Total positivity of sums, Hadamard products and Hadamard powers: Results and counterexamples

We show that, for Hankel matrices, total nonnegativity (resp. total positivity) of order r is preserved by sum, Hadamard product, and Hadamard power with real exponent t \ge r-2. We give examples to show that our results are sharp relative to matrix size and structure (general, symmetric or Hankel). Some of these examples also resolve the Hadamard critical-exponent problem for totally positive and totally nonnegative matrices

Total Positivity in Markov Structures

We discuss properties of distributions that are multivariate totally positive of order two (MTP$_{2}$) related to conditional independence. In particular, we show that any independence model generated by an MTP$_{2}$ distribution is a compositional semigraphoid which is upward-stable and singleton-transitive. In addition, we prove that any MTP$_{2}$ distribution satisfying an appropriate support condition is faithful to its concentration graph. Finally, we analyze factorization properties of MTP$_{2}$ distributions and discuss ways of constructing MTP$_{2}$ distributions; in particular we give conditions on the log-linear parameters of a discrete distribution which ensure MTP$_{2}$ and characterize conditional Gaussian distributions which satisfy MTP$_{2}$.SF was supported in part by an NSERC Discovery Research Grant, KS by grant #FA9550-12-1-0392 from the U.S. Air Force Office of Scientic Research (AFOSR) and the Defense Advanced Research Projects Agency (DARPA), CU by the Austrian Science Fund (FWF) Y 903-N35, and PZ by the European Union Seventh Framework Programme PIOF-GA-2011-300975

A fast Metropolis-Hastings method for generating random correlation matrices

We propose a novel Metropolis-Hastings algorithm to sample uniformly from the space of correlation matrices. Existing methods in the literature are based on elaborated representations of a correlation matrix, or on complex parametrizations of it. By contrast, our method is intuitive and simple, based the classical Cholesky factorization of a positive definite matrix and Markov chain Monte Carlo theory. We perform a detailed convergence analysis of the resulting Markov chain, and show how it benefits from fast convergence, both theoretically and empirically. Furthermore, in numerical experiments our algorithm is shown to be significantly faster than the current alternative approaches, thanks to its simple yet principled approach.Comment: 8 pages, 3 figures, 2018 conferenc

A family of diameter-based eigenvalue bounds for quantum graphs

We establish a sharp lower bound on the first non-trivial eigenvalue of the Laplacian on a metric graph equipped with natural (i.e., continuity and Kirchhoff) vertex conditions in terms of the diameter and the total length of the graph. This extends a result of, and resolves an open problem from, [J. B. Kennedy, P. Kurasov, G. Malenov\'a and D. Mugnolo, Ann. Henri Poincar\'e 17 (2016), 2439--2473, Section 7.2], and also complements an analogous lower bound for the corresponding eigenvalue of the combinatorial Laplacian on a discrete graph. We also give a family of corresponding lower bounds for the higher eigenvalues under the assumption that the total length of the graph is sufficiently large compared with its diameter. These inequalities are sharp in the case of trees.Comment: Substantial revision of v1. The main result, originally for the first eigenvalue, has been generalised to the higher ones. The title has been changed and the proofs substantially reorganised to reflect the new result, and a section containing concluding remarks has been adde

Genetic testing of children for adult-onset conditions: opinions of the British adult population and implications for clinical practice

This study set out to explore the attitudes of a representative sample of the British public towards genetic testing in children to predict disease in the future. We sought opinions about genetic testing for adult-onset conditions for which no prevention/treatment is available during childhood, and about genetic 'carrier' status to assess future reproductive risks. The study also examined participants' level of agreement with the reasons professional organisations give in favour of deferring such testing. Participants (n=2998) completed a specially designed questionnaire, distributed by email. Nearly half of the sample (47%) agreed that parents should be able to test their child for adult-onset conditions, even if there is no treatment or prevention at time of testing. This runs contrary to professional guidance about genetic testing in children. Testing for carrier status was supported by a larger proportion (60%). A child's future ability to decide for her/himself if and when to be tested was the least supported argument in favour of deferring testing.European Journal of Human Genetics advance online publication, 5 November 2014; doi:10.1038/ejhg.2014.221

Quantum Tomography under Prior Information

We provide a detailed analysis of the question: how many measurement settings or outcomes are needed in order to identify a quantum system which is constrained by prior information? We show that if the prior information restricts the system to a set of lower dimensionality, then topological obstructions can increase the required number of outcomes by a factor of two over the number of real parameters needed to characterize the system. Conversely, we show that almost every measurement becomes informationally complete with respect to the constrained set if the number of outcomes exceeds twice the Minkowski dimension of the set. We apply the obtained results to determine the minimal number of outcomes of measurements which are informationally complete with respect to states with rank constraints. In particular, we show that 4d-4 measurement outcomes (POVM elements) is enough in order to identify all pure states in a d-dimensional Hilbert space, and that the minimal number is at most 2 log_2(d) smaller than this upper bound.Comment: v3: There was a mistake in the derived finer upper bound in Theorem 3. The corrected upper bound is +1 to the earlier versio

Understanding the Operative Experience of the Practicing Pediatric Surgeon: Implications for Training and Maintaining Competency

Importance The number of practicing pediatric surgeons has increased rapidly in the past 4 decades, without a significant increase in the incidence of rare diseases specific to the field. Maintenance of competency in the index procedures for these rare diseases is essential to the future of the profession. Objective To describe the demographic characteristics and operative experiences of practicing pediatric surgeons using Pediatric Surgery Board recertification case log data. Design, Setting, and Participants We performed a retrospective review of 5 years of pediatric surgery certification renewal applications submitted to the Pediatric Surgery Board between 2009 and 2013. A surgeon’s location was defined by population as urban, large rural, small rural, or isolated. Case log data were examined to determine case volume by category and type of procedures. Surgeons were categorized according to recertification at 10, 20, or 30 years. Main Outcome and Measure Number of index cases during the preceding year. Results Of 308 recertifying pediatric surgeons, 249 (80.8%) were men, and 143 (46.4%) were 46 to 55 years of age. Most of the pediatric surgeons (304 of 308 [98.7%]) practiced in urban areas (ie, with a population >50 000 people). All recertifying applicants were clinically active. An appendectomy was the most commonly performed procedure (with a mean [SD] number of 49.3 [35.0] procedures per year), nonoperative trauma management came in second (with 20.0 [33.0] procedures per year), and inguinal hernia repair for children younger than 6 months of age came in third (with 14.7 [13.8] procedures per year). In 6 of 10 “rare” pediatric surgery cases, the mean number of procedures was less than 2. Of 308 surgeons, 193 (62.7%) had performed a neuroblastoma resection, 170 (55.2%) a kidney tumor resection, and 123 (39.9%) an operation to treat biliary atresia or choledochal cyst in the preceding year. Laparoscopy was more frequently performed in the 10-year recertification group for Nissen fundoplication, appendectomy, splenectomy, gastrostomy/jejunostomy, orchidopexy, and cholecystectomy (P < .05) but not lung resection (P = .70). It was more frequently used by surgeons recertifying in the 10-year group (used in 11 375 of 14 456 procedures [78.7%]) than by surgeons recertifying in the 20-year (used in 6214 of 8712 procedures [71.3%]) or 30-year group (used in 2022 of 3805 procedures [53.1%]). Conclusions and Relevance Practicing pediatric surgeons receive limited exposure to index cases after training. With regard to maintaining competency in an era in which health care outcomes have become increasingly important, these results are concerning