Semidefinite Approximations of the Matrix Logarithm

© 2018, SFoCM. The matrix logarithm, when applied to Hermitian positive definite matrices, is concave with respect to the positive semidefinite order. This operator concavity property leads to numerous concavity and convexity results for other matrix functions, many of which are of importance in quantum information theory. In this paper we show how to approximate the matrix logarithm with functions that preserve operator concavity and can be described using the feasible regions of semidefinite optimization problems of fairly small size. Such approximations allow us to use off-the-shelf semidefinite optimization solvers for convex optimization problems involving the matrix logarithm and related functions, such as the quantum relative entropy. The basic ingredients of our approach apply, beyond the matrix logarithm, to functions that are operator concave and operator monotone. As such, we introduce strategies for constructing semidefinite approximations that we expect will be useful, more generally, for studying the approximation power of functions with small semidefinite representations

Diagonal and Low-Rank Matrix Decompositions, Correlation Matrices, and Ellipsoid Fitting

In this paper we establish links between, and new results for, three problems that are not usually considered together. The first is a matrix decomposition problem that arises in areas such as statistical modeling and signal processing: given a matrix $X$ formed as the sum of an unknown diagonal matrix and an unknown low rank positive semidefinite matrix, decompose $X$ into these constituents. The second problem we consider is to determine the facial structure of the set of correlation matrices, a convex set also known as the elliptope. This convex body, and particularly its facial structure, plays a role in applications from combinatorial optimization to mathematical finance. The third problem is a basic geometric question: given points $v_1,v_2,...,v_n\in \R^k$ (where $n > k$) determine whether there is a centered ellipsoid passing \emph{exactly} through all of the points. We show that in a precise sense these three problems are equivalent. Furthermore we establish a simple sufficient condition on a subspace $U$ that ensures any positive semidefinite matrix $L$ with column space $U$ can be recovered from $D+L$ for any diagonal matrix $D$ using a convex optimization-based heuristic known as minimum trace factor analysis. This result leads to a new understanding of the structure of rank-deficient correlation matrices and a simple condition on a set of points that ensures there is a centered ellipsoid passing through them.Comment: 20 page

Sensory testing in leprosy:Comparison of ballpoint pen and monofilaments

The 10 g monofilament has been replaced by the ballpoint pen in routine sensory testing of nerves in leprosy control in Ethiopia. Results of sensory testing between the ballpoint pen and different monofilaments on hands and feet were compared. Ballpoint pen underdiagnosis of loss of sensation was defined to occur when the pen was felt and the monofilament was not. Differences were evaluated both for individual test points (test point level) and for the test points of extremities collectively (extremity level). An extremity (either a hand or a foot) was defined as having sensory nerve function impairment (SNFI) if a supplying nerve had SNFI, which was the case when sensation was absent in two or more test points in the area supplied by that nerve. At test point level, the percentages with ballpoint pen underdiagnosis relative to the 2, 10, 20 and 50 g monofilaments were 40, 21, 9 and 7%, respectively, in the hands, and 47, 30, 15 and 7% in the feet. Ballpoint pen underdiagnosis percentages of SNFI at extremity level were 32, 18, 8 and 9% in the hands, and 37, 26, 14 and 6% in the feet. The risk of ballpoint pen underdiagnosis appears to be higher in extremities without visible damage. In conclusion, substantial levels of underdiagnosis of sensory loss with the ballpoint pen were observed. However, the consequences for the prognosis of treatment with corticosteroids in patients with the more subtle sensation loss noted here need to be established. Development and testing of guidelines is a prerequisite for the use of the ballpoint pen

Sensory testing in leprosy:Comparison of ballpoint pen and monofilaments

The 10 g monofilament has been replaced by the ballpoint pen in routine sensory testing of nerves in leprosy control in Ethiopia. Results of sensory testing between the ballpoint pen and different monofilaments on hands and feet were compared. Ballpoint pen underdiagnosis of loss of sensation was defined to occur when the pen was felt and the monofilament was not. Differences were evaluated both for individual test points (test point level) and for the test points of extremities collectively (extremity level). An extremity (either a hand or a foot) was defined as having sensory nerve function impairment (SNFI) if a supplying nerve had SNFI, which was the case when sensation was absent in two or more test points in the area supplied by that nerve. At test point level, the percentages with ballpoint pen underdiagnosis relative to the 2, 10, 20 and 50 g monofilaments were 40, 21, 9 and 7%, respectively, in the hands, and 47, 30, 15 and 7% in the feet. Ballpoint pen underdiagnosis percentages of SNFI at extremity level were 32, 18, 8 and 9% in the hands, and 37, 26, 14 and 6% in the feet. The risk of ballpoint pen underdiagnosis appears to be higher in extremities without visible damage. In conclusion, substantial levels of underdiagnosis of sensory loss with the ballpoint pen were observed. However, the consequences for the prognosis of treatment with corticosteroids in patients with the more subtle sensation loss noted here need to be established. Development and testing of guidelines is a prerequisite for the use of the ballpoint pen

Superconducting double transition and substantial Knight shift in Sr2RuO₄

Recent nuclear magnetic resonance experiments measuring the Knight shift in Sr2RuO4 have challenged the widely accepted picture of chiral pairing in this superconductor. Here we study the implications of helical pairing on the superconducting state while comparing our results with the available experimental data on the upper critical field and Knight shift. We solve the Bogoliubov–de Gennes equation employing a realistic three-dimensional tight-binding model that captures the experimental Fermi surface very well. In agreement with experiments we find a Pauli limiting to the upper critical field and, at low temperatures and high fields, a second superconducting transition. These transitions, which form a superconducting subphase in the H-T phase diagram are first-order in nature and merge into a single second-order transition at a bicritical point (T∗,H∗),for which we find (0.8 K, 2.4 T) with experiment reporting (0.8 K,∼1.2 T) [Phys. Rev. B93, 184513 (2016)]. Furthermore, we find a substantial drop in the Knight shift in agreement with recent experiments

Impact of infectious diseases consultation on the management of Staphylococcus aureus bacteraemia in children.

OBJECTIVES: Infectious diseases consultation (IDC) in adults with Staphylococcus aureus bacteraemia (SAB) has been shown to improve management and outcome. The aim of this study was to evaluate the impact of IDC on the management of SAB in children. STUDY DESIGN: Observational cohort study of children with SAB. SETTING: Cambridge University Hospitals National Health Service (NHS) Foundation Trust, a large acute NHS Trust in the UK. PARTICIPANTS: All children with SAB admitted to the Cambridge University Hospitals NHS Foundation Trust between 16 July 2006 and 31 December 2012. METHODS: Children with SAB between 2006 and 31 October 2009 were managed by routine clinical care (pre-IDC group) and data were collected retrospectively by case notes review. An IDC service for SAB was introduced in November 2009. All children with SAB were reviewed regularly and data were collected prospectively (IDC group) until 31 December 2012. Baseline characteristics, quality metrics and outcome were compared between the pre-IDC group and IDC group. RESULTS: There were 66 episodes of SAB in 63 children-28 patients (30 episodes) in the pre-IDC group, and 35 patients (36 episodes) in the IDC group. The median age was 3.4 years (IQR 0.2-10.7 years). Patients in the IDC group were more likely to have echocardiography performed, a removable focus of infection identified and to receive a longer course of intravenous antimicrobial therapy. There were no differences in total duration of antibiotic therapy, duration of hospital admission or outcome at 30 or 90 days following onset of SAB. CONCLUSIONS: IDC resulted in improvements in the investigation and management of SAB in children.This work was supported by grants from the UK Clinical Research Collaboration (UKCRC) Translational Infection Research Initiative (TIRI); the Medical Research Council (G1000803), with contributions from the Biotechnology and Biological Sciences Research Council, the National Institute for Health Research (NIHR) on behalf of the UK Department of Health, and the Chief Scientist of the Scottish Government Health Directorate; the Public Health England; and the NIHR Cambridge Biomedical Research Centre

Molecular and Epigenetic Mechanisms Underlying Cognitive and Adaptive Responses to Stress

Consolidation of contextual memories after a stressful encounter is essential for the survival of an organism and in allowing a more appropriate response to be elicited should the perceived threat reoccur. Recent evidence has explored the complex role that epigenetic mechanisms play in the formation of such memories, and the underlying signaling pathways are becoming more apparent. The glucocorticoid receptor (GR) has been shown to play a key role in these events having both genomic and non-genomic actions in the brain. GR has been shown to interact with the extracellular signal-regulated kinase mitogen-activated protein kinase (ERK MAPK) signaling pathway which, in concert, drives epigenetic modifications and chromatin remodeling, resulting in gene induction and memory consolidation. Evidence indicates that stressful events can have an effect on the offspring in utero, and that epigenetic marks altered early in life may persist into adulthood. A new and controversial area of research, however, suggests that epigenetic modifications could be inherited through the germline, a concept known as transgenerational epigenetics. This review explores the role that epigenetic processes play in the central nervous system, specifically in the consolidation of stress-induced memories, the concept of transgenerational epigenetic inheritance, and the potential role of epigenetics in revolutionizing the treatment of stress-related disorders through the emerging field of pharmacoepigenetics and personalized medical treatment