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

Preserving entanglement under decoherence and sandwiching all separable states

Every entangled state can be perturbed, for instance by decoherence, and stay entangled. For a large class of pure entangled states, we show how large the perturbation can be. Our class includes all pure bipartite and all maximally entangled states. For an entangled state, E, the constucted neighborhood of entangled states is the region outside two parallel hyperplanes, which sandwich the set of all separable states. The states for which these neighborhoods are largest are the maximally entangled ones. As the number of particles, or the dimensions of the Hilbert spaces for two of the particles increases, the distance between two of the hyperplanes which sandwich the separable states goes to zero. It is easy to decide if a state Q is in the neighborhood of entangled states we construct for an entangled state E. One merely has to check if the trace of EQ is greater than a constant which depends upon E and which we determine.Comment: Corrected first author's e-mail address. All the rest remains unchange

In utero antihypertensive medication exposure and neonatal outcomes : A data linkage cohort study

We acknowledge the support from the Farr Institute @ Scotland. The Farr Institute @ Scotland is supported by a 10-funder consortium: Arthritis Research UK, the British Heart Foundation, Cancer Research UK, the Economic and Social Research Council, the Engineering and Physical Sciences Research Council, the Medical Research Council, the National Institute of Health Research, the National Institute for Social Care and Health Research (Welsh Assembly Government), the Chief Scientist Office (Scottish Government Health Directorates), the Wellcome Trust, (MRC grant number MR/K007017/1).Peer reviewedPublisher PD

Exact, E=0, Solutions for General Power-Law Potentials. I. Classical Orbits

For zero energy, $E=0$, we derive exact, classical solutions for {\em all} power-law potentials, $V(r)=-\gamma/r^\nu$, with $\gamma>0$ and $-\infty <\nu<\infty$. When the angular momentum is non-zero, these solutions lead to the orbits $\r(t)= [\cos \mu (\th(t)-\th_0(t))]^{1/\mu}$, for all $\mu \equiv \nu/2-1 \ne 0$. When $\nu>2$, the orbits are bound and go through the origin. This leads to discrete discontinuities in the functional dependence of $\th(t)$ and $\th_0(t)$, as functions of $t$, as the orbits pass through the origin. We describe a procedure to connect different analytic solutions for successive orbits at the origin. We calculate the periods and precessions of these bound orbits, and graph a number of specific examples. Also, we explain why they all must violate the virial theorem. The unbound orbits are also discussed in detail. This includes the unusual orbits which have finite travel times to infinity and also the special $\nu = 2$ case.Comment: LaTeX, 27 pages with 12 figures available from the authors or can be generated from Mathematica instructions at end of the fil

Educational and health outcomes of children and adolescents receiving antidepressant medication : Scotland-wide retrospective record linkage cohort study of 766 237 schoolchildren

Funding Health Data Research UK (grant reference number MR/S003800/1). Acknowledgements The study was sponsored by Health Data Research UK (www.hdruk.ac.uk), which is a joint investment led by the Medical Research Council, together with the National Institute for Health Research (England), the Chief Scientist Office (Scotland), Health and Care Research Wales, Health and Social Care Research and Development Division (Public Health Agency, Northern Ireland), the Engineering and Physical Sciences Research Council, the Economic and Social Research Council, the British Heart Foundation and Wellcome (grant reference number MR/S003800/1). The sponsor and funders had no role in the design and conduct of the study; collection, management, analysis and interpretation of the data; preparation, review or approval of the manuscript; or decision to submit the manuscript for publication. This study formed part of a wider PhD thesis undertaken by the lead author within the University of Glasgow and was published in 2017. Certain sections of this paper appear in the thesis, which is accessible and downloadable from the following link: http://theses.gla.ac.uk/8594/1/2017flemingphd.pdf. Author Contributions J.P.P. had the original concept. All authors agreed the study design. D.C. and A.K. provided data and undertook record linkage. M.F. and D.F.M. undertook the statistical analyses. All authors interpreted the results. M.F. and J.P.P. drafted the manuscript and all other authors contributed revisions. All authors reviewed and approved the final version of the manuscript. M.F. is guarantor for the study. Approvals The authors applied for permission to access, link and analyse these data and undertook mandatory training in data protection, IT security and information governance. Therefore, the datasets generated and analysed during the study are not publicly available. The study was approved by the National Health Service Privacy Advisory Committee and covered by a data-processing agreement between Glasgow University and ISD, and a data-sharing agreement between Glasgow University and ScotXed. All data were linked by the Electronic Data Research and Innovation Service (eDRIS), part of NHS National Services Scotland. Ethics The NHS West of Scotland Research Ethics Service confirmed that formal NHS ethics approval was not required, since the study involved anonymized extracts of routinely collected data with an acceptably negligible risk of identification. Conflict of interest: None declaredPeer reviewedPublisher PD

Educational and health outcomes of children and adolescents receiving antiepileptic medication : Scotland-wide record linkage study of 766 244 schoolchildren

Acknowledgements The study was sponsored by Health Data Research UK (www.hdruk.ac.uk) which is a joint investment led by the Medical Research Council, together with the National Institute for Health Research (England), the Chief Scientist Office (Scotland), Health and Care Research Wales, Health and Social Care Research and Development Division (Public Health Agency, Northern Ireland), the Engineering and Physical Sciences Research Council, the Economic and Social Research Council, the British Heart Foundation and Wellcome. This study formed part of a wider PhD thesis undertaken by the lead author within the University of Glasgow, which was published in 2017. Therefore, certain sections of this paper appear in the thesis, which is accessible and downloadable from the following link: http://theses.gla.ac.uk/8594/1/2017flemingphd.pdf. Funding The study was sponsored by Health Data Research UK. The sponsor and funders had no role in the design and conduct of the study; collection, management, analysis, and interpretation of the data; preparation, review or approval of the manuscript, or decision to submit the manuscript for publication. Availability of data and materials The authors applied for permission to access, link and analyse these data and undertook mandatory training in data protection, IT security and information governance. Therefore, the datasets generated and analysed during the study are not publicly available.Peer reviewedPublisher PD

Too Young for the Death Penalty: An Empirical Examination of Community Conscience and the Juvenile Death Penalty from the Perspective of Capital Jurors

As our analysis of jury decisionmaking in juvenile capital trials was nearing completion, the Missouri Supreme Court declared the juvenile death penalty unconstitutional in Simmons v. Roper. The court held that the execution of persons younger than eighteen years of age at the time of their crime violates the Eighth and Fourteenth Amendments to the United States Constitution. This decision patently rejected the U.S. Supreme Court\u27s ruling in Stanford v. Kentucky, which permitted the execution of sixteen- and seventeen-year-olds. In deciding Simmons, the Missouri Supreme Court applied the U.S. Supreme Court\u27s reasoning in Atkins v. Virginia to the juvenile death penalty. In Atkins, the Supreme Court found that there was a national consensus against the death penalty for the mentally retarded that made their execution constitutionally unacceptable. Similarly, in Simmons, the Missouri Supreme Court found that there was a national consensus against the death penalty for juveniles and ruled that juveniles could no longer be executed as a matter of federal constitutional law. The dissenting judges in this four-to-three decision did not take issue with the substantive findings of the majority, but objected instead to what they regarded as the impropriety of their state supreme court making a federal constitutional ruling that contravened an earlier U.S. Supreme Court decision. The U.S. Supreme Court has granted certiorari for a Fall 2004 review of the Simmons case in which it will have an opportunity to consider new evidence on the constitutionality of the juvenile death penalty--evidence of a kind it has lamented not having in earlier death penalty challenges, including specifically a challenge to the juvenile death penalty\u27s constitutionality. This is evidence regarding real capital jurors\u27 exercise of sentencing discretion in cases where they have actually decided whether defendants, whose crimes were committed when they were juveniles, should live or die. This article presents this newly available evidence from capital jurors in an assessment of the constitutionality of the juvenile death penalty. Drawing upon interviews conducted by the Capital Jury Project with persons who served as capital jurors, we are able to compare jurors who served on cases where the defendant was less than eighteen years of age at the time of the crime with those who served on cases with defendants eighteen years of age and older. We can contrast how capital jurors in these cases assessed the culpability of the defendants and the appropriateness of the death sentence

Educational and health outcomes of children treated for type 1 diabetes: Scotland-wide record linkage study of 766,047 children

Objective: This study was conducted to determine the association between childhood type 1 diabetes and educational and health outcomes. Research Design and Methods: Record linkage of nine Scotland-wide databases (diabetes register, dispensed prescriptions, maternity records, hospital admissions, death certificates, annual pupil census, school absences/exclusions, school examinations, and unemployment) produced a cohort of 766,047 singleton children born in Scotland who attended Scottish schools between 2009 and 2013. We compared the health and education outcomes of schoolchildren receiving insulin with their peers, adjusting for potential confounders. Results: The 3,330 children (0.47%) treated for type 1 diabetes were more likely to be admitted to the hospital (adjusted hazard ratio [HR] 3.97, 95% CI 3.79–4.16), die (adjusted HR 3.84, 95% CI 1.98–7.43), be absent from school (adjusted incidence rate ratio [IRR] 1.34, 95% CI 1.30–1.39), and have learning difficulties (adjusted odds ratio [OR] 1.19, 95% CI 1.03–1.38). Among children with type 1 diabetes, higher mean HbA1c (particularly HbA1c in the highest quintile) was associated with greater absenteeism (adjusted IRR 1.75, 95% CI 1.56–1.96, P &lt; 0.001), increased school exclusion (adjusted IRR 2.82, 95% CI 1.14–6.98), poorer attainment (adjusted OR 3.52, 95% CI 1.72–7.18), and higher risk of unemployment (adjusted OR 2.01, 95% CI 1.05–3.85). Conclusions: Children with type 1 diabetes fare worse than their peers in respect of education and health outcomes, especially if they have higher mean HbA1c. Interventions are required to minimize school absence and ensure that it does not affect educational attainment

A Checklist for Medication Compliance and Persistence Studies Using Retrospective Databases

The increasing number of retrospective database studies related to medication compliance and persistence (C&P), and the inherent variability within each, has created a need for improvement in the quality and consistency of medication C&P research. This article stems from the International Society of Pharmacoeconomics and Outcomes Research (ISPOR) efforts to develop a checklist of items that should be either included, or at least considered, when a retrospective database analysis of medication compliance or persistence is undertaken. This consensus document outlines a systematic approach to designing or reviewing retrospective database studies of medication C&P. Included in this article are discussions on data sources, measures of C&P, results reporting, and even conflict of interests. If followed, this checklist should improve the consistency and quality of C&P analyses, which in turn will help providers and payers understand the impact of C&P on health outcomes.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/75562/1/j.1524-4733.2006.00139.x.pd