automatic differentiation is used (whereas the "forward mode" introduces an additional factor corresponding to the problem dimension). However, in practise performance on large problems can be significantly (and unacceptably) worse than predicted. In this paper we illustrate that when natural structure is exploited fast gradient computation can be recovered, even for large dimensional problems

Osteoarthritis of the Knee in European-Descent Populations

Abstract Osteoarthritis (OA) is the most common form of arthritis and a major cause of disability. This study evaluates the association in Caucasian populations of two single nucleotide polymorphisms (SNPs) mapping to the Human Leukocyte Antigen (HLA) region and deriving from a genome wide association scan (GWAS) of knee OA in Japanese populations. The frequencies for rs10947262 were compared in 36,408 controls and 5,749 knee OA cases from European-descent populations. rs7775228 was tested in 32,823 controls and 1,837 knee OA cases of European descent. The risk (major) allele at rs10947262 in Caucasian samples was not significantly associated with an odds ratio (OR) = 1.07 (95%CI 0.94 -1.21; p = 0.28). For rs7775228 the metaanalysis resulted in OR = 0.94 (95%CI 0.81-1.09; p = 0.42) for the allele associated with risk in the Japanese GWAS. In Japanese individuals these two SNPs are in strong linkage disequilibrium (LD) (r 2 = 0.86) with the HLA class II haplotype DRB1*1502 DQA1*0103 DQB1*0601 (frequency 8%). In Caucasian and Chinese samples, using imputed data, these SNPs appear not to be in LD with that haplotype (r 2 ,0.07). The rs10947262 and rs7775228 variants are not associated with risk of knee OA in European descent populations and they do not appear tag the same HLA class II haplotype as they do in Japanese individuals

The impact of regulations on firms: a case study of the biotech industry

Osteoarthritis (OA) is the most common form of arthritis and a major cause of disability. This study evaluates the association in Caucasian populations of two single nucleotide polymorphisms (SNPs) mapping to the Human Leukocyte Antigen (HLA) region and deriving from a genome wide association scan (GWAS) of knee OA in Japanese populations. The frequencies for rs10947262 were compared in 36,408 controls and 5,749 knee OA cases from European-descent populations. rs7775228 was tested in 32,823 controls and 1,837 knee OA cases of European descent. The risk (major) allele at rs10947262 in Caucasian samples was not significantly associated with an odds ratio (OR) = 1.07 (95%CI 0.94 -1.21; p = 0.28). For rs7775228 the metaanalysis resulted in OR = 0.94 (95%CI 0.81-1.09; p = 0.42) for the allele associated with risk in the Japanese GWAS. In Japanese individuals these two SNPs are in strong linkage disequilibrium (LD) (r2 = 0.86) with the HLA class II haplotype DRB1*1502 DQA1*0103 DQB1*0601 (frequency 8%). In Caucasian and Chinese samples, using imputed data, these SNPs appear not to be in LD with that haplotype (r2,0.07). The rs10947262 and rs7775228 variants are not associated with risk of knee OA in European descent populations and they do not appear tag the same HLA class II haplotype as they do in Japanese individuals

Enzyme immunoassays in brown snake (Pseudonaja spp.) envenoming: Detecting venom, antivenom and venom-antivenom complexes

To access publisher's full text version of this article click on the hyperlink at the bottom of the pageIn a small fraction of patients with schizophrenia or autism, alleles of copy-number variants (CNVs) in their genomes are probably the strongest factors contributing to the pathogenesis of the disease. These CNVs may provide an entry point for investigations into the mechanisms of brain function and dysfunction alike. They are not fully penetrant and offer an opportunity to study their effects separate from that of manifest disease. Here we show in an Icelandic sample that a few of the CNVs clearly alter fecundity (measured as the number of children by age 45). Furthermore, we use various tests of cognitive function to demonstrate that control subjects carrying the CNVs perform at a level that is between that of schizophrenia patients and population controls. The CNVs do not all affect the same cognitive domains, hence the cognitive deficits that drive or accompany the pathogenesis vary from one CNV to another. Controls carrying the chromosome 15q11.2 deletion between breakpoints 1 and 2 (15q11.2(BP1-BP2) deletion) have a history of dyslexia and dyscalculia, even after adjusting for IQ in the analysis, and the CNV only confers modest effects on other cognitive traits. The 15q11.2(BP1-BP2) deletion affects brain structure in a pattern consistent with both that observed during first-episode psychosis in schizophrenia and that of structural correlates in dyslexia.info:eu-repo/grantAgreement/EC/FP/286213/EU//OpenAIREplusinfo:eu-repo/grantAgreement/EC/FP7/28621

Map of the HLA class II region in harboring markers rs7775228 and rs10947262.

<p>The frequency of haplotype DRB1*1502 DQA1*0103 DQB1*0601 and its linkage disequilibrium with rs7775228 and rs10947262 in three ethnic groups is shown.</p

Characteristics of cases and controls of unpublished studies included in the analysis.

<p>MAF =  minor allele frequency.</p><p>(1) <b>Minor allele = C, major allele = T.</b></p><p>(2) <b>Minor allele = T, major allele = C.</b></p

Meta-analysis of genetic association between two HLA SNPs and knee OA.

<p>Forest plot of study-specific estimates and random-effects summary effect size and 95% confidence intervals (95%CIs) for the genetic association with knee OA of (<b>A</b>) the major allele (T) at rs7775228 in three published studies and the deCODE case-control study (<b>B</b>) the major allele (C) at rs10947262 in three published studies and seven additional European studies. Square sizes are proportional to the number of cases in each study.</p